# Descriptive Statistics

## Skewness and the Mean, Median, and Mode

Consider the following data set:
4, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 10
This data set can be represented by the following histogram. Each interval has width 1, and each value is located in the middle of an interval.
The histogram displays a symmetrical distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other. The mean, the median, and the mode are each seven for these data. In a perfectly symmetrical distribution, the mean and the median are the same. This example has one mode (unimodal), and the mode is the same as the mean and median.  In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median.
The histogram for the data: 4, 5, 6, 6, 6, 7, 7, 7, 7, 8 is not symmetrical. The right-hand side seems chopped off compared to the left-hand side. A distribution of this type is called skewed to the left because it is pulled out to the left. A skewed left distribution has more high values.
The mean is 6.3, the median is 6.5, and the mode is seven.  Notice that the mean is less than the median, and they are both less than the mode. The mean and the median both reflect the skewing, but the mean reflects it more so. The mean is pulled toward the tail in a skewed distribution.
The histogram for the data: 6, 7, 7, 7, 7, 8, 8, 8, 9, 10 is also not symmetrical. It is skewed to the right. A skewed right distribution has more low values.
The mean is 7.7, the median is 7.5, and the mode is seven. Of the three statistics, the mean is the largest, while the mode is the smallest. Again, the mean reflects the skewing the most.
To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean.
Skewness and symmetry become important when we discuss probability distributions in later chapters.

### Chapter Review

Looking at the distribution of data can reveal a lot about the relationship between the mean, the median, and the mode. There are three types of distributions. A right (or positive) skewed distribution has a shape like . A left (or negative) skewed distribution has a shape like . A symmetrical distribution looks like .
Use the following information to answer the next three exercises. State whether the data are symmetrical, skewed to the left, or skewed to the right.

### Homework


# Descriptive Statistics

## Measures of the Spread of the Data

An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the mean. The most common measure of variation, or spread, is the standard deviation. The standard deviation is a number that measures how far data values are from their mean.

### The standard deviation

1. provides a numerical measure of the overall amount of variation in a data set and
2. can be used to determine whether a particular data value is close to or far from the mean.

### The standard deviation provides a measure of the overall variation in a data set.

The standard deviation is always positive or zero. The standard deviation is small when all the data are concentrated close to the mean, exhibiting little variation or spread. The standard deviation is larger when the data values are more spread out from the mean, exhibiting more variation.
Suppose that we are studying the amount of time customers wait in line at the checkout at Supermarket A and Supermarket B. The average wait time at both supermarkets is five minutes. At Supermarket A, the standard deviation for the wait time is two minutes; at Supermarket B, the standard deviation for the wait time is four minutes.
Because Supermarket B has a higher standard deviation, we know that there is more variation in the wait times at Supermarket B. Overall, wait times at Supermarket B are more spread out from the average whereas wait times at  Supermarket A are more concentrated near the average.

### The standard deviation can be used to determine whether a data value is close to or far from the mean.

Suppose that both Rosa and Binh shop at Supermarket A. Rosa waits at the checkout counter for seven minutes, and Binh waits for one minute. At Supermarket A, the mean waiting time is five minutes, and the standard deviation is two minutes. The standard deviation can be used to determine whether a data value is close to or far from the mean. A z-score is a standardized score that lets us compare data sets. It tells us how many standard deviations a data value is from the mean and is calculated as the ratio of the difference in a particular score and the population mean to the population standard deviation.
We can use the given information to create the table below.
Since Rosa and Binh only shop at Supermarket A, we can ignore the row for Supermarket B.
We need the values from the first row to determine the number of standard deviations above or below the mean each individual wait time is; we can do so by calculating two different z-scores.
Rosa waited for seven minutes, so the z-score representing this deviation from the population mean may be calculated as
The z-score of one tells us that Rosa’s wait time is one standard deviation above the mean wait time of five minutes.
Binh waited for one minute, so the z-score representing this deviation from the population mean may be calculated as
The z-score of −2 tells us that Binh’s wait time is two standard deviations below the mean wait time of five minutes.
A data value that is two standard deviations from the average is just on the borderline for what many statisticians would consider to be far from the average. Considering data to be far from the mean if they are more than two standard deviations away is more of an approximate rule of thumb than a rigid rule. In general, the shape of the distribution of the data affects how much of the data is farther away than two standard deviations. You will learn more about this in later chapters.
The number line may help you understand standard deviation. If we were to put five and seven on a number line, seven is to the right of five. We say, then, that seven is one standard deviation to the right of five because 5 + (1)(2) = 7.
If one were also part of the data set, then one is two standard deviations to the left of five because 5 + (–2)(2) = 1.
1. In general, a value = mean + (#ofSTDEV)(standard deviation)
2. where #ofSTDEVs = the number of standard deviations
3. #ofSTDEV does not need to be an integer
4. One is two standard deviations less than the mean of five because 1 = 5 + (–2)(2).
The equation value = mean + (#ofSTDEVs)(standard deviation) can be expressed for a sample and for a population as follows:

The lowercase letter s represents the sample standard deviation and the Greek letter σ (lower case) represents the population standard deviation.

The symbol
 is the sample mean, and the Greek symbol
 is the population mean.

### Calculating the Standard Deviation

If x is a number, then the difference x – mean is called its deviation. In a data set, there are as many deviations as there are items in the data set. The deviations are used to calculate the standard deviation. If the numbers belong to a population, in symbols, a deviation is x – μ. For sample data, in symbols, a deviation is x –
 .
The procedure to calculate the standard deviation depends on whether the numbers are the entire population or are data from a sample.  The calculations are similar but not identical.  Therefore, the symbol used to represent the standard deviation depends on whether it is calculated from a population or a sample. The lowercase letter s represents the sample standard deviation and the Greek letter σ (lowercase sigma) represents the population standard deviation. If the sample has the same characteristics as the population, then s should be a good estimate of σ.
To calculate the standard deviation, we need to calculate the variance first.  The variance is the average of the squares of the deviations (the x –
  values for a sample or the x – μ values for a population). The symbol σ2 represents the population variance; the population standard deviation σ is the square root of the population variance. The symbol s2 represents the sample variance; the sample standard deviation s is the square root of the sample variance. You can think of the standard deviation as a special average of the deviations.
If the numbers come from a census of the entire population and not a sample, when we calculate the average of the squared deviations to find the variance, we divide by N, the number of items in the population. If the data are from a sample rather than a population, when we calculate the average of the squared deviations, we divide by , one less than the number of items in the sample.

### Formulas for the Sample Standard Deviation

1.
  or

2.  For the sample standard deviation, the denominator is ; that is, the sample size minus 1.

### Formulas for the Population Standard Deviation

1.

  or

2. For the population standard deviation, the denominator is N, the number of items in the population.
In these formulas, f represents the frequency with which a value appears. For example, if a value appears once, f is one. If a value appears three times in the data set or population, f is three.

### Types of Variability in Samples

When researchers study a population, they often use a sample, either for convenience or because it is not possible to access the entire population. Variability is the term used to describe the differences that may occur in these outcomes. Common types of variability include the following:
1. Observational or measurement variability
2. Natural variability
3. Induced variability
4. Sample variability
Here are some examples to describe each type of variability:
Example 1: Measurement variability
Measurement variability occurs when there are differences in the instruments used to measure or in the people using those instruments. If we are gathering data on how long it takes for a ball to drop from a height by having students measure the time of the drop with a stopwatch, we may experience measurement variability if the two stopwatches used were made by different manufacturers. For example, one stopwatch measures to the nearest second, whereas the other one measures to the nearest tenth of a second. We also may experience measurement variability because two different people are gathering the data. Their reaction times in pressing the button on the stopwatch may differ; thus, the outcomes will vary accordingly. The differences in outcomes may be affected by measurement variability.
Example 2: Natural variability
Natural variability arises from the differences that naturally occur because members of a population differ from each other. For example, if we have two identical corn plants and we expose both plants to the same amount of water and sunlight, they may still grow at different rates simply because they are two different corn plants. The difference in outcomes may be explained by natural variability.
Example 3: Induced variability
Induced variability is the counterpart to natural variability. This occurs because we have artificially induced an element of variation that, by definition, was not present naturally. For example, we assign people to two different groups to study memory, and we induce a variable in one group by limiting the amount of sleep they get. The difference in outcomes may be affected by induced variability.
Example 4: Sample variability
Sample variability occurs when multiple random samples are taken from the same population. For example, if I conduct four surveys of 50 people randomly selected from a given population, the differences in outcomes may be affected by sample variability.

### Sampling Variability of a Statistic

The statistic of a sampling distribution was discussed in Descriptive Statistics: Measures of the Center of the Data. How much the statistic varies from one sample to another is known as the  sampling variability of a statistic. You typically measure the sampling variability of a statistic by its standard error. The standard error of the mean is an example of a standard error. The standard error is the standard deviation of the sampling distribution. In other words, it is the average standard deviation that results from repeated sampling. You will cover the standard error of the mean in the chapter The Central Limit Theorem (not now). The notation for the standard error of the mean is
 , where σ is the standard deviation of the population and n is the size of the sample.

### Explanation of the standard deviation calculation shown in the table

The deviations show how spread out the data are about the mean. The data value 11.5 is farther from the mean than is the data value 11, which is indicated by the deviations .97 and .47. A positive deviation occurs when the data value is greater than the mean, whereas a negative deviation occurs when the data value is less than the mean. The deviation is –1.525 for the data value nine. If you add the deviations, the sum is always zero. We can sum the products of the frequencies and deviations to show that the sum of the deviations is always zero.

   For , there are n = 20 deviations. So you cannot simply add the deviations to get the spread of the data. By squaring the deviations, you make them positive numbers, and the sum will also be positive. The variance, then, is the average squared deviation.
The variance is a squared measure and does not have the same units as the data. Taking the square root solves the problem. The standard deviation measures the spread in the same units as the data.
Notice that instead of dividing by n = 20, the calculation divided by n – 1 = 20 – 1 = 19 because the data is a sample. For the sample variance, we divide by the sample size minus one (n – 1). Why not divide by n?  The answer has to do with the population variance. The sample variance is an estimate of the population variance. Based on the theoretical mathematics that lies behind these calculations, dividing by (n – 1) gives a better estimate of the population variance.
The standard deviation, s or σ, is either zero or larger than zero. Describing the data with reference to the spread is called variability.  The variability in data depends on the method by which the outcomes are obtained, for example, by measuring or by random sampling. When the standard deviation is zero, there is no spread; that is, all the data values are equal to each other. The standard deviation is small when all the data are concentrated close to the mean and larger when the data values show more variation from the mean. When the standard deviation is a lot larger than zero, the data values are very spread out about the mean; outliers can make s or σ very large.
The standard deviation, when first presented, can seem unclear.  By graphing your data, you can get a better feel for the deviations and the standard deviation.  You will find that in symmetrical distributions, the standard deviation can be very helpful, but in skewed distributions, the standard deviation may not be much help. The reason is that the two sides of a skewed distribution have different spreads. In a skewed distribution, it is better to look at the first quartile, the median, the third quartile, the smallest value, and the largest value. Because numbers can be confusing, always graph your data. Display your data in a histogram or a box plot.

### Standard deviation of Grouped Frequency Tables

Recall that for grouped data we do not know individual data values, so we cannot describe the typical value of the data with precision. In other words, we cannot find the exact mean, median, or mode. We can, however, determine the best estimate of the measures of center by finding the mean of the grouped data with the formula

where
   interval frequencies and m = interval midpoints.
Just as we could not find the exact mean, neither can we find the exact standard deviation. Remember that standard deviation describes numerically the expected deviation a data value has from the mean. In simple English, the standard deviation allows us to compare how unusual individual data are when compared to the mean.

### Comparing Values from Different Data Sets

As explained before, a z-score allows us to compare statistics from different data sets. If the data sets have different means and standard deviations, then comparing the data values directly can be misleading.
1. For each data value, calculate how many standard deviations away from its mean the value is.
2. In symbols, the formulas for calculating z-scores become the following.

As shown in the table, when only a sample mean and sample standard deviation are given, the top formula is used. When the population mean and population standard deviation are given, the bottom formula is used.
The following lists give a few facts that provide a little more insight into what the standard deviation tells us about the distribution of the data.
2. At least 75 percent of the data is within two standard deviations of the mean.
3. At least 89 percent of the data is within three standard deviations of the mean.
4. At least 95 percent of the data is within 4.5 standard deviations of the mean.
5. This is known as Chebyshev's Rule.
A bell-shaped distribution is one that is normal and symmetric, meaning the curve can be folded along a line of symmetry drawn through the median, and the left and right sides of the curve would fold on each other symmetrically.. With a bell-shaped distribution, the mean, median, and mode are all located at the same place.
2. Approximately 68 percent of the data is within one standard deviation of the mean.
3. Approximately 95 percent of the data is within two standard deviations of the mean.
4. More than 99 percent of the data is within three standard deviations of the mean.
5. This is known as the Empirical Rule.
6. It is important to note that this rule applies only when the shape of the distribution of the data is bell-shaped and symmetric; we will learn more about this when studying the Normal or Gaussian probability distribution in later chapters.

### References

King, B. (2005, Dec.). Graphically Speaking. Retrieved from http://www.ltcc.edu/web/about/institutional-research
Microsoft Bookshelf. (n.d.).

### Chapter Review

The standard deviation can help you calculate the spread of data. There are different equations to use if you are calculating the standard deviation of a sample or of a population.
1. The standard deviation allows us to compare individual data or classes to the data set mean numerically.
2. s =
  or s =
  is the formula for calculating the standard deviation of a sample. To calculate the standard deviation of a population, we would use the population mean, μ, and the formula σ =
  or σ =
 .

### Formula Review

 where

   and

###

For each of the examples given below, tell whether the differences in outcomes may be explained by measurement variability, natural variability, induced variability, or sampling variability.
Use the following information to answer the next two exercises. The following data are the distances between 20 retail stores and a large distribution center. The distances are in miles.
29, 37, 38, 40, 58, 67, 68, 69, 76, 86, 87, 95, 96, 96, 99, 106, 112, 127, 145, 150

### Homework

Use the following information to answer the next nine exercises: The population parameters below describe the full-time equivalent number of students (FTES) each year at Lake Tahoe Community College from 1976–1977 through 2004–2005.
1. μ = 1,000 FTES
2. median = 1,014 FTES
3. σ = 474 FTES
4. first quartile = 528.5 FTES
5. third quartile = 1,447.5 FTES
6. n = 29 years

### Bringing It Together

Use the following information to answer the next two exercises. X = the number of days per week that 100 clients use a particular exercise facility.


# Probability Topics

## Introduction

It is often necessary to guess about the outcome of an event in order to make a decision. Politicians study polls to guess their likelihood of winning an election. Teachers choose a particular course of study based on what they think students can comprehend. Doctors choose the treatments needed for various diseases based on their assessment of likely results. You may have visited a casino where people play games chosen because of the belief that the likelihood of winning is good. You may have chosen your course of study based on the probable availability of jobs.
You have, more than likely, used probability. In fact, you probably have an intuitive sense of probability. Probability deals with the chance of an event occurring. Whenever you weigh the odds of whether or not to do your homework or to study for an exam, you are using probability. In this chapter, you will learn how to solve probability problems using a systematic approach.


# Probability Topics

## Terminology

Probability is a measure that is associated with how certain we are of results, or outcomes, of a particular activity. When the activity is a planned operation carried out under controlled conditions, it is called an experiment. If the result is not predetermined, then the experiment is said to be a chance experiment. Each time the experiment is attempted is called a trial.
Examples of chance experiments include the following:
1. flipping a fair coin,
2. spinning a spinner,
3. drawing a marble at random from a bag, and
4. rolling a pair of dice.
A result of an experiment is called an outcome. The sample space of an experiment is the set, or collection, of all possible outcomes.
There are four main ways to represent a sample space:
*We will investigate tree diagrams and Venn diagrams in Section 3.5.
Note—when represented as a set, the sample space is denoted with an uppercase S.
An event is any combination of outcomes. It is a subset of the sample space, so uppercase letters like A and B are commonly used to represent events. For example, if the experiment is to flip three fair coins, event A might be getting at most one head.
The probability of an event A is written P(A), and

       means the event A can never happen. P(A) = 1 means the event A always happens.

       means the event A is equally likely to occur or not to occur.
If two outcomes or events are equally likely, then they have equal probability. For example, if you toss a fair, six-sided die, each face (1, 2, 3, 4, 5, or 6) is as likely to occur as any other face. If you toss a fair coin, a Head (H) and a Tail (T) are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer.
To calculate the probability of an event A when all outcomes in the sample space are equally likely, count the number of outcomes for event A and divide by the total number of outcomes in the sample space. This is known as the theoretical probability of A.
Theoretical Probability of Event A
For example, if you toss a fair dime and a fair nickel, the sample space is {HH, TH, HT, TT} where T = tails and H = heads. The sample space has four outcomes. Let A represent the outcome getting one head. There are two outcomes that meet this condition {HT, TH}, so

Theoretical probability is not sufficient in all situations, however. Suppose we want to calculate the probability that a randomly selected car will run a red light at a given intersection. In this case, we need to look at events that have occurred, not theoretical possibilities. We could install a traffic camera and count the number of times that cars failed to stop when the light was red and the total number of cars that passed through the intersection for a period of time. These data will allow us to calculate the experimental, or empirical, probability that a car runs the red light.
Experimental Probability of Event A
While theoretical and experimental methods provide two different ways to calculate probability, these methods are closely related. If you flip one fair coin, there is one way to obtain heads and two possible outcomes. So, the theoretical probability of heads is

      . Probability does not predict short-term results, however. If an experiment involves flipping a coin 10 times, you should not expect exactly five heads and five tails. The probability of any outcome measures the long-term relative frequency of that outcome. If you continue to flip the coin (from 20 to 2,000 to 20,000 times) the relative frequency of heads approaches .5 (the probability of heads).This important characteristic of probability experiments is known as the law of large numbers, which states that as the number of repetitions of an experiment is increased, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes do not happen according to any set pattern or order, overall, the long-term observed, or empirical, relative frequency will approach the theoretical probability.
Suppose you roll one fair, six-sided die with the numbers {1, 2, 3, 4, 5, 6} on its faces. Let event E = rolling a number that is at least five. There are two outcomes {5, 6}.

       If you were to roll the die only a few times, you would not be surprised if your observed results did not match the probability. If you were to roll the die a very large number of times, you would expect that, overall,

       of the rolls would result in an outcome of at least five. You would not expect exactly

      , but the long-term relative frequency of obtaining this result would approach the theoretical probability of

       as the number of repetitions grows larger and larger.
It is important to realize that in many situations, the outcomes are not equally likely. A coin or die may be unfair, or biased. Two math professors in Europe had their statistics students test the Belgian one-euro coin and discovered that in 250 trials, a head was obtained 56 percent of the time and a tail was obtained 44 percent of the time. The data seem to show that the coin is not a fair coin; more repetitions would be helpful to draw a more accurate conclusion about such bias. Some dice may be biased. Look at the dice in a game you have at home; the spots on each face are usually small holes carved out and then painted to make the spots visible. Your dice may or may not be biased; it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling casinos make a lot of money depending on outcomes from rolling dice, so casino dice are made differently to eliminate bias. Casino dice have flat faces; the holes are completely filled with paint having the same density as the material that the dice are made out of so that each face is equally likely to occur. Later we will learn techniques to use to work with probabilities for events that are not equally likely.
OR EventAn outcome is in the event A OR B if the outcome is in A or is in B or is in both A and B. For example, let A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}. A OR B = {1, 2, 3, 4, 5, 6, 7, 8}. Notice that 4 and 5 are not listed twice.
AND EventAn outcome is in the event A AND B if the outcome is in both A and B at the same time. For example, let A and B be {1, 2, 3, 4, 5} and {4, 5, 6, 7, 8}, respectively. Then A AND B = {4, 5}.
The complement of event A is denoted A′ (read "A prime"). A′ consists of all outcomes that are not in A. Notice that P(A) + P(A′) = 1. For example, let S = {1, 2, 3, 4, 5, 6} and let A = {1, 2, 3, 4}. Then, A′ = {5, 6}. P(A) = , P(A′) = , and P(A) + P(A′) =
  = 1.
The conditional probability of A given B is written P(A|B), read "the probability of A, given B." P(A|B) is the probability that event A will occur given that the event B has already occurred. A conditional probability reduces the sample space. We calculate the probability of A from the reduced sample space B. The formula to calculate P(A|B) is P(A|B) =
  where P(B) is greater than zero.
For example, suppose we toss one fair, six-sided die. The sample space S = {1, 2, 3, 4, 5, 6}. Let A = {2, 3} and B = {2, 4, 6}. P(A|B) represents the probability that a randomly selected outcome is in A given that it is in B. We know that the outcome must lie in B, so there are three possible outcomes. There is only one outcome in B that also lies in A, so P(A|B) =

      .
We get the same result by using the formula. Remember that S has six outcomes.
P(A|B) =

Understanding Terminology and SymbolsIt is important to read each problem carefully to think about and understand what the events are. Understanding the wording is the first very important step in solving probability problems. Reread the problem several times if necessary. Clearly identify the event of interest. Determine whether there is a condition stated in the wording that would indicate that the probability is conditional; carefully identify the condition, if any.

### References

Worldatlas. (2013). Countries list by continent. Retrieved from http://www.worldatlas.com/cntycont.htm

### Chapter Review

In this module we learned the basic terminology of probability. The set of all possible outcomes of an experiment is called the sample space. Events are subsets of the sample space, and they are assigned a probability that is a number between zero and one, inclusive.

### Formula Review

A and B are events
P(S) = 1 where S is the sample space
0 ≤ P(A) ≤ 1
P(A|B) =

Use the following information to answer the next four exercises. A box is filled with several party favors. It contains 12 hats, 15 noisemakers, 10 finger traps, and five bags of confetti.
Let H = the event of getting a hat.
Let N = the event of getting a noisemaker.
Let F = the event of getting a finger trap.
Let C = the event of getting a bag of confetti.
Use the following information to answer the next six exercises. A jar of 150 jelly beans contains 22 red jelly beans, 38 yellow, 20 green, 28 purple, 26 blue, and the rest are orange.
Let B = the event of getting a blue jelly bean
Let G = the event of getting a green jelly bean.
Let O = the event of getting an orange jelly bean.
Let P = the event of getting a purple jelly bean.
Let R = the event of getting a red jelly bean.
Let Y = the event of getting a yellow jelly bean.
Use the following information to answer the next six exercises. There are 23 countries in North America, 12 countries in South America, 47 countries in Europe, 44 countries in Asia, 54 countries in Africa, and 14 countries in Oceania (Pacific Ocean region).
Let A = the event that a country is in Asia.
Let E = the event that a country is in Europe.
Let F = the event that a country is in Africa.
Let N = the event that a country is in North America.
Let O = the event that a country is in Oceania.
Let S = the event that a country is in South America.
Use the following information to answer the next two exercises. You see a game at a local fair. You have to throw a dart at a color wheel. Each section on the color wheel is equal in area.

Let B = the event of landing on blue.
Let R = the event of landing on red.
Let G = the event of landing on green.
Let Y = the event of landing on yellow.

Use the following information to answer the next 10 exercises. On a baseball team, there are infielders and outfielders. Some players are great hitters, and some players are not great hitters.
Let I = the event that a player in an infielder.
Let O = the event that a player is an outfielder.
Let H = the event that a player is a great hitter.
Let N = the event that a player is not a great hitter.
Use the following information to answer the next two exercises. You are rolling a fair, six-sided number cube. Let E = the event that it lands on an even number. Let M = the event that it lands on a multiple of three.

### Homework


# Probability Topics

## Independent and Mutually Exclusive Events

Independent and mutually exclusive do not mean the same thing.

### Independent Events

Two events are independent if the following are true:
1. P(A|B) = P(A)
2. P(B|A) = P(B)
3. P(A AND B) = P(A)P(B)
Two events A and B are independent events if the knowledge that one occurred does not affect the chance the other occurs. For example, the outcomes of two roles of a fair die are independent events. The outcome of the first roll does not change the probability for the outcome of the second roll. To show two events are independent, you must show only one of the above conditions. If two events are not independent, then we say that they are dependent events.
Sampling may be done with replacement or without replacement.
1. With replacement: If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.
A bag contains four blue and three white marbles. James draws one marble from the bag at random, records the color, and replaces the marble. The probability of drawing blue is

  . When James draws a marble from the bag a second time, the probability of drawing blue is still

  . James replaced the marble after the first draw, so there are still four blue and three white marbles.
1. Without replacement: When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.
The bag still contains four blue and three white marbles. Maria draws one marble from the bag at random, records the color, and sets the marble aside. The probability of drawing blue on the first draw is

  . Suppose Maria draws a blue marble and sets it aside. When she draws a marble from the bag a second time, there are now three blue and three white marbles. So, the probability of drawing blue is now

  . Removing the first marble without replacing it influences the probabilities on the second draw.
If it is not known whether A and B are independent or dependent, assume they are dependent until you can show otherwise.

### Mutually Exclusive Events

A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B) = 0.
For example, suppose the sample space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, and C = {7, 9}. A AND B = {4, 5}. P(A AND B) =  and is not equal to zero. Therefore, A and B are not mutually exclusive. A and C do not have any numbers in common so P(A AND C) = 0. Therefore, A and C are mutually exclusive.
If it is not known whether A and B are mutually exclusive, assume they are not until you can show otherwise. The following examples illustrate these definitions and terms.

### References

Gallup. (n.d.). Retrieved from www.gallup.com/
Lopez, S., and Sidhu, P. (2013, March 28). U.S. teachers love their lives, but struggle in the workplace. Gallup Wellbeing. http://www.gallup.com/poll/161516/teachers-love-lives-struggle-workplace.aspx

### Chapter Review

Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs. If two events are not independent, then we say that they are dependent
In sampling with replacement, each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered not to be independent. When events do not share outcomes, they are mutually exclusive of each other.

### Formula Review

If A and B are independent, P(A AND B) = P(A)P(B), P(A|B) = P(A), and P(B|A) = P(B).
If A and B are mutually exclusive, P(A OR B) = P(A) + P(B) and P(A AND B) = 0.

### Homework

Use the following information to answer the next 12 exercises. The graph shown is based on more than 170,000 interviews that took place from January through December 2012. The sample consists of employed Americans 18 years of age or older. The Health Index Scores are the sample space. We randomly sample one type of Health Index Score, the emotional well-being score.

### Bringing It Together


# Probability Topics

## Two Basic Rules of Probability

In calculating probability, there are two rules to consider when you are determining if two events are independent or dependent and if they are mutually exclusive or not.

### The Multiplication Rule

If A and B are two events defined on a sample space, then P(A AND B) = P(B)P(A|B).
This equation can be rewritten as P(A AND B) = P(B)P(A|B), the multiplication rule.
If A and B are independent, then P(A|B) = P(A). In this special case, P(A AND B) = P(A|B)P(B) becomes P(A AND B) = P(A)P(B).
A bag contains four green marbles, three red marbles, and two yellow marbles. Mark draws two marbles from the bag without replacement. The probability that he draws a yellow marble and then a green marble is
Notice that

  . After the yellow marble is drawn, there are four green marbles in the bag and eight marbles in all.

### The Addition Rule

If A and B are defined on a sample space, then P(A OR B) = P(A) + P(B) − P(A AND B).
Draw one card from a standard deck of playing cards. Let H = the card is a heart, and let J = the card is a jack. These events are not mutually exclusive because a card can be both a heart and a jack.
If A and B are mutually exclusive, then P(A AND B) = 0. Then P(A OR B) = P(A) + P(B) − P(A AND B) becomes P(A OR B) = P(A) + P(B).
Draw one card from a standard deck of playing cards. Let H = the card is a heart and S = the card is a spade. These events are mutually exclusive because a card cannot be a heart and a spade at the same time. The probability that the card is a heart or a spade is

### References

Baseball Almanac. (2013). Retrieved from www.baseball-almanac.com
DiCamillo, Mark, and Field, M. The file poll. Field Research Corporation. Retrieved from http://www.field.com/fieldpollonline/subscribers/Rls2443.pdf
Field Research Corporation. (n.d.). Retrieved from www.field.com/fieldpollonline
Forum Research. (n.d.). Mayors approval down. Retrieved from http://www.forumresearch.com/forms/News Archives/News Releases/74209_TO_Issues_-_Mayoral_Approval_%28Forum_Research%29%2820130320%29.pdf
Rider, D. (2011, Sept. 14). Ford support plummeting, poll suggests. The Star. Retrieved from http://www.thestar.com/news/gta/2011/09/14/ford_support_plummeting_poll_suggests.html
Shin, H. B., and Kominski, R. A. (2010 April). Language use in the United States: 2007 (American Community Survey Reports, ACS-12). Washington, DC: United States Census Bureau. Retrieved from http://www.census.gov/hhes/socdemo/language/data/acs/ACS-12.pdf
The Roper Center for Public Opinion Research. (n.d.). Archives. Retrieved from http://www.ropercenter.uconn.edu/
The Wall Street Journal. (n.d.). Retrieved from https://www.wsj.com/
U.S. Census Bureau. (n.d.). Retrieved from https://www.census.gov/
Wikipedia. (n.d.). Roulette. Retrieved from http://en.wikipedia.org/wiki/Roulette

### Chapter Review

The multiplication rule and the addition rule are used for computing the probability of A and B, as well as the probability of A or B for two given events A, B defined on the sample space. In sampling with replacement, each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered to be not independent. The events A and B are mutually exclusive events when they do not have any outcomes in common.

### Formula Review

The multiplication rule—P(A AND B) = P(A|B)P(B)
The addition rule—P(A OR B) = P(A) + P(B) − P(A AND B)
Use the following information to answer the next 10 exercises. Forty-eight percent of all voters of a certain state prefer life in prison without parole over the death penalty for a person convicted of first-degree murder. Among Latino registered voters in this state, 55 percent prefer life in prison without parole over the death penalty for a person convicted of first-degree murder. Of all citizens in this state, 37.6 percent are Latino.
In this problem, let

Suppose that one citizen is randomly selected.

### Homework

.  A local restaurant sells pork chops and chicken breasts.  The given values below are the weights (in ounces) of pork chops and chicken breasts listed on the menu.  Your server will randomly select one piece of meat (pork chop or chicken breast) that you will be served.


# Probability Topics

## Contingency Tables

A two-way table provides a way of portraying data that can facilitate calculating probabilities. When used to calculate probabilities, a two-way table is often called a contingency table. The table helps in determining conditional probabilities quite easily. The table displays sample values in relation to two different variables that may be dependent or contingent on one another. We used two-way tables in Chapters 1 and 2 to calculate marginal and conditional distributions. These tables organize data in a way that supports the calculation of relative frequency and, therefore, experimental (empirical) probability. Later on, we will use contingency tables again, but in another manner.

### References

American Red Cross. (2013). Blood Types. Retrieved from http://www.redcrossblood.org/learn-about-blood/bloodtypes
Centers for Disease Control and Prevention/National Center for Health Statistics, United States Department of Health and Human Services. (n.d.). Retrieved from https://www.cdc.gov/nchs/
Haiman, C. A., et al. (2006, Jan. 26). Ethnic and racial differences in the smoking-related risk of lung cancer. The New England Journal of Medicine. Retrieved from http://www.nejm.org/doi/full/10.1056/NEJMoa033250
Samuels, T. M. (2013). Strange facts about RH negative blood. eHow Health. Retrieved from http://www.ehow.com/facts_5552003_strange-rh-negative-blood.html
The Disaster Center Crime Pages. (n.d.). United States: Uniform crime report  state statistics from 19602011. Retrieved from http://www.disastercenter.com/crime/
United Blood Services. (2011). Human blood types. Retrieved from http://www.unitedbloodservices.org/learnMore.aspx
United States Senate. (n.d.). Retrieved from www.senate.gov

### Chapter Review

There are several tools you can use to help organize and sort data when calculating probabilities. Contingency tables, also known as two-way tables, help display data and are particularly useful when calculating probabilites that have multiple dependent variables.

###

Use the following information to answer the next four exercises.  shows a random sample of musicians and how they learned to play their instruments.

### Bringing It Together

Use the following information to answer the next seven exercises. An article in the , reported about a study of people who use a product in California and Hawaii. In one part of the report, the self-reported ethnicity and using the product levels per day were given. Of the people using the product at most 10 times a day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans, and 7,650 whites. Of the people using the product 11 to 20 times per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people using the product 21 to 30 times per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 Whites. Of the people using the product at least 31 times per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites.

### Homework

Use the information in the  The table shows the political party affiliation of each of 67 members of the U.S. Senate in June 2012, and when they would next be up for reelection.

Use the following information to answer the next two exercises. The table of data obtained from  shows hit information for four well-known baseball players. Suppose that one hit from the table is randomly selected.


# Probability Topics

## Tree and Venn Diagrams

Sometimes, when the probability problems are complex, it can be helpful to graph the situation. Tree diagrams and Venn diagrams are two tools that can be used to visualize and solve conditional probabilities.

### Tree Diagrams

A tree diagram is a special type of graph used to determine the outcomes of an experiment. It consists of branches that are labeled with either frequencies or probabilities. Tree diagrams can make some probability problems easier to visualize and solve. The following example illustrates how to use a tree diagram:

### Venn Diagram

A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events.

### References

American Cancer Society. (n.d.). Retrieved from https://www.cancer.org/
Clara County Public Health Department. (n.d.). Retrieved from https://www.sccgov.org/sites/sccphd/en-us/pages/phd.aspx
Federal Highway Administration, U.S. Department of Transportation. (n.d.). Retrieved from https://www.fhwa.dot.gov/
The Data and Story Library. (1996). Retrieved from http://lib.stat.cmu.edu/DASL/
The Roper Center for Public Opinion Research. (2013). Search for datasets. Retrieved from http://www.ropercenter.uconn.edu/data_access/data/search_for_datasets.html
USA Today. (n.d.). Retrieved from https://www.usatoday.com/
U.S. Census Bureau. (n.d.). Retrieved from https://www.census.gov/
World Bank Group. (2013). Environment. Available online at http://data.worldbank.org/topic/environment

### Chapter Review

A tree diagram uses branches to show the different outcomes of experiments and makes complex probability questions easy to visualize.
A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events. A Venn diagram is especially helpful for visualizing the OR event, the AND event, and the complement of an event and for understanding conditional probabilities.

###

### Homework

Use the following information to answer the next two exercises. This tree diagram shows the tossing of an unfair coin followed by drawing one bead from a cup containing three red (R), four yellow (Y), and five blue (B) beads. For the coin, P(H) =
  and P(T) =
  where H is heads and T is tails.

### Bringing It Together

Use the following information to answer the next two exercises. Suppose that you have eight cards. Five are green and three are yellow. The cards are well shuffled.
Use the following information to answer the next two exercises. The percent of licensed U.S. drivers (from a recent year) who are female is 48.60. Of the females, 5.03 percent are age 19 and under; 81.36 percent are age 20–64; 13.61 percent are age 65 or over. Of the licensed U.S. male drivers, 5.04 percent are age 19 and under; 81.43 percent are age 20–64; 13.53 percent are age 65 or over.


# Discrete Random Variables

## Introduction

A student takes a 10-question, true-false quiz. Because the student had such a busy schedule, he or she could not study and guesses randomly at each answer. What is the probability of the student passing the test with at least a 70 percent?
Small companies might be interested in the number of long-distance phone calls their employees make during the peak time of the day. Suppose the average is 20 calls. What is the probability that the employees make more than 20 long-distance phone calls during the peak time?
These two examples illustrate two different types of probability problems involving discrete random variables. Recall that discrete data are data that you can count. A random variable is a variable whose values are numerical outcome of a probability experiment.  We always describe a random variable in words and its values in numbers. The values of a random variable can vary with each repetition of an experiment.

### Random Variable Notation

Uppercase letters such as X or Y denote a random variable. Lowercase letters like x or y denote the value of a random variable. If
The following are examples of random variables:
Example 1: Suppose a jar contains three marbles, one blue, one red, and one white. Randomly draw one marble from the jar. Let X = the possible number of red marbles to be drawn. The sample space for the drawing is red, white, and blue. Then, x = 0,1. If the marble we draw is red, then x = 1; otherwise, x = 0.
Example 2: Let X = the number of female children in a randomly selected family with only two kids. Here we are only interested in families with two kids, not families with one kid or more than two kids. The sample space for the genders of two-kid families is MM, MF, FM, FF. Here the first letter represents the gender of the older child and the second letter represents the gender of the younger child. F represents a female child and M represents a male child. For example, FM represents that the older child is a girl and the younger child is a boy, while MF represents that the older child is a boy and the younger child is a girl. Then, x = 0,1,2. A family has 0 female children if it has two boys (MM), a family has one female child if it has one boy and one girl (MF or FM), and a family has two female children if both kids are girls (FF).
Example 3: Let X = the number of heads you get when you toss three fair coins. The sample space for the toss of three fair coins is TTT, THH, HTH, HHT, HTT, THT, TTH, HHH. Here the first letter represents the result of the first toss, the second letter represents the result of the second toss, and the third letter represents the result of the third toss. T represents a tail and H represents a head. For example, THH means we get a tail in the first toss but a head in the second and third toss, while HHT means we get a head in the first and second toss but a tail in the third toss. Then, x = 0, 1, 2, 3. There are 0 heads if the result is TTT, one head if the result is THT, TTH, or HTT, two heads if the result is THH, HTH, or HHT, and three heads if the result is HHH.


# Discrete Random Variables

## Probability Distribution Function (PDF) for a Discrete Random Variable

There are two types of random variables, discrete random variables and continuous random variables. The values of a discrete random variable are countable, which means the values are obtained by counting. All random variables we discussed in previous examples are discrete random variables. We counted the number of red balls, the number of heads, or the number of female children to get the corresponding random variable values. The values of a continuous random variable are uncountable, which means the values are not obtained by counting. Instead, they are obtained by measuring. For example, let X = temperature of a randomly selected day in June in a city. The value of X can be 68°, 71.5°, 80.6°, or 90.32°. These values are obtained by measuring by a thermometer.  Another example of a continuous random variable is the height of a randomly selected high school student. The value of this random variable can be 5'2", 6'1", or 5'8". Those values are obtained by measuring by a ruler.
A discrete probability distribution function has two characteristics:
1. Each probability is between zero and one, inclusive.
2. The sum of the probabilities is one.

### Chapter Review

The characteristics of a probability distribution function (PDF) for a discrete random variable are as follows:
1. Each probability is between zero and one, inclusive (inclusive means to include zero and one)
2. The sum of the probabilities is one

###

Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, or in other words, how long new hires stay with the company. Over the years, the company has established the following probability distribution:
Let X = the number of years a new hire will stay with the company.
Let P(x) = the probability that a new hire will stay with the company x years.
Use the following information to answer the next four exercises: A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker has established a probability distribution.
Use the following information to answer the next two exercises: Ellen has music practice three days a week. She practices for all of the three days 85 percent of the time, two days 8 percent of the time, one day 4 percent of the time, and no days 3 percent of the time. One week is selected at random.
Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35 percent of the time, four events 25 percent of the time, three events 20 percent of the time, two events 10 percent of the time, one event 5 percent of the time, and no events 5 percent of the time.

### Homework


# Discrete Random Variables

## Mean or Expected Value and Standard Deviation

The expected value of a discrete random variable  X, symbolized as E(X), is often referred to as the long-term average or  (symbolized as μ). This means that over the long term of doing an experiment over and over, you would expect this average. For example, let X = the number of heads you get when you toss three fair coins. If you repeat this experiment (toss three fair coins) a large number of times, the expected value of X is the number of heads you expect to get for each three tosses on average.
As you learned in Chapter 3, if you toss a fair coin, the probability that the result is heads is 0.5. This probability is a theoretical probability, which is what we expect to happen. This probability does not describe the short-term results of an experiment. If you flip a coin two times, the probability does not tell you that these flips will result in one head and one tail. Even if you flip a coin 10 times or 100 times, the probability does not tell you that you will get half tails and half heads. The probability gives information about what can be expected in the long term. To demonstrate this, Karl Pearson once tossed a fair coin 24,000 times! He recorded the results of each toss, obtaining heads 12,012 times. The relative frequency of heads is 12,012/24,000 = .5005, which is very close to the theoretical probability .5. In his experiment, Pearson illustrated the law of large numbers.
The law of large numbers states that, as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency approaches zero (the theoretical probability and the relative frequency get closer and closer together). The relative frequency is also called the experimental probability, a term that means what actually happens.
In the next example, we will demonstrate how to find the expected value and standard deviation of a discrete probability distribution by using relative frequency.
Like data, probability distributions have variances and standard deviations. The variance of a probability distribution is symbolized as

   and the standard deviation of a probability distribution is symbolized as σ. Both are parameters since they summarize information about a population. To find the variance

   of a discrete probability distribution, find each deviation from its expected value, square it, multiply it by its probability, and add the products. To find the standard deviation σ of a probability distribution, simply take the square root of variance

  . The formulas are given as below.
Generally for probability distributions, we use a calculator or a computer to calculate μ and σ to reduce rounding errors. For some probability distributions, there are shortcut formulas for calculating μ and σ.
Some of the more common discrete probability functions are binomial, geometric, hypergeometric, and Poisson. Most elementary courses do not cover the geometric, hypergeometric, and Poisson. Your instructor will let you know if he or she wishes to cover these distributions.
A probability distribution function is a pattern. You try to fit a probability problem into a pattern or distribution in order to perform the necessary calculations. These distributions are tools to make solving probability problems easier. Each distribution has its own special characteristics. Learning the characteristics enables you to distinguish among the different distributions.

### References

Florida State University. (n.d.). Class catalogue at the Florida State University. Retrieved from https://apps.oti.fsu.edu/RegistrarCourseLookup/SearchFormLegacy
World Earthquakes. (2012). World earthquakes: Live earthquake news and highlights. Retrieved from http://www.worldearthquakes.com/index.php?option=ethq_prediction

### Chapter Review

The expected value, or mean, of a discrete random variable predicts the long-term results of a statistical experiment that has been repeated many times. The standard deviation of a probability distribution is used to measure the variability of possible outcomes.

### Formula Review

Mean or Expected Value:

Standard Deviation:

Use the following information to answer the next five exercises: A physics professor wants to know what percent of physics majors will spend the next several years doing postgraduate research. He has the following probability distribution:
Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next so that she can plan what classes to offer. Over the years, she has established the following probability distribution:
1. Let X = the number of years a student will study ballet with the teacher.
2. Let P(x) = the probability that a student will study ballet x years.

### HOMEWORK


# Discrete Random Variables

## Binomial Distribution (Optional)

There are three characteristics of a binomial experiment:

The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X = the number of successes obtained in the n independent trials.
There are shortcut formulas for calculating mean μ, variance σ2, and standard deviation σ of a binomial probability distribution. The formulas are given as below. The deriving of these formulas will not be discussed in this book.
Here n is the number of trials, p is the probability of a success, and q is the probability of a failure.

### Notation for the Binomial: B = Binomial Probability Distribution Function

X ~ B(n, p)
Read this as X is a random variable with a binomial distribution. The parameters are n and p: n = number of trials, p = probability of a success on each trial.
The following is the interpretation of the mean

   and standard deviation

  :
If you randomly select 20 adult workers, and do that over and over, you expect around eight adult workers out of 20 to have a high school diploma but do not pursue any further education on average. And you expect that to vary by about two workers on average.

### References

American Cancer Society. (2013). What are the key statistics about pancreatic cancer? Retrieved from http://www.cancer.org/cancer/pancreaticcancer/detailedguide/pancreatic-cancer-key-statistics
Central Intelligence Agency. (n.d.). The world factbook. Retrieved from https://www.cia.gov/library/publications/theworld-factbook/geos/af.html
ESPN NBA. (2013). NBA statistics  2013. Retrieved from http://espn.go.com/nba/statistics/_/seasontype/2
Newport, F. (2013). Americans still enjoy saving rather than spending: Few demographic differences seen in these views other than by income. GALLUP Economy. Retrieved from http://www.gallup.com/poll/162368/americansenjoy-saving-rather-spending.aspx
Pryor, J. H., et al. (2011). The American freshman: National norms fall 2011. Los Angeles, CA: Cooperative Institutional Research Program, Higher Education Research Institute. Retrieved from http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/TheAmericanFreshman2011.pdf
Wikipedia. (n.d.). Distance education. Retrieved from http://en.wikipedia.org/wiki/Distance_education
World Bank Group. (2013). Access to electricity (% of population). Retrieved from http://data.worldbank.org/indicator/EG.ELC.ACCS.ZS?order=wbapi_data_value_2009%20wbapi_data_value%20wbapi_data_value-first&sort=asc

### Chapter Review

A statistical experiment can be classified as a binomial experiment if the following conditions are met:
1. There are a fixed number of trials, n
2. There are only two possible outcomes, called success and failure, for each trial; the letter p denotes the probability of a success on one trial and q denotes the probability of a failure on one trial
3. The n trials are independent and are repeated using identical conditions
The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X = the number of successes obtained in the n independent trials. The mean of X can be calculated using the formula μ = np, and the standard deviation is given by the formula σ =
 .

### Formula Review

X ~ B(n, p) means that the discrete random variable X has a binomial probability distribution with n trials and probability of success p.
X = the number of successes in n independent trials
n = the number of independent trials
X takes on the values x = 0, 1, 2, 3, . . . , n
p = the probability of a success for any trial
q = the probability of a failure for any trial
p + q = 1
q = 1 – p
The mean of X is μ = np. The standard deviation of X is σ =
 .
Use the following information to answer the next eight exercises: Researchers collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agreed with a recent federal law that was passed.
Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number who agreed with that law.

### HOMEWORK

Use the following information to answer the next four exercises: Recently, a nurse commented that when a patient calls the medical advice line claiming to have the flu, the chance that he or she truly has the flu (and not just a nasty cold) is only about 4 percent. Of the next 25 patients calling in claiming to have the flu, we are interested in how many actually have the flu.
Use the following information to answer the next three exercises: The probability that a local hockey team will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games.
Let X = the number of games won in that upcoming month.


# Discrete Random Variables

## Geometric Distribution (Optional)

There are three main characteristics of a geometric experiment:
1. Repeating independent Bernoulli trials until a success is obtained. Recall that a Bernoulli trial is a binomial experiment with number of trials n = 1. In other words, you keep repeating what you are doing until the first success. Then you stop. For example, you throw a dart at a bull's-eye until you hit the bull's-eye. The first time you hit the bull's-eye is a success so you stop throwing the dart. It might take six tries until you hit the bull's-eye. You can think of the trials as failure, failure, failure, failure, failure, success, stop.
2. In theory, the number of trials could go on forever. There must be at least one trial.
3. The probability, p, of a success and the probability, q, of a failure do not change from trial to trial. p + q = 1 and q = 1 − p. For example, the probability of rolling a three when you throw one fair die is
 . This is true no matter how many times you roll the die. Suppose you want to know the probability of getting the first three on the fifth roll. On rolls one through four, you do not get a face with a three. The probability for each of the rolls is q =
 , the probability of a failure. The probability of getting a three on the fifth roll is
  = .0804.
X = the number of independent trials until the first success.
p = the probability of a success, q = 1  p = the probability of a failure.
There are shortcut formulas for calculating mean μ, variance σ2, and standard deviation σ of a geometric probability distribution. The formulas are given as below. The deriving of these formulas will not be discussed in this book.

### Notation for the Geometric: G = Geometric Probability Distribution Function

X ~ G(p)
Read this as X is a random variable with a . The parameter is p; p = the probability of a success for each trial.

### References

Central Intelligence Agency. (n.d.). The world factbook. Retrieved from https://www.cia.gov/library/publications/theworld-factbook/geos/af.html
Pew Research Center. (n.d.). Millennials: A portrait of generation next. Retrieved from http://www.pewsocialtrends.org/files/2010/10/millennials-confident-connected-open-to-change.pdf
Pew Research. (2013). Millennials: confident. Executive Summary: Pew Research Social & Demographic Trends. Retrieved from http://www.pewsocialtrends.org/2010/02/24/millennials-confident-connected-open-tochange/
Pryor, J. H., et al. (2011). The American freshman: National norms fall 2011. Los Angeles: Cooperative Institutional Research Program, Higher Education Research Institute. Retrieved from http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/ TheAmericanFreshman2011.pdf
The European Union and ICON-Institute. (2007/8). Summary of the national risk and vulnerability assessment 2007/8: A profile of Afghanistan. Retrieved from http://ec.europa.eu/europeaid/where/asia/documents/afgh_brochure_summary_en.pdf
The World Bank. (2013). Prevalence of HIV, total (% of populations ages 15-49). Retrieved from http://data.worldbank.org/indicator/SH.DYN.AIDS.ZS?order=wbapi_data_value_2011+wbapi_data_value+wbapi_data_value-last&sort=desc
UNICEF Television. (n.d.). UNICEF reports on female literacy centers in Afghanistan established to teach women and girls basic reading and writing skills. (Video). Retrieved from http://www.unicefusa.org/assets/video/afghan-femaleliteracy-centers.html

### Chapter Review

There are three characteristics of a geometric experiment:
1. There are one or more Bernoulli trials with all failures except the last one, which is a success
2. In theory, the number of trials could go on forever; there must be at least one trial
3. The probability, p, of a success and the probability, q, of a failure are the same for each trial
In a geometric experiment, define the discrete random variable X as the number of independent trials until the first success. We say that X has a geometric distribution and write X ~ G(p) where p is the probability of success in a single trial.
The mean of the geometric distribution X ~ G(p) is μ =
  =
 .

### Formula Review

X ~ G(p) means that the discrete random variable X has a geometric probability distribution with probability of success in a single trial p.
X = the number of independent trials until the first success
X takes on the values x = 1, 2, 3, . . .
p = the probability of a success for any trial
q = the probability of a failure for any trial p + q = 1 : q = 1 – p
The mean is μ =
 .
The standard deviation is σ =

 =

.
Use the following information to answer the next six exercises: Researchers collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agree with a recent law that was passed. Suppose that you randomly select freshman from the study until you find one who replies yes. You are interested in the number of freshmen you must ask.

### HOMEWORK


# Discrete Random Variables

## Hypergeometric Distribution (Optional)

There are five characteristics of a hypergeometric experiment:
1. You take samples from two groups.
2. You are concerned with a group of interest, called the first group.
3. You sample without replacement from the combined groups. For example, you want to choose a softball team from a combined group of 11 men and 13 women. The team consists of 10 players.
4. Each pick is not independent, since sampling is without replacement. In the softball example, the probability of picking a woman first is
 . The probability of picking a man second is
  if a woman was picked first. It is
  if a man was picked first. The probability of the second pick depends on what happened in the first pick.
5. You are not dealing with Bernoulli trials.
The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution. The random variable X = the number of items from the group of interest.

### Notation for the Hypergeometric: H = Hypergeometric Probability Distribution Function

X ~ H(r, b, n)
Read this as X is a random variable with a hypergeometric distribution. The parameters are r, b, and n: r = the size of the group of interest (first group), b = the size of the second group, n = the size of the chosen sample.

### Chapter Review

A hypergeometric experiment is a statistical experiment with the following properties:
1. You take samples from two groups
2. You are concerned with a group of interest, called the first group
3. You sample without replacement from the combined groups
4. Each pick is not independent, since sampling is without replacement
5. You are not dealing with Bernoulli trials
The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution. The random variable X = the number of items from the group of interest. The distribution of X is denoted X ~ H(r, b, n), where r = the size of the group of interest (first group), b = the size of the second group, and n = the size of the chosen sample. It follows that n ≤ r + b. The mean of X is μ =
  and the standard deviation is σ =
 .

### Formula Review

X ~ H(r, b, n) means that the discrete random variable X has a hypergeometric probability distribution with r = the size of the group of interest (first group), b = the size of the second group, and n = the size of the chosen sample.
X = the number of items from the group of interest that are in the chosen sample, and X may take on the values x = 0, 1, . . . , up to the size of the group of interest. The minimum value for X may be larger than zero in some instances.
n ≤ r + b
The mean of X is given by the formula μ =
  and the standard deviation is =
 .
Use the following information to answer the next five exercises: Suppose that a group of statistics students is divided into two groups: business majors and non-business majors. There are 16 business majors in the group and seven non-business majors in the group. A random sample of nine students is taken. We are interested in the number of business majors in the sample.

### HOMEWORK


# Discrete Random Variables

## Poisson Distribution (Optional)

There are two main characteristics of a Poisson experiment.
1. The Poisson probability distribution gives the probability of a number of events occurring in a fixed interval of time or space if these events happen with a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on the average, there are five words spelled incorrectly in 100 pages. The interval is the 100 pages.
2. The Poisson distribution may be used to approximate the binomial if the probability of success is small (such as .01) and the number of trials is large (such as 1,000). You will verify the relationship in the homework exercises. n is the number of trials, and p is the probability of a success.
The random variable X = the number of occurrences in the interval of interest.

### Notation for the Poisson: P = Poisson Probability Distribution Function

X ~ P(μ)
Read this as X is a random variable with a Poisson distribution. The parameter is μ (or λ); μ (or λ) = the mean for the interval of interest.

### References

Centers for Disease Control and Prevention. (2012, Oct. 2). Teen drivers: Get the facts. Retrieved from http://www.cdc.gov/Motorvehiclesafety/Teen_Drivers/teendrivers_factsheet.html
Daily Mail. (2011, June 9). One born every minute: the maternity unit where mothers are THREE to a bed. Retrieved from http://www.dailymail.co.uk/news/article-2001422/Busiest-maternity-ward-planet-averages-60-babies-day-mothersbed.html
Department of Aviation at the Hartsfield-Jackson Atlanta International Airport. (2013). ATL fact sheet. Retrieved from http://www.atlanta-airport.com/Airport/ATL/ATL_FactSheet.aspx
Lenhart, A. (2012). Teens, smartphones & testing: Texting volume is up while the frequency of voice calling is down. About one in four teens say they own smartphones. Pew Internet. Retrieved from http://www.pewinternet.org/~/media/Files/Reports/2012/PIP_Teens_Smartphones_and_Texting.pdf
Ministry of Health, Labour, and Welfare. (n.d.). Children and childrearing. Retrieved from http://www.mhlw.go.jp/english/policy/children/children-childrearing/index.html
Pew Internet. (2013). How Americans use text messaging. Retrieved from http://pewinternet.org/Reports/2011/Cell-Phone-Texting-2011/Main-Report.aspx
South Carolina Department of Mental Health. (2006). Eating disorder statistics. Retrieved from http://www.state.sc.us/dmh/anorexia/statistics.htm
The Guardian. (2011, June 8). Giving birth in Manila: The maternity ward at the Dr Jose Fabella Memorial Hospital in Manila, the busiest in the Philippines, where there is an average of 60 births a day. Retrieved from http://www.theguardian.com/world/gallery/2011/jun/08/philippines-health#/?picture=375471900&index=2
Vanderkam, L. (2012, Oct. 8). Stop checking your email, now. CNNMoney. Retrieved from http://management.fortune.cnn.com/2012/10/08/stop-checking-your-email-now/
World Earthquakes. (2012). World earthquakes: Live earthquake news and highlights. Retrieved from http://www.worldearthquakes.com/index.php?option=ethq_prediction

### Chapter Review

A Poisson probability distribution of a discrete random variable gives the probability of a number of events occurring in a fixed interval of time or space, if these events happen at a known average rate and independently of the time since the last event. The Poisson distribution may be used to approximate the binomial, if the probability of success is small (less than or equal to .05) and the number of trials is large (greater than or equal to 20).

### Formula Review

X ~ P(μ) means that X has a Poisson probability distribution where X = the number of occurrences in the interval of interest.
X takes on the values x = 0, 1, 2, 3, . . .
The mean μ is typically given.
The variance is σ2 = μ, and the standard deviation is
 .
When P(μ) is used to approximate a binomial distribution, μ = np where n represents the number of independent trials and p represents the probability of success in a single trial.
Use the following information to answer the next six exercises: On average, a clothing store gets 120 customers per day.
Use the following information to answer the next six exercises: On average, eight teens in the United States die from motor vehicle injuries per day. As a result, states across the country are debating raising the driving age.

### HOMEWORK

Use the following information to answer the next two exercises: The average number of times per week that Mrs. Plum’s cats wake her up at night because they want to play is 10. We are interested in the number of times her cats wake her up each week.


# Continuous Random Variables

## Introduction

Continuous random variables have many applications. Baseball batting averages, IQ scores, the length of time a long-distance telephone call lasts, the amount of money a person carries, the length of time a computer chip lasts, and SAT scores are just a few. The field of reliability depends on a variety of continuous random variables.

### Properties of Continuous Probability Distributions

The graph of a continuous probability distribution is a curve. Probability is represented by the area under the curve.
The curve is called the probability density function (abbreviated as pdf). We use the symbol f(x) to represent the curve. f(x) is the function that corresponds to the graph; we use the density function f(x) to draw the graph of the probability distribution.
Area under the curve is given by a different function called the  cumulative distribution function  (abbreviated as cdf). The cumulative distribution function is used to evaluate probability as area.

We will find the area that represents probability by using geometry, formulas, technology, or probability tables.  In general, calculus is needed to find the area under the curve for many probability density functions.  When we use formulas to find the area in this textbook, we are using formulas that were found by using the techniques of integral calculus.  However, because most students taking this course have not studied calculus, we will not be using calculus in this textbook.
There are many continuous probability distributions. When probability is modeled by use of a continuous probability distribution, the distribution used is selected to model and fit the particular situation in the best way.
In this chapter and the next, we will study the uniform distribution, the exponential distribution, and the normal distribution. The following graphs illustrate these distributions:


# Continuous Random Variables

## Continuous Probability Functions

We begin by defining a continuous probability density function. We use the function notation f(x). Intermediate algebra may have been your first formal introduction to functions. In the study of probability, the functions we study are special. We define the function f(x) so that the area between it and the x-axis is equal to a probability. Since the maximum probability is one, the maximum area is also one. For continuous probability distributions, PROBABILITY = AREA.

### Chapter Review

The probability density function (pdf) is used to describe probabilities for continuous random variables. The area under the density curve between two points corresponds to the probability that the variable falls between those two values. In other words, the area under the density curve between points a and b is equal to P(a < x < b). The cumulative distribution function (cdf) gives the probability as an area. If X is a continuous random variable, the probability density function (pdf), f(x), is used to draw the graph of the probability distribution. The total area under the graph of f(x) is one. The area under the graph of f(x) and between values a and b gives the probability P(a < x < b).
The cumulative distribution function (cdf) of X is defined by P (X ≤ x). It is a function of x that gives the probability that the random variable is less than or equal to x.

### Formula Review

Probability density function (pdf) f(x):
1. f(x) ≥ 0
2. The total area under the curve f(x) is one.
Cumulative distribution function (cdf): P(X ≤ x)

### Homework

For each probability and percentile problem, draw the picture.


# Continuous Random Variables

## The Uniform Distribution

The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data are inclusive or exclusive of endpoints.

### Chapter Review

If X has a uniform distribution where a < x < b or a ≤ x ≤ b, then X takes on values between a and b (may include a and b). All values x are equally likely. We write X ∼ U(a, b). The mean of X is
. The standard deviation of X is
. The probability density function of X is
 for a ≤ x ≤ b. The cumulative distribution function of X is P(X ≤ x) =
. X is continuous.
The probability P(c < X < d) may be found by computing the area under f(x), between c and d. Since the corresponding area is a rectangle, the area may be found simply by multiplying the width and the height.

### Formula Review

X = a real number between a and b (in some instances, X can take on the values a and b). a = smallest X, b = largest X
X ~ U(a, b)
The mean is

The standard deviation is

Probability density function:

 for

Area to the left of  P(X < x) = (x – a)

Area to the right of  P(X > x) = (b – x)

Area between  P(c < x < d) = (base)(height) = (d – c)

Uniform: X ~ U(a, b) where a < x < b
1. pdf:

for a ≤ x ≤ b

2. cdf: P(X ≤ x) =

3. mean µ =

4. standard deviation σ

5. P(c < X < d) = (d – c)

### References

McDougall, J. A. (1995). The McDougall program for maximum weight loss. New York: Plume
Use the following information to answer the next 10 questions. The data that follow are the square footage (in 1,000 feet squared) of 28 homes:
The sample mean = 2.50 and the sample standard deviation = 0.8302.
The distribution can be written as X ~ U(1.5, 4.5).
Use the following information to answer the next eight exercises. A distribution is given as X ~ U(0, 12).
Use the following information to answer the next 12 exercises. The age of cars in the staff parking lot of a suburban college is uniformly distributed from six months (0.5 years) to 9.5 years.

### Homework

For each probability and percentile problem, draw the picture.
Use the following information to answer the next three exercises. The Sky Train from the terminal to the rental–car and long–term parking center is supposed to arrive every eight minutes. The waiting times for the train are known to follow a uniform distribution.


# Continuous Random Variables

## The Exponential Distribution (Optional)

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long-distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.
Values for an exponential random variable occur in the following way. There are fewer large values and more small values. For example, the amount of money customers spend in one trip to the supermarket follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.
Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts.

### Memorylessness of the Exponential Distribution

In  recall that the amount of time between customers is exponentially distributed with a mean of two minutes (X ~ Exp(0.5)). Suppose that five minutes have elapsed since the last customer arrived. Since an unusually long amount of time has now elapsed, it would seem to be more likely for a customer to arrive within the next minute. With the exponential distribution, this is not the case—the additional time spent waiting for the next customer does not depend on how much time has already elapsed since the last customer. This is referred to as the memoryless property. Specifically, the memoryless property says the following
For example, if five minutes have elapsed since the last customer arrived, then the probability that more than one minute will elapse before the next customer arrives is computed by using r = 5 and t = 1 in the foregoing equation.
This is the same probability as that of waiting more than one minute for a customer to arrive after the previous arrival.
The exponential distribution is often used to model the longevity of an electrical or a mechanical device. In , the lifetime of a certain computer part has the exponential distribution with a mean of ten years (X ~ Exp(0.1)). The memoryless property says that knowledge of what has occurred in the past has no effect on future probabilities. In this case it means that an old part is not any more likely to break down at any particular time than a brand new part. In other words, the part stays as good as new until it suddenly breaks. For example, if the part has already lasted ten years, then the probability that it lasts another seven years is P(X > 17|X > 10) = P(X > 7) = 0.4966.

### Relationship Between the Poisson and the Exponential Distribution

There is an interesting relationship between the exponential distribution and the Poisson distribution. Suppose that the time that elapses between two successive events follows the exponential distribution with a mean of μ units of time. Also assume that these times are independent, meaning that the time between events is not affected by the times between previous events. If these assumptions hold, then the number of events per unit time follows a Poisson distribution with mean λ = 1/μ. Recall from the chapter on Discrete Random Variables that if X has the Poisson distribution with mean λ, then
. Conversely, if the number of events per unit time follows a Poisson distribution, then the amount of time between events follows the exponential distribution. (k! = k*(k–1*)(k–2)*(k–3)…3*2*1)

### Chapter Review

If X has an exponential distribution with mean μ, then the decay parameter is m =
, and we write X ~ Exp(m) where x ≥ 0 and m > 0 . The probability density function of X is f(x) = me (or equivalently
 . The cumulative distribution function of X is P(X ≤ x) = 1 – e–.
The exponential distribution has the memoryless property, which says that future probabilities do not depend on any past information. Mathematically, it says that P(X > x + k|X > x) = P(X > k).
If T represents the waiting time between events, and if T ~ Exp(λ), then the number of events X per unit time follows the Poisson distribution with mean λ. The probability density function of X is
. This may be computed using a TI-83, 83+, 84, 84+ calculator with the command poissonpdf(λ, k). The cumulative distribution function P(X ≤ k) may be computed using the TI-83, 83+,84, 84+ calculator with the command poissoncdf(λ, k).

### Formula Review

Exponential: X ~ Exp(m) where m = the decay parameter
1. pdf: f(x) = me(– where x ≥ 0 and m > 0
2. cdf: P(X ≤ x) = 1 – e(–
3. mean µ =

4. standard deviation σ = µ
5. percentile k: k =

6. Additionally

7. Memoryless property: P(X > x + k|X > x) = P (X > k)
8. Poisson probability:
 with mean λ
9. k! = k*(k−1)*(k−2)*(k−3)*…3*2*1

### References

Baseball-Reference.com. (2013). No-hitter. Retrieved from http://www.baseball-reference.com/bullpen/No-hitter
U.S. Census Bureau. (n.d.). Retrieved from https://www.census.gov/
World Earthquakes. (2013). Earthquake data for Papua New Guinea. Retrieved from http://www.world-earthquakes.com/
Zhou, Rick. (2013). Exponential distribution lecture slides. Retrieved from www.public.iastate.edu/~riczw/stat330s11/lecture/lec13.pdf
Use the following information to answer the next 10 exercises. A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be modeled by the following distribution: X ~ Exp(0.2)
Use the following information to answer the next eight exercises. A distribution is given as X ~ Exp(0.75).
Use the following information to answer the next eight exercises. Carbon-14 is a radioactive element with a half-life of about 5,730 years. Carbon-14 is said to decay exponentially. The decay rate is 0.000121. We start with one gram of carbon-14. We are interested in the time (years) it takes to decay carbon-14.

### Homework

Use the following information to answer the next three exercises. The average lifetime of a certain new cell phone is three years. The manufacturer will replace any cell phone failing within two years of the date of purchase. The lifetime of these cell phones is known to follow an exponential distribution.


# The Normal Distribution

## Introduction

The normal, a continuous distribution, is the most important of all the distributions. It is widely used and even more widely abused. Its graph is bell-shaped. You see the bell curve in almost all disciplines, including psychology, business, economics, the sciences, nursing, and, of course, mathematics. Some of your instructors may use the normal distribution to help determine your grade. Most IQ scores are normally distributed. Often, real-estate prices fit a normal distribution. The normal distribution is extremely important, but it cannot be applied to everything in the real world.
In this chapter, you will study the normal distribution, the standard normal distribution, and applications associated with them.
The normal distribution has two parameters: the mean (μ) and the standard deviation (σ). If X is a quantity to be measured that has a normal distribution with mean (μ) and standard deviation (σ), we designate this by writing
The curve is symmetric about a vertical line drawn through the mean, μ. In theory, the mean is the same as the median, because the graph is symmetric about μ. With a normal distribution, the mean, median, and mode all lie at the same point. The normal distribution depends only on the mean and the standard deviation. The location of the mean simply indicates the location of the line of symmetry, in a normal distribution. Since the area under the curve must equal one, a change in the standard deviation, σ, causes a change in the shape of the curve; the curve becomes fatter or skinnier depending on σ. A change in μ causes the graph to shift to the left or right. The location of the mean simply indicates the location of the line of symmetry, in a normal distribution. This means there are an infinite number of normal probability distributions. One distribution of special interest is called the standard normal distribution.

### Formula Review

X ∼ N(μ, σ)
μ = the mean, σ = the standard deviation


# The Normal Distribution

## The Standard Normal Distribution

The standardized normal distribution is a type of normal distribution, with a mean of 0 and standard deviation of 1. It represents a distribution of standardized scores, called , as opposed to raw scores (the actual data values). A  indicates the number of standard deviation a score falls above or below the mean. Z-scores allow for comparison of scores, occurring in different data sets, with different means and standard deviations. It would not make sense to compare apples and oranges. Likewise, it does not make sense to compare scores from two different samples that have different means and standard deviations. Z-scores can be looked up in a Z-Table of Standard Normal Distribution, in order to find the area under the standard normal curve, between a score and the mean, between two scores, or above or below a score. The standard normal distribution allows us to interpret standardized scores and provides us with one table that we may use, in order to compute areas under the normal curve, for an infinite number of data sets, no matter what the mean or standard deviation.
A z-score is calculated as

  . The score itself can be found by using algebra and solving for x. Multiplying both sides of the equation by σ gives:

  . Adding μ to both sides of the equation gives

  .
Suppose we have a data set with a mean of 5 and standard deviation of 2. We want to determine the number of standard deviations the score of 11 falls above the mean. We can find this answer (or z-score) by writing
or
we can solve for z.
We have determined that the score of 11 falls 3 standard deviations above the mean of 5.
With a standard normal distribution, we indicate the distribution by writing Z ~ N(0, 1) which shows the normal distribution has a mean of 0 and standard deviation of 1. This notation simply indicates that a standard normal distribution is being used.

### Z-Scores

As described previously, if X is a normally distributed random variable and X ~ N(μ, σ), then the z-score is
The z-score tells you how many standard deviations the value x is above, to the right of, or below, to the left of, the mean, μ. Values of x that are larger than the mean have positive z-scores, and values of x that are smaller than the mean have negative z-scores. If x equals the mean, then x has a z-score of zero.
When determining the z-score for an x-value, for a normal distribution, with a given mean and standard deviation, the notation above for a normal distribution, will be given.

The Empirical Rule If X is a random variable and has a normal distribution with mean µ and standard deviation σ, then the Empirical Rule states the following:
1. About 68 percent of the x values lie between –1σ and +1σ of the mean µ (within one standard deviation of the mean).
2. About  95 percent of the x values lie between –2σ and +2σ of the mean µ (within two standard deviations of the mean).
3. About 99.7 percent of the x values lie between –3σ and +3σ of the mean µ (within three standard deviations of the mean). Notice that almost all the x values lie within three standard deviations of the mean.
4. The z-scores for +1σ and –1σ are +1 and –1, respectively.
5. The z-scores for +2σ  and –2σ are +2 and –2, respectively.
6. The z-scores for +3σ  and –3σ are +3 and –3, respectively.
So, in other words, this is that about 68 percent of the values lie between z-scores of –1 and 1, about 95% of the values lie between z-scores of –2 and 2, and about 99.7 percent of the values lie between z-scores of -3 and 3. These facts can be checked, by looking up the mean to z area in a z-table for each positive z-score and multiplying by 2.
The empirical rule is also known as the 68–95–99.7 rule.

### References

CollegeBoard. (2012). 2012 College-bound seniors: Total group profile report. Retrieved from http://media.collegeboard.com/digitalServices/pdf/research/TotalGroup-2012.pdf
Joyce, C. A., Janssen, S., & Liu, M. L. (2010). The world almanac and book of facts, 2010. New York, NY: World Almanac Books.
London School of Hygiene and Tropical Medicine. (2009). Calculation of z-scores. Retrieved from http://conflict.lshtm.ac.uk/page_125.htm
National Center for Education Statistics. (2009). ACT score averages and standard deviations, by sex and race/ethnicity, and percentage of ACT test takers, by selected composite score ranges and planned fields of study: Selected years, 1995 through 2009. Retrieved from http://nces.ed.gov/programs/digest/d09/tables/dt09_147.asp
NBA.com. (2013). NBA Media Ventures. Retrieved from http://www.nba.com
StatCrunch. (2010). Blood pressure of males and females. Retrieved from http://www.statcrunch.com/5.0/viewreport.php?reportid=11960
The Mercury News. (n.d.). Retrieved from http://www.mercurynews.com/
Wikipedia. (2013). List of stadiums by capacity - Wikipedia. Retrieved from https://en.wikipedia.org/wiki/List_of_stadiums_by_capacity

### Chapter Review

A z-score is a standardized value. Its distribution is the standard normal, Z ~ N(0, 1). The mean of the z-scores is zero and the standard deviation is one. If z is the z-score for a value x from the normal distribution N(µ, σ), then z tells you how many standard deviations x is abovegreater thanor belowless thanµ.

### Formula Review

Z ~ N(0, 1)
z = a standardized value (z-score)
mean = 0, standard deviation = 1
To find the kth percentile of X when the z-score is known,k = μ + (z)σ
z-score: z =

Z = the random variable for z-scores
Use the following information to answer the next two exercises: The life of Sunshine CD players is normally distributed with mean of 4.1 years and a standard deviation of 1.3 years. A CD player is guaranteed for three years. We are interested in the length of time a CD player lasts.

### Homework

Use the following information to answer the next two exercises: The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days.