User: The machine scored parts of the SAT exam (critical reading and mathematics) are designed so that scores approximately follow the given model. What percent of students are expected to score between 400 and 600, which is within one standard deviation of the mean? 
Assistant: To determine the percentage of students expected to score between 400 and 600 on the SAT exam based on the given model, we can use the properties of the normal distribution, assuming the scores follow a normal distribution.

The SAT exam scores typically have a mean (μ) of 500 and a standard deviation (σ) of 100. 

1. **Identify the range**: The range given is from 400 to 600. 
   - The lower bound (400) is one standard deviation below the mean: \( \mu - \sigma = 500 - 100 = 400 \).
   - The upper bound (600) is one standard deviation above the mean: \( \mu + \sigma = 500 + 100 = 600 \).

2. **Use the empirical rule**: The empirical rule (or 68-95-99.7 rule) states that:
   - Approximately **68%** of data falls within one standard deviation of the mean in a normal distribution.

Since the range from 400 to 600 encompasses one standard deviation below and above the mean, we can conclude that approximately **68% of students** are expected to score between 400 and 600.

Therefore, the expected percentage of students scoring between 400 and 600 is **68%**.
User: A density curve can be obtained by smoothing a histogram of data because _____.  histograms are used for quantitative variables  histograms are used for categorical variables  all types of bar graphs can be smoothed into a density curve  density curves can be attributed to any type of graph
Assistant: A density curve can be obtained by smoothing a histogram of data because **histograms are used for quantitative variables**.

Histograms represent the distribution of quantitative data by dividing the data into bins (intervals) and displaying the frequency of data points in each bin. By smoothing the histogram, you can create a density curve that provides a continuous representation of the distribution of the data, allowing for better visualization and analysis of the underlying data patterns. In contrast, bar graphs (used for categorical variables) do not lend themselves to the same kind of smoothing to create a density representation.
User: Three Normal distributions all have mean 20. Distribution A has standard deviation 1, distribution B has standard deviation 5, and distribution C has standard deviation 10. The distribution with the flattest peak is __________.  Distribution A  Distribution C  Distribution B  They will all have the same shape.
Assistant: The distribution with the flattest peak is **Distribution C**.

In a normal distribution, the standard deviation affects the spread of the data. A smaller standard deviation results in a steeper and narrower peak, while a larger standard deviation results in a flatter peak. 

- **Distribution A** has a standard deviation of 1: narrow and steep peak.
- **Distribution B** has a standard deviation of 5: wider than A but not as flat as C.
- **Distribution C** has a standard deviation of 10: the flattest peak among the three distributions.

Therefore, Distribution C, with the largest standard deviation, has the flattest peak of the three.
User: Mensa is a society for "geniuses." One way to qualify for membership is having an IQ at least 2.5 standard deviations above average. One typical IQ scale has a mean of 100 and standard deviation 15. An IQ of 137.5 or _____ qualifies a person for membership in Mensa.  62.5  less  145  greater
Assistant: