Sihangli/scibench-rl huggingface dataset:
Problem:  If a ball is thrown into the air with a velocity of $40 \mathrm{ft} / \mathrm{s}$, its height in feet $t$ seconds later is given by $y=40 t-16 t^2$. Find the average velocity for the time period beginning when $t=2$ and lasting 0.5 second.
Answer:  -32 $\mathrm{ft} / \mathrm{s}$
Numeric value: -32
Problem: A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced $1.5 \mathrm{~cm}$ apart. The liver is $15 \mathrm{~cm}$ long and the cross-sectional areas, in square centimeters, are $0,18,58,79,94,106,117,128,63, 39, 0$. Use the Midpoint Rule to estimate the volume of the liver.

Answer:  1110  $\mathrm{cm}^3$
Numeric value: 1110
Problem: A manufacturer of corrugated metal roofing wants to produce panels that are $28 \mathrm{in}$. wide and $2 \mathrm{in}$. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has equation $y=\sin (\pi x / 7)$ and find the width $w$ of a flat metal sheet that is needed to make a 28-inch panel. (Use your calculator to evaluate the integral correct to four significant digits.)

Answer:  29.36  $\mathrm{in}$
Numeric value: 29.36
Problem: The dye dilution method is used to measure cardiac output with $6 \mathrm{mg}$ of dye. The dye concentrations, in $\mathrm{mg} / \mathrm{L}$, are modeled by $c(t)=20 t e^{-0.6 t}, 0 \leqslant t \leqslant 10$, where $t$ is measured in seconds. Find the cardiac output.
Answer:  6.6  $\mathrm{L}/\mathrm{min}$
Numeric value: 6.6
Problem: A planning engineer for a new alum plant must present some estimates to his company regarding the capacity of a silo designed to contain bauxite ore until it is processed into alum. The ore resembles pink talcum powder and is poured from a conveyor at the top of the silo. The silo is a cylinder $100 \mathrm{ft}$ high with a radius of $200 \mathrm{ft}$. The conveyor carries ore at a rate of $60,000 \pi \mathrm{~ft}^3 / \mathrm{h}$ and the ore maintains a conical shape whose radius is 1.5 times its height. If, at a certain time $t$, the pile is $60 \mathrm{ft}$ high, how long will it take for the pile to reach the top of the silo?
Answer:  9.8  $\mathrm{h}$
Numeric value: 9.8
Problem: A boatman wants to cross a canal that is $3 \mathrm{~km}$ wide and wants to land at a point $2 \mathrm{~km}$ upstream from his starting point. The current in the canal flows at $3.5 \mathrm{~km} / \mathrm{h}$ and the speed of his boat is $13 \mathrm{~km} / \mathrm{h}$. How long will the trip take?

Answer:  20.2  $\mathrm{min}$
Numeric value: 20.2
Problem: Find the area bounded by the curves $y=\cos x$ and $y=\cos ^2 x$ between $x=0$ and $x=\pi$.
Answer:  2  
Numeric value: 2
Problem: A sled is pulled along a level path through snow by a rope. A 30-lb force acting at an angle of $40^{\circ}$ above the horizontal moves the sled $80 \mathrm{ft}$. Find the work done by the force.
Answer:  $2400\cos({40}^{\circ})$  $\mathrm{ft-lb}$
Numeric value: 1838.50666349
Problem:  If $R$ is the total resistance of three resistors, connected in parallel, with resistances $R_1, R_2, R_3$, then
$$
\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}
$$
If the resistances are measured in ohms as $R_1=25 \Omega$, $R_2=40 \Omega$, and $R_3=50 \Omega$, with a possible error of $0.5 \%$ in each case, estimate the maximum error in the calculated value of $R$.
Answer:  $\frac{1}{17}$  $\Omega$
Numeric value: 0.05882352941
Problem: The length and width of a rectangle are measured as $30 \mathrm{~cm}$ and $24 \mathrm{~cm}$, respectively, with an error in measurement of at most $0.1 \mathrm{~cm}$ in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.

Answer:  5.4  $\mathrm{cm^2}$
Numeric value: 5.4
Problem: The planet Mercury travels in an elliptical orbit with eccentricity 0.206 . Its minimum distance from the sun is $4.6 \times 10^7 \mathrm{~km}$. Find its maximum distance from the sun.
Answer:  7  $\mathrm{10^7} \mathrm{~km}$
Numeric value: 7
Problem: Use differentials to estimate the amount of tin in a closed tin can with diameter $8 \mathrm{~cm}$ and height $12 \mathrm{~cm}$ if the tin is $0.04 \mathrm{~cm}$ thick.
Answer:  16  $\mathrm{cm^3}$
Numeric value: 16
Problem: Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it.  A cable that weighs $2 \mathrm{~lb} / \mathrm{ft}$ is used to lift $800 \mathrm{~lb}$ of coal up a mine shaft $500 \mathrm{~ft}$ deep. Find the work done.

Answer:  650000  $\mathrm{ft-lb}$
Numeric value: 650000
Problem:  A patient takes $150 \mathrm{mg}$ of a drug at the same time every day. Just before each tablet is taken, 5$\%$ of the drug remains in the body. What quantity of the drug is in the body after the third tablet? 
Answer:  157.875  $\mathrm{mg}$
Numeric value: 157.875
Problem: A $360-\mathrm{lb}$ gorilla climbs a tree to a height of $20 \mathrm{~ft}$. Find the work done if the gorilla reaches that height in 5 seconds.
Answer:  7200  $\mathrm{ft-lb}$
Numeric value: 7200
Problem: Find the area of triangle $A B C$, correct to five decimal places, if
$$
|A B|=10 \mathrm{~cm} \quad|B C|=3 \mathrm{~cm} \quad \angle A B C=107^{\circ}
$$
Answer:  14.34457  $\mathrm{cm^2}$
Numeric value: 14.34457
Problem: Use Stokes' Theorem to evaluate $\int_C \mathbf{F} \cdot d \mathbf{r}$, where $\mathbf{F}(x, y, z)=x y \mathbf{i}+y z \mathbf{j}+z x \mathbf{k}$, and $C$ is the triangle with vertices $(1,0,0),(0,1,0)$, and $(0,0,1)$, oriented counterclockwise as viewed from above.

Answer:  $-\frac{1}{2}$  
Numeric value: -0.5
Problem: A hawk flying at $15 \mathrm{~m} / \mathrm{s}$ at an altitude of $180 \mathrm{~m}$ accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation
$$
y=180-\frac{x^2}{45}
$$
until it hits the ground, where $y$ is its height above the ground and $x$ is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter.
Answer:  209.1  $\mathrm{m}$
Numeric value: 209.1
Problem: The intensity of light with wavelength $\lambda$ traveling through a diffraction grating with $N$ slits at an angle $\theta$ is given by $I(\theta)=N^2 \sin ^2 k / k^2$, where $k=(\pi N d \sin \theta) / \lambda$ and $d$ is the distance between adjacent slits. A helium-neon laser with wavelength $\lambda=632.8 \times 10^{-9} \mathrm{~m}$ is emitting a narrow band of light, given by $-10^{-6}<\theta<10^{-6}$, through a grating with 10,000 slits spaced $10^{-4} \mathrm{~m}$ apart. Use the Midpoint Rule with $n=10$ to estimate the total light intensity $\int_{-10^{-6}}^{10^{-6}} I(\theta) d \theta$ emerging from the grating.
Answer:  59.4  
Numeric value: 59.4
Problem: A model for the surface area of a human body is given by $S=0.1091 w^{0.425} h^{0.725}$, where $w$ is the weight (in pounds), $h$ is the height (in inches), and $S$ is measured in square feet. If the errors in measurement of $w$ and $h$ are at most $2 \%$, use differentials to estimate the maximum percentage error in the calculated surface area.
Answer:  2.3  $\%$
Numeric value: 2.3
Problem: The temperature at the point $(x, y, z)$ in a substance with conductivity $K=6.5$ is $u(x, y, z)=2 y^2+2 z^2$. Find the rate of heat flow inward across the cylindrical surface $y^2+z^2=6$, $0 \leqslant x \leqslant 4$
Answer: $1248\pi$  
Numeric value: 3920.70763168
Problem: If a ball is thrown into the air with a velocity of $40 \mathrm{ft} / \mathrm{s}$, its height (in feet) after $t$ seconds is given by $y=40 t-16 t^2$. Find the velocity when $t=2$.
Answer:  -24  $\mathrm{ft} / \mathrm{s}$
Numeric value: -24
Problem: A woman walks due west on the deck of a ship at $3 \mathrm{mi} / \mathrm{h}$. The ship is moving north at a speed of $22 \mathrm{mi} / \mathrm{h}$. Find the speed of the woman relative to the surface of the water.
Answer:  $\sqrt{493}$  $\mathrm{mi}/\mathrm{h}$
Numeric value: 22.2036033112
Problem: A $360-\mathrm{lb}$ gorilla climbs a tree to a height of $20 \mathrm{~ft}$. Find the work done if the gorilla reaches that height in 10 seconds. 
Answer:  7200 $\mathrm{ft-lb}$
Numeric value: 7200
Problem:  A ball is thrown eastward into the air from the origin (in the direction of the positive $x$-axis). The initial velocity is $50 \mathrm{i}+80 \mathrm{k}$, with speed measured in feet per second. The spin of the ball results in a southward acceleration of $4 \mathrm{ft} / \mathrm{s}^2$, so the acceleration vector is $\mathbf{a}=-4 \mathbf{j}-32 \mathbf{k}$. What speed does the ball land?

Answer:  $10\sqrt{93}$  $\mathrm{ft}/\mathrm{s}$
Numeric value: 96.4365076099
Problem: The demand function for a commodity is given by
$$
p=2000-0.1 x-0.01 x^2
$$
Find the consumer surplus when the sales level is 100 .
Answer:  7166.67  $\$$
Numeric value: 7166.67
Problem: The linear density in a rod $8 \mathrm{~m}$ long is $12 / \sqrt{x+1} \mathrm{~kg} / \mathrm{m}$, where $x$ is measured in meters from one end of the rod. Find the average density of the rod.
Answer:  6  $\mathrm{~kg} / \mathrm{m}$
Numeric value: 6
Problem: A variable force of $5 x^{-2}$ pounds moves an object along a straight line when it is $x$ feet from the origin. Calculate the work done in moving the object from $x=1 \mathrm{~ft}$ to $x=10 \mathrm{~ft}$.
Answer:  4.5  $\mathrm{ft-lb}$
Numeric value: 4.5
Problem: One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 5000 inhabitants, 160 people have a disease at the beginning of the week and 1200 have it at the end of the week. How long does it take for $80 \%$ of the population to become infected?
Answer:  15  $\mathrm{days}$
Numeric value: 15
Problem: Find the volume of the described solid S. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths $3 \mathrm{~cm}$, $4 \mathrm{~cm}$, and $5 \mathrm{~cm}$
Answer:  10  $\mathrm{cm}^3$
Numeric value: 10
Problem: The base of a solid is a circular disk with radius 3 . Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.
Answer:  36  
Numeric value: 36
Problem: A projectile is fired with an initial speed of $200 \mathrm{~m} / \mathrm{s}$ and angle of elevation $60^{\circ}$. Find the speed at impact.
Answer:  200  $\mathrm{m}/\mathrm{s}$
Numeric value: 200
Problem: A force of $30 \mathrm{~N}$ is required to maintain a spring stretched from its natural length of $12 \mathrm{~cm}$ to a length of $15 \mathrm{~cm}$. How much work is done in stretching the spring from $12 \mathrm{~cm}$ to $20 \mathrm{~cm}$ ?

Answer:  3.2  $\mathrm{J}$
Numeric value: 3.2
Problem: Use Poiseuille's Law to calculate the rate of flow in a small human artery where we can take $\eta=0.027, R=0.008 \mathrm{~cm}$, $I=2 \mathrm{~cm}$, and $P=4000$ dynes $/ \mathrm{cm}^2$.
Answer:  1.19  $\mathrm{10^{-4}} \mathrm{~cm}^3/\mathrm{s}$
Numeric value: 1.19
Problem: If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=100$.
Answer:  $0.25$  
Numeric value: 0.25
Problem: A device contains three components, each of which has a lifetime in hours with the pdf
$$
f(x)=\frac{2 x}{10^2} e^{-(x / 10)^2}, \quad 0 < x < \infty .
$$
The device fails with the failure of one of the components. Assuming independent lifetimes, what is the probability that the device fails in the first hour of its operation? HINT: $G(y)=P(Y \leq y)=1-P(Y>y)=1-P$ (all three $>y$ ).
Answer:  $0.03$  
Numeric value: 0.03
Problem: The tensile strength $X$ of paper, in pounds per square inch, has $\mu=30$ and $\sigma=3$. A random sample of size $n=100$ is taken from the distribution of tensile strengths. Compute the probability that the sample mean $\bar{X}$ is greater than 29.5 pounds per square inch.
Answer:  $0.9522$  
Numeric value: 0.9522
Problem: Let $\bar{X}$ be the mean of a random sample of size 36 from an exponential distribution with mean 3 . Approximate $P(2.5 \leq \bar{X} \leq 4)$
Answer:  $0.8185$  
Numeric value: 0.8185
Problem: Let $X_1, X_2$ be a random sample of size $n=2$ from a distribution with pdf $f(x)=3 x^2, 0 < x < 1$. Determine $P\left(\max X_i < 3 / 4\right)=P\left(X_1<3 / 4, X_2<3 / 4\right)$
Answer:  $\frac{729}{4096}$  
Numeric value: 0.178
Problem: Let $X$ equal the tarsus length for a male grackle. Assume that the distribution of $X$ is $N(\mu, 4.84)$. Find the sample size $n$ that is needed so that we are $95 \%$ confident that the maximum error of the estimate of $\mu$ is 0.4 .
Answer:  $117$  
Numeric value: 117
Problem: In a study concerning a new treatment of a certain disease, two groups of 25 participants in each were followed for five years. Those in one group took the old treatment and those in the other took the new treatment. The theoretical dropout rate for an individual was $50 \%$ in both groups over that 5 -year period. Let $X$ be the number that dropped out in the first group and $Y$ the number in the second group. Assuming independence where needed, give the sum that equals the probability that $Y \geq X+2$. HINT: What is the distribution of $Y-X+25$ ?
Answer:  $0.3359$  
Numeric value: 0.3359
Problem: Let $X$ and $Y$ have a bivariate normal distribution with correlation coefficient $\rho$. To test $H_0: \rho=0$ against $H_1: \rho \neq 0$, a random sample of $n$ pairs of observations is selected. Suppose that the sample correlation coefficient is $r=0.68$. Using a significance level of $\alpha=0.05$, find the smallest value of the sample size $n$ so that $H_0$ is rejected.
Answer:  $9$  
Numeric value: 9
Problem: In order to estimate the proportion, $p$, of a large class of college freshmen that had high school GPAs from 3.2 to 3.6 , inclusive, a sample of $n=50$ students was taken. It was found that $y=9$ students fell into this interval. Give a point estimate of $p$.
Answer: $0.1800$  
Numeric value: 0.1800
Problem: If $\bar{X}$ and $\bar{Y}$ are the respective means of two independent random samples of the same size $n$, find $n$ if we want $\bar{x}-\bar{y} \pm 4$ to be a $90 \%$ confidence interval for $\mu_X-\mu_Y$. Assume that the standard deviations are known to be $\sigma_X=15$ and $\sigma_Y=25$.
Answer:  $144$  
Numeric value: 144
Problem: For a public opinion poll for a close presidential election, let $p$ denote the proportion of voters who favor candidate $A$. How large a sample should be taken if we want the maximum error of the estimate of $p$ to be equal to 0.03 with $95 \%$ confidence?
Answer:  $1068$  
Numeric value: 1068
Problem: Let the distribution of $T$ be $t(17)$. Find $t_{0.01}(17)$.
Answer: $2.567$  
Numeric value: 2.567
Problem: Let $X_1, X_2, \ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute $P(77<\bar{X}<79.5)$.
Answer:  $0.4772$  
Numeric value: 0.4772
Problem: 5.4-19. A doorman at a hotel is trying to get three taxicabs for three different couples. The arrival of empty cabs has an exponential distribution with mean 2 minutes. Assuming independence, what is the probability that the doorman will get all three couples taken care of within 6 minutes?

Answer:  $0.5768$  
Numeric value: 0.5768
Problem: Consider the following two groups of women: Group 1 consists of women who spend less than $\$ 500$ annually on clothes; Group 2 comprises women who spend over $\$ 1000$ annually on clothes. Let $p_1$ and $p_2$ equal the proportions of women in these two groups, respectively, who believe that clothes are too expensive. If 1009 out of a random sample of 1230 women from group 1 and 207 out of a random sample 340 from group 2 believe that clothes are too expensive, Give a point estimate of $p_1-p_2$.
Answer:  $0.2115$  
Numeric value: 0.2115
Problem: Given below example: Approximate $P(39.75 \leq \bar{X} \leq 41.25)$, where $\bar{X}$ is the mean of a random sample of size 32 from a distribution with mean $\mu=40$ and variance $\sigma^2=8$. In the above example, compute $P(1.7 \leq Y \leq 3.2)$ with $n=4$
Answer:  $0.6749$  
Numeric value: 0.6749
Problem: If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=1000$.
Answer:  $0.925$  
Numeric value: 0.925
Problem: Let $Y_1 < Y_2 < Y_3 < Y_4 < Y_5 < Y_6$ be the order statistics of a random sample of size $n=6$ from a distribution of the continuous type having $(100 p)$ th percentile $\pi_p$. Compute $P\left(Y_2 < \pi_{0.5} < Y_5\right)$.
Answer:  $0.7812$  
Numeric value: 0.7812
Problem: Let $X_1, X_2$ be independent random variables representing lifetimes (in hours) of two key components of a
device that fails when and only when both components fail. Say each $X_i$ has an exponential distribution with mean 1000. Let $Y_1=\min \left(X_1, X_2\right)$ and $Y_2=\max \left(X_1, X_2\right)$, so that the space of $Y_1, Y_2$ is $ 0< y_1 < y_2 < \infty $ Find $G\left(y_1, y_2\right)=P\left(Y_1 \leq y_1, Y_2 \leq y_2\right)$.
Answer: 0.5117   
Numeric value: 0.5117
Problem: Let $Z_1, Z_2, \ldots, Z_7$ be a random sample from the standard normal distribution $N(0,1)$. Let $W=Z_1^2+Z_2^2+$ $\cdots+Z_7^2$. Find $P(1.69 < W < 14.07)$
Answer:  $0.925$  
Numeric value: 0.925
Problem: Let $X_1$ and $X_2$ be independent Poisson random variables with respective means $\lambda_1=2$ and $\lambda_2=3$. Find $P\left(X_1=3, X_2=5\right)$. HINT. Note that this event can occur if and only if $\left\{X_1=1, X_2=0\right\}$ or $\left\{X_1=0, X_2=1\right\}$.
Answer:  0.0182  
Numeric value: 0.0182
Problem: Let $Y$ be the number of defectives in a box of 50 articles taken from the output of a machine. Each article is defective with probability 0.01 . Find the probability that $Y=0,1,2$, or 3 By using the binomial distribution.
Answer:  $0.9984$  
Numeric value: 0.9984
Problem: Some dentists were interested in studying the fusion of embryonic rat palates by a standard transplantation technique. When no treatment is used, the probability of fusion equals approximately 0.89 . The dentists would like to estimate $p$, the probability of fusion, when vitamin A is lacking. How large a sample $n$ of rat embryos is needed for $y / n \pm 0.10$ to be a $95 \%$ confidence interval for $p$ ?
Answer:  $38$  
Numeric value: 38
Problem: To determine the effect of $100 \%$ nitrate on the growth of pea plants, several specimens were planted and then watered with $100 \%$ nitrate every day. At the end of
two weeks, the plants were measured. Here are data on seven of them:
$$
\begin{array}{lllllll}
17.5 & 14.5 & 15.2 & 14.0 & 17.3 & 18.0 & 13.8
\end{array}
$$
Assume that these data are a random sample from a normal distribution $N\left(\mu, \sigma^2\right)$. Find the value of a point estimate of $\mu$.
Answer:  $15.757$  
Numeric value: 15.757
Problem: Suppose that the distribution of the weight of a prepackaged '1-pound bag' of carrots is $N\left(1.18,0.07^2\right)$ and the distribution of the weight of a prepackaged '3-pound bag' of carrots is $N\left(3.22,0.09^2\right)$. Selecting bags at random, find the probability that the sum of three 1-pound bags exceeds the weight of one 3-pound bag. HInT: First determine the distribution of $Y$, the sum of the three, and then compute $P(Y>W)$, where $W$ is the weight of the 3-pound bag.
Answer: $0.9830$   
Numeric value: 0.9830
Problem: The distributions of incomes in two cities follow the two Pareto-type pdfs $$ f(x)=\frac{2}{x^3}, 1 < x < \infty , \text { and } g(y)= \frac{3}{y^4} ,  \quad 1 < y < \infty,$$ respectively. Here one unit represents $ 20,000$. One person with income is selected at random from each city. Let $X$ and $Y$ be their respective incomes. Compute $P(X < Y)$.
Answer:  $\frac{2}{5}$  
Numeric value: 0.4
Problem: Let $p$ equal the proportion of triathletes who suffered a training-related overuse injury during the past year. Out of 330 triathletes who responded to a survey, 167 indicated that they had suffered such an injury during the past year. Use these data to give a point estimate of $p$.
Answer:  $0.5061$  
Numeric value: 0.5061
Problem: One characteristic of a car's storage console that is checked by the manufacturer is the time in seconds that it takes for the lower storage compartment door to open completely. A random sample of size $n=5$ yielded the following times:
$\begin{array}{lllll}1.1 & 0.9 & 1.4 & 1.1 & 1.0\end{array}$ Find the sample mean, $\bar{x}$.
Answer: $1.1$   
Numeric value: 1.1
Problem: Let $X_1, X_2, \ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute $P(74.2<\bar{X}<78.4)$.
Answer:  $0.8561$  
Numeric value: 0.8561
Problem: Let $X_1$ and $X_2$ be independent random variables with probability density functions $f_1\left(x_1\right)=2 x_1, 0 < x_1 <1 $, and $f_2 \left(x_2\right) = 4x_2^3$ , $0 < x_2 < 1 $, respectively. Compute $P \left(0.5 < X_1 < 1\right.$ and $\left.0.4 < X_2 < 0.8\right)$.
Answer:  $\frac{36}{125}$
  
Numeric value: 1.44
Problem: If $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find A lower bound for $P(23 < X < 43)$.
Answer:  $0.84$  
Numeric value: 0.84
Problem: Let $Y_1 < Y_2 < \cdots < Y_8$ be the order statistics of eight independent observations from a continuous-type distribution with 70 th percentile $\pi_{0.7}=27.3$. Determine $P\left(Y_7<27.3\right)$.
Answer:  $0.2553$  
Numeric value: 0.2553
Problem: Let $X$ and $Y$ be independent with distributions $N(5,16)$ and $N(6,9)$, respectively. Evaluate $P(X>Y)=$ $P(X-Y>0)$.
Answer:  $0.4207$  
Numeric value: 0.4207
Problem: A quality engineer wanted to be $98 \%$ confident that the maximum error of the estimate of the mean strength, $\mu$, of the left hinge on a vanity cover molded by a machine is 0.25 . A preliminary sample of size $n=32$ parts yielded a sample mean of $\bar{x}=35.68$ and a standard deviation of $s=1.723$. How large a sample is required?
Answer:  $257$  
Numeric value: 257
Problem: Let the distribution of $W$ be $F(8,4)$. Find the following: $F_{0.01}(8,4)$.
Answer:  14.80  
Numeric value: 14.80
Problem: If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=500$.
Answer:  $0.85$  
Numeric value: 0.85
Problem: Let $\bar{X}$ be the mean of a random sample of size 12 from the uniform distribution on the interval $(0,1)$. Approximate $P(1 / 2 \leq \bar{X} \leq 2 / 3)$.
Answer: $0.4772$  
Numeric value: 0.4772
Problem: Determine the constant $c$ such that $f(x)= c x^3(1-x)^6$, $0 < x < 1$ is a pdf.
Answer:  840  
Numeric value: 840
Problem: Three drugs are being tested for use as the treatment of a certain disease. Let $p_1, p_2$, and $p_3$ represent the probabilities of success for the respective drugs. As three patients come in, each is given one of the drugs in a random order. After $n=10$ 'triples' and assuming independence, compute the probability that the maximum number of successes with one of the drugs exceeds eight if, in fact, $p_1=p_2=p_3=0.7$ 
Answer:  $0.0384$  
Numeric value: 0.0384
Problem: Evaluate
$$
\int_0^{0.4} \frac{\Gamma(7)}{\Gamma(4) \Gamma(3)} y^3(1-y)^2 d y
$$ Using integration.
Answer:  0.1792  
Numeric value: 0.1792
Problem: Let $X$ equal the maximal oxygen intake of a human on a treadmill, where the measurements are in milliliters of oxygen per minute per kilogram of weight. Assume that, for a particular population, the mean of $X$ is $\mu=$ 54.030 and the standard deviation is $\sigma=5.8$. Let $\bar{X}$ be the sample mean of a random sample of size $n=47$. Find $P(52.761 \leq \bar{X} \leq 54.453)$, approximately.
Answer:  $0.6247$  
Numeric value: 0.6247
Problem: Two components operate in parallel in a device, so the device fails when and only when both components fail. The lifetimes, $X_1$ and $X_2$, of the respective components are independent and identically distributed with an exponential distribution with $\theta=2$. The cost of operating the device is $Z=2 Y_1+Y_2$, where $Y_1=\min \left(X_1, X_2\right)$ and $Y_2=\max \left(X_1, X_2\right)$. Compute $E(Z)$.
Answer:  $5$  
Numeric value: 5