Q: Scores on a police academy fitness test are normally distributed with mean $$64$$ and standard deviation $$9. $$Only the top 20% of applicants are selected. What is the minimum cutoff score required to be selected?

$$z = 0.84 \implies x = 64 + 0.84 \times 9 = 71.56$$

Q: In a study of 500 athletes, 20 say they train every day. What is the 95% confidence interval for the population proportion of athletes who train every day? (Use $$\hat{p} = 0.04$$ and $$z^* = 1.96.)$$

$$0.04 \pm 1.96 \times \sqrt{\frac{0.04 \times 0.96}{500}} \implies (0.016, 0.064)$$

Q: A random sample of 16 bookcases has a mean height of $$67.5$$ inches and a standard deviation of $$1.7$$ inches. Find the 99% confidence interval for the population standard deviation. (Use $$\$$chi^2$$ distribution with df=15$.)

$$\left( \sqrt{\frac{15 \times 1.7^2$}{34.528}}, \sqrt{\frac{15 \times 1.7^2$}{5.229}} \right)$$

Q: A raffle sells 1,000 tickets at $$3 each. Prizes: 1st place 1$$,000 (1 winner), 2nd place $$300 (2 winners), 3rd place 10 (20 $$winners). What is the expected net earning for a single ticket buyer?

$$E(X) = (1/1000) \times 997 + (2/1000) \times 297 + (20/1000) \times 7 + (977/1000) \times (-3) = -\0.66$

Question 1

A pizza shop owner records the number of pizzas delivered each day, with the following probability distribution: $$X = 6, 8, 10, 12, 14$$ and $$P(X) = 0.15, k, 0.35, 0.1, 0.1. $$What is the value of $$k$$?

Accepted Answer

$$k = 0.3$$

Question 2

Teacher salaries in California are normally distributed with mean $$\37$$,764$$ and standard deviation $$\$$5,100. $$What is the probability that a randomly selected teacher earns more than $$\45$$,000$? (Round your answer to four decimal places.)

Accepted Answer

$$P(X \u003e 45,000) \approx 0.0778$$

Question 3

Scores on a police academy fitness test are normally distributed with mean $$64$$ and standard deviation $$9. $$Only the top 20% of applicants are selected. What is the minimum cutoff score required to be selected?

Accepted Answer

$$z = 0.84 \implies x = 64 + 0.84 	imes 9 = 71.56$$

Question 4

A research economist finds that 40% of businesses fail in their first year. In a random sample of 15 businesses, what is the probability that exactly 10 fail? (Use the binomial formula.)

Accepted Answer

$$P(X=10) = {15 \choose 10} (0.4$$)^{10}$$ (0.6)^5$ \approx 0.0146$$

Question 5

A roadside stand has 529 tomatoes available. The daily demand for tomatoes is normally distributed with mean $$571$$ and standard deviation $$30. $$What is the probability that all tomatoes will be sold in a day?

Accepted Answer

$$P(X \geq 529) = P(Z \geq \frac{529-571}{30}) = P(Z \geq -1.4) = 0.9192$$

Question 6

The mean annual precipitation for a city is $$30.85$$ inches with a standard deviation of $$3.6$$ inches. What is the probability that a randomly selected month will have at least $$30$$ inches of precipitation?

Accepted Answer

$$P(X \geq 30) = P(Z \geq \frac{30-30.85}{3.6}) = P(Z \geq -0.236) = 0.5939$$

Question 7

Which of the following is NOT a property of the normal distribution curve?

Accepted Answer

The curve is skewed to the right

Question 8

According to the Empirical Rule, approximately what percent of the data lies within two standard deviations of the mean in a normal distribution?

Accepted Answer

Approximately $$95\%$$

Question 9

In a study of 500 athletes, 20 say they train every day. What is the 95% confidence interval for the population proportion of athletes who train every day? (Use $$\hat{p} = 0.04$$ and $$z^* = 1.96.)$$

Accepted Answer

$$0.04 \pm 1.96 	imes \sqrt{\frac{0.04 	imes 0.96}{500}} \implies (0.016, 0.064)$$

Question 10

A random sample of 16 bookcases has a mean height of $$67.5$$ inches and a standard deviation of $$1.7$$ inches. Find the 99% confidence interval for the population standard deviation. (Use $$\$$chi^2$$ distribution with df=15$.)

Accepted Answer

$$\left( \sqrt{\frac{15 	imes 1.7^2$}{34.528}}, \sqrt{\frac{15 	imes 1.7^2$}{5.229}} ight)$$

Question 11

A basketball player makes 80% of his free throws. If he shoots 10 free throws, what is the probability that he makes exactly 3 of them?

Accepted Answer

$$P(X=3) = {10 \choose 3} (0.8)^3$ (0.2)^7$ \approx 0.0008$$

Question 12

A raffle sells 1,000 tickets at $$3 each. Prizes: 1st place 1$$,000 (1 winner), 2nd place $$300 (2 winners), 3rd place 10 (20 $$winners). What is the expected net earning for a single ticket buyer?

Accepted Answer

$$E(X) = (1/1000) 	imes 997 + (2/1000) 	imes 297 + (20/1000) 	imes 7 + (977/1000) 	imes (-3) = -\0.66$

Question 13

A soft-drink machine fills cups with a normal distribution, standard deviation $$0.5 oz. $$What mean $$Y$$ should be set so that 8-oz cups overflow only 3% of the time?

Accepted Answer

$$Y = 8 - z_{0.03} 	imes 0.5 = 8 - (-1.88) 	imes 0.5 = 8.94$$

Question 14

Jack tosses 3 fair coins simultaneously, 10 times. What is the probability that exactly 3 of the 10 tosses result in all 3 coins showing heads?

Accepted Answer

$$P = {10 \choose 3} (1/8)^3$ (7/8)^7$ \approx 0.201$$

Question 15

A meteorologist samples 13 thunderstorms, mean speed $$15$$ mph, standard deviation $$1.7$$ mph. What is the 99% confidence interval for the population mean speed? (Use $$t^* = 3.012$$ for $$df=12.)$$

Accepted Answer

$$15 \pm 3.012 	imes \frac{1.7}{\sqrt{13}} \implies (13.57, 16.43)$$

Question 16

Which of the following changes will DECREASE the width of a confidence interval for a population mean, all else held constant?

Accepted Answer

Increasing the sample size

Question 17

A factory's filling machine dispenses soda with mean $$11.97 oz $$and standard deviation $$0.15 oz. $$What fill amount marks the highest 4% of fill amounts?

Accepted Answer

$$x = 11.97 + 1.75 	imes 0.15 = 12.23 oz$$

Question 18

A basketball player makes 80% of his free throws. If he shoots 10 free throws, what is the probability that he misses less than 2 shots?

Accepted Answer

$$P(X \geq 9) = P(X=9) + P(X=10) = {10 \choose 9} (0.8)^9$ (0.2)^1$ + (0.8$$)^{10}$$ \approx 0.3758$$

Question 19

A factory's filling machine dispenses soda with mean $$11.97 oz $$and standard deviation $$0.15 oz. If 5$$,000 bottles contain more than $$11.85 oz$$, what is the total number of bottles $$K$$ produced?

Accepted Answer

$$P(X \u003e 11.85) = P(Z \u003e \frac{11.85-11.97}{0.15}) = P(Z \u003e -0.8) = 0.7881;\ K = 5000 / 0.7881 \approx 6344$$

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