$\$ 200,000$if the home is destroyed by fire. Let$Y=$the profit made by the company on a single policy. From previous data, the probability that a home in this area will be destroyed by fire is 0.0002 . (a) Make a table that shows the probability distribution of Y. (b) Compute the expected value of Y. Explain what this result means for the insurance company. Xiaomeng Z. Numerade Educator ### Problem 11 Refer to Exercise 3. Calculate the mean of the random variable X and interpret this result in context. Xiaomeng Z. Numerade Educator ### Problem 12 Refer to Exercise 4. Calculate the mean of the random variable X and interpret this result in context. Xiaomeng Z. Numerade Educator ### Problem 13 A not-so-clever employee decided to fake his monthly expense report. He believed that the first digits of his expense amounts should be equally likely to be any of the numbers from 1 to 9. In that case, the first digit Y of a randomly selected expense amount would have the probability distribution shown in the histogram. (a) Explain why the mean of the random variable$Y$is located at the solid red line in the figure. (b) The first digits of randomly selected expense amounts actually follow Benford's law (Exercise 5 ) What's the expected value of the first digit? Explain how this information could be used to detect a fake expense report. (c) What's$P(Y>6) ?$According to Benford's law, what proportion of first digits in the employee's expense amounts should be greater than 6$?$How could this information be used to detect a fake expense report? Xiaomeng Z. Numerade Educator ### Problem 14 A life insurance company sells a term insurance policy to a 21-year-old male that pays$\$100,000$ if the insured dies within the next 5 years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of $\$ 250$each year as payment for the insurance. The amount Y that the company earns on this policy is$\$250$ per year, less the $\$ 100,000$that it must pay if the insured dies. Here is a partially completed table that shows information about risk of mortality and the values of$Y=$profit earned by the company: Age at death: 21 22 23 24 25 26 or more Profit: ?$99,750 ?$99,500 Probability: 0.00183 0.00186 0.00189 0.00191 0.00193 (a) Copy the table onto your paper. Fill in the missing values of$Y .$(b) Find the missing probability. Show your work. (c) Calculate the mean$\mu_{Y}$. Interpret this value in context. Xiaomeng Z. Numerade Educator ### Problem 15 Refer to Exercise 3. Calculate and interpret the standard deviation of the random variable X. Show your work. Sheryl E. Numerade Educator ### Problem 16 Refer to Exercise 4. Calculate and interpret the standard deviation of the random variable X. Show your work. Xiaomeng Z. Numerade Educator ### Problem 17 Refer to Exercise 13. It might also be possible to detect an employee’s fake expense records by looking at the variability in the first digits of those expense amounts. (a) Calculate the standard deviation$\sigma_{\mathrm{Y}}$. This gives us an idea of how much variation we'd expect in the employee's expense records if he assumed that first digits from 1 to 9 were equally likely. (b) Now calculate the standard deviation of first digits that follow Benford's law (Exercise 5). Would using standard deviations be a good way to detect fraud? Explain. Xiaomeng Z. Numerade Educator ### Problem 18 (a) It would be quite risky for you to insure the life of a 21 -year-old friend under the terms of Exercise 14. There is a high probability that your friend would live and you would gain$\$1250$ in premiums. But if he were to die, youl would lose almost $\$ 100,000$. Explain carefully why selling insurance is not risky for an insurance company that insures many thousands of 21 -year-old men. (b) The risk of an investment is often measured by the standard deviation of the return on the investment. The more variable the return is, the riskier the investment. We can measure the great risk of insuring a single person's life in Exercise 14 by computing the standard deviation of the income$Y$that the insurer will receive. Find$\sigma_{Y}$using the distribution and mean found in Exercise 14 . Xiaomeng Z. Numerade Educator ### Problem 19 How do rented housing units differ from units occupied by their owners? Here are the distributions of the number of rooms for owner occupied units and renter-occupied units in San Jose, California: Number of Rooms 1 2 3 4 5 6 7 8 9 10 Owned 0.003 0.002 0.023 0.104 0.210 0.224 0.197 0.149 0.053 0.035 Rented 0.008 0.027 0.287 0.363 0.164 0.093 0.039 0.013 0.003 0.003 Let$X=$the number of rooms in a randomly selected owner-occupied unit and$Y=$the number of rooms in a randomly chosen renter-occupied unit. (a) Make histograms suitable for comparing the probability distributions of$X$and$Y .$Describe any differences that you observe. (b) Find the mean number of rooms for both types of housing unit. Explain why this difference makes sense. (c) Find the standard deviations of both$X$and$Y$Explain why this difference makes sense. Xiaomeng Z. Numerade Educator ### Problem 20 In government data, a houschold consists of all occupants of a dwelling unit, while a family consists of two or more persons who live together and are related by blood or marriage. So all families form households, but some households are not families. Here are the distributions of household size and family size in the United States: Number of Persons 1 2345 6 7 Household probability 0.25 0.32 0.17 0.15 0.07 0.03 0.01 Family probability 0 0.42 0.23 0.21 0.09 0.03 0.02 Let$X=$the number of people in a randomly selected U.S. household and$Y=$the number of people in a randomly chosen U.S. family. (a) Make histograms suitable for comparing the probability distributions of$X$and$Y .$Describe any differences that you observe. (b) Find the mean for each random variable. Explain why this difference makes sense. (c) Find the standard deviations of both$X$and$Y .$Explain why this difference makes sense. Xiaomeng Z. Numerade Educator ### Problem 21 Random numbers Let$X$be a number between 0 and 1 produced by a random number generator. Assuming that the random variable$X$has a uniform distribution, find the following probabilities: (a)$P(X > 0.49)$(b)$P(X \geq 0.49)$(c)$P(0.19 \leq X < 0.37 \text { or } 0.84 < X \leq 1.27)$Xiaomeng Z. Numerade Educator ### Problem 22 Random numbers Let Y be a number between 0 and 1 produced by a random number generator. Assuming that the random variable Y has a uniform distribution, find the following probabilities: (a)$P(Y \leq 0.4)$(b)$P(Y < 0.4)$(c)$P(0.1 < Y \leq 0.15 \text { or } 0.77 \leq Y < 0.88)$Xiaomeng Z. Numerade Educator ### Problem 23 ITBS scores The Normal distribution with mean$\mu=6.8$and standard deviation$\sigma=1.6$is a good description of the lowa Test of Basic Skills (ITBS) vocabulary scores of seventh-grade students in Gary, Indiana. Call the score of a randomly chosen student$X$for short. Find$P(X \geq 9)$and interpret the result. Follow the four-step process. Xiaomeng Z. Numerade Educator ### Problem 24 Running a mile A study of$12,000$able-bodied male students at the University of Illinois found that their times for the mile run were approximately Normal with mean 7.11 minutes and standard deviation 0.74 minute.$^{8}$Choose a student at random from this group and call his time for the mile Y. Find$P(Y<6)$and interpret the result. Follow the four-step process. Xiaomeng Z. Numerade Educator ### Problem 25 A sample survey contacted an SRS of 663 registered voters in Oregon shortly after an clection and asked respondents whether they had voted. Voter records show that 56$\%$of registered voters had actually voted. We will see later that in repeated random samples of size 663 , the proportion in the sample who voted (call this proportion With vary according to the Normal distribution with mean$\mu=0.56$and standard deviation$\sigma=0.019$(a) If the respondents answer truthfully, what is$\mathrm{P}(0.52 \leq V \leq 0.60)$? This is the probability that the sample proportion$V$estimates the population proportion 0.56 within$\pm 0.04$(b) In fact, 72$\%$of the respondents said they had voted$(V=0.72) .$If respondents answer truthfully, what is$P(V \geq 0.72) ?$This probability is so small that it is good evidence that some people who did not vote claimed that they did vote. Xiaomeng Z. Numerade Educator ### Problem 26 How many close friends do you have? An opinion poll asks this question of an SRS of 1100 adults. Suppose that the number of close friends adults claim to have varies from person to person with mean$\mu=9$and standard deviation$\sigma=2.5$We will see later that in repeated random samples of size 1100 , the mean response$\overline{x}$will vary according to the Normal distribution with mean 9 and standard deviation 0.075 . What is$P(8.9 \leq \overline{x} \leq 9.1)$, the probability that the sample result$\overline{x}$estimates the population truth$\mu=9$to within$\pm 0.1 ?$Willis J. Numerade Educator ### Problem 27 Exercises 27 and 28 refer to the following setting. Choose an American household at random and let the random variable X be the number of cars (including SUVs and light trucks) they own. Here is the probability model if we ignore the few households that own more than 5 cars: Number of cars X: 012345 Probability: 0.09 0.36 0.35 0.13 0.05 0.02 A housing company builds houses with two-car garages. What percent of households have more cars than the garage can hold? (a) 13$\%$(b) 20$\%$(c) 45$\%$(d) 55$\%$(e) 80$\%$Xiaomeng Z. Numerade Educator ### Problem 28 Exercises 27 and 28 refer to the following setting. Choose an American household at random and let the random variable X be the number of cars (including SUVs and light trucks) they own. Here is the probability model if we ignore the few households that own more than 5 cars: Number of cars X: 012345 Probability: 0.09 0.36 0.35 0.13 0.05 0.02 What’s the expected number of cars in a randomly selected American household? (a) Between 0 and 5 (b) 1.00 (c) 1.75 (d) 1.84 (e) 2.00 Xiaomeng Z. Numerade Educator ### Problem 29 A deck of cards contains 52 cards, of which 4 are aces. You are offered the following wager: Draw one card at random from the deck. You win$\$10$ if the card drawn is an ace. Otherwise, you lose $\$ 1 .$If you make this wager very many times, what will be the mean amount you win? (a) About$-\$1$ , because you will lose most of the time.
(b) About $\$ 9,$because you win$\$10$ but lose only $\$ 1$. (c) About$-\$0.15 ;$ that is, on average you lose about 15 cents.
(d) About $\$ 0.77 ;$that is, on average you win about 77 cents. (e) About$\$0,$ because the random draw gives you a fair bet.

Xiaomeng Z.

### Problem 30

The deck of 52 cards contains 13 hearts. Here is another wager: Draw one card at random from the
deck. If the card drawn is a heart, you win $\$ 2$. Otherwise, you lose$\$1$ . Compare this wager (call it Wager 2 ) with that of the previous exercise (call it Wager 1 ). Which one should you prefer?
(a) Wager 1 , because it has a higher expected value.
(b) Wager $2,$ because it has a higher expected value.
(c) Wager 1 , because it has a higher probability of winning.
(d) Wager 2 , because it has a higher probability of winning.
(e) Both wagers are equally favorable.

Xiaomeng Z.

### Problem 31

Exercises 31 to 34 refer to the following setting. Many chess masters and chess advocates believe that chess play develops general intelligence, analytical skill, and the ability to concentrate. According to such beliefs, improved reading skills should result from study to improve chess-playing
skills. To investigate this belief, researchers conducted a study. All of the subjects in the study participated in a comprehensive chess program, and their reading performances were measured before and after the program. The graphs and numerical summaries below provide information on the subjects’ pretest scores, posttest scores, and the difference (post – pre) between these two scores.

Xiaomeng Z.

### Problem 32

Exercises 31 to 34 refer to the following setting. Many chess masters and chess advocates believe that chess play develops general intelligence, analytical skill, and the ability to concentrate. According to such beliefs, improved reading skills should result from study to improve chess-playing
skills. To investigate this belief, researchers conducted a study. All of the subjects in the study participated in a comprehensive chess program, and their reading performances were measured before and after the program. The graphs and numerical summaries below provide information on the subjects’ pretest scores, posttest scores, and the difference (post – pre) between these two scores.

Chess and reading (4.3) If the study found a statistically significant improvement in reading scores,

Bryan M.

### Problem 33

Exercises 31 to 34 refer to the following setting. Many chess masters and chess advocates believe that chess play develops general intelligence, analytical skill, and the ability to concentrate. According to such beliefs, improved reading skills should result from study to improve chess-playing
skills. To investigate this belief, researchers conducted a study. All of the subjects in the study participated in a comprehensive chess program, and their reading performances were measured before and after the program. The graphs and numerical summaries below provide information on the subjects’ pretest scores, posttest scores, and the difference (post – pre) between these two scores.

Predicting posttest scores (3.2) What is the equation of the linear regression model relating posttest
and pretest scores? Define any variables used.

Xiaomeng Z.
How well does it fit? $(3.2)$ Discuss what $s, r^{2},$ and the residual plot tell you about this linear regression model.