M108 5.1 Intro. to Rand. Var. and Prob. Distri (Homework)

The college student senate is sponsoring a spring break Caribbean cruise raffle. The proceeds are to be donated to the Samaritan Center for the Homeless. A local travel agency donated the cruise, valued at $2000. The students sold 2578 raffle tickets at $5 per ticket.

(a) Kevin bought twenty-nine tickets. What is the probability that Kevin will win the spring break cruise to the Caribbean? (Round your answer to five decimal places.) 0.01125 Correct: Your answer is correct. What is the probability that Kevin will not win the cruise? (Round your answer to five decimal places.) 0.98875 Correct: Your answer is correct. (b) Expected earnings can be found by multiplying the value of the cruise by the probability that Kevin will win. What are Kevin's expected earnings? (Round your answer to two decimal places.) $ 22.50 Correct: Your answer is correct. Is this more or less than the amount Kevin paid for the twenty-nine tickets? Less Correct: Your answer is correct. How much did Kevin effectively contriute to the Samaritan Center for the Homeless? (Round your answer to two decimal places.) $ 122.50 Correct: Your answer is correct.

Consider a binomial distribution with n = 5 trials. Use the probabilities given in the Binomial Probability Distribution table in the Appendix to make histograms showing the probabilities of r = 0, 1, 2, 3, 4, and 5 successes for each of the following. Comment on the skewness of each distribution.

(a) The probability of success is p = 0.50. This distribution is symmetric. (b) The probability of success is p = 0.25. This distribution is skewed to the right. (c) The probability of success is p = 0.75. This distribution is skewed to the left. (d) What is the relationship between the distributions shown in parts (b) and (c)? The distributions are mirror images of each other. (e) If the probability of success is p = 0.73, do you expect the distribution to be skewed to the right or to the left? Why? Skewed to the left, since p > 0.50.

What is the age distribution of promotion-sensitive shoppers? A supermarket super shopper is defined as a shopper for whom at least 70% of the items purchased were on sale or purchased with a coupon. Age range, years 18-28 29-39 40-50 51-61 62 and over Midpoint x 23 34 45 56 67 Percent of super shoppers 6% 47% 21% 12% 14% For the 62-and-over group, use the midpoint 67 years.

(a) Using the age midpoints x and the percentage of super shoppers, do we have a valid probability distribution? Explain. Yes. The events are distinct and the probabilities sum to 1. (b) Use a histogram to graph the probability (c) Compute the expected age μ of a super shopper. (Round your answer to two decimal places.) μ = 42.91

What is the income distribution of super shoppers? A supermarket super shopper is defined as a shopper for whom at least 70% of the items purchased were on sale or purchased with a coupon. In the following table, income units are in thousands of dollars, and each interval goes up to but does not include the given high value. The midpoints are given to the nearest thousand dollars. Income range 5-15 15-25 25-35 35-45 45-55 55 or more Midpoint x 10 20 30 40 50 60 Percent of super shoppers 20% 15% 22% 17% 20% 6%

(a) Using the income midpoints x and the percent of super shoppers, do we have a valid probability distribution? Explain. Yes. The events are distinct and the probabilities sum to 1. (c) Compute the expected income μ of a super shopper. (Round your answer to two decimal places.) μ = 32.00 thousands of dollars (d) Compute the standard deviation σ for the income of super shoppers. (Round your answer to two decimal places.) σ = 15.56 thousands of dollars

Jim is a 60-year-old Anglo male in reasonably good health. He wants to take out a $50,000 term (that is, straight death benefit) life insurance policy until he is 65. The policy will expire on his 65th birthday. The probability of death in a given year is provided by the Vital Statistics Section of the Statistical Abstract of the United States (116th Edition). x = age 60 61 62 63 64 P(death at this age) 0.01171 0.01435 0.01636 0.02005 0.02308 Jim is applying to Big Rock Insurance Company for his term insurance policy.

(a) What is the probability that Jim will die in his 60th year? (Enter your answer to five decimal places.) 0.01171 Correct: Your answer is correct. Using this probability and the $50,000 death benefit, what is the expected cost to Big Rock Insurance? (Round your answer to two decimal places.) $ 585.50 Correct: Your answer is correct. (b) Repeat part (a) for years 61, 62, 63, and 64. (Round your answers to two decimal places.) Year Expected Cost 61 $ 717.50 Correct: Your answer is correct. 62 $ 818.00 Correct: Your answer is correct. 63 $ 1002.50 Correct: Your answer is correct. 64 $ 1154.00 Correct: Your answer is correct. What would be the total expected cost to Big Rock Insurance over the years 60 through 64? (Round your answer to two decimal places.) $ 4277.50 Correct: Your answer is correct. (c) If Big Rock Insurance wants to make a profit of $700 above the expected total cost paid out for Jim's death, how much should it charge for the policy? (Round your answer to two decimal places.) $ 4977.50 Correct: Your answer is correct. (d) If Big Rock Insurance Company charges $5000 for the policy, how much profit does the company expect to make? (Round your answer to two decimal places.) $ 722.50 Correct: Your answer is correct.

Sara is a 60-year-old Anglo female in reasonably good health. She wants to take out a $50,000 term (that is, straight death benefit) life insurance policy until she is 65. The policy will expire on her 65th birthday. The probability of death in a given year is provided by the Vital Statistics Section of the Statistical Abstract of the United States (116th Edition). x = age 60 61 62 63 64 P(death at this age) 0.00661 0.00881 0.00908 0.01032 0.01162

(a) What is the probability that Sara will die in her 60th year? (Use 5 decimal places.) 0.00661 Correct: Your answer is correct. Using this probability and the $50,000 death benefit, what is the expected cost to Big Rock Insurance? $ 330.50 Correct: Your answer is correct. (b) Repeat part (a) for ages 61, 62, 63, and 64. Age Expected Cost 61 $ 440.50 Correct: Your answer is correct. 62 $ 454.00 Correct: Your answer is correct. 63 $ 516.00 Correct: Your answer is correct. 64 $ 581.00 Correct: Your answer is correct. What would be the total expected cost to Big Rock Insurance over the years 60 through 64? $ 2322 Correct: Your answer is correct. (c) If Big Rock Insurance wants to make a profit of $700 above the expected total cost paid out for Sara's death, how much should it charge for the policy? $ 3022 Correct: Your answer is correct. (d) If Big Rock Insurance Company charges $5000 for the policy, how much profit does the company expect to make? $ 2678 Correct: Your answer is correct.

Richard has just been given a 8-question multiple-choice quiz in his history class. Each question has four answers, of which only one is correct. Since Richard has not attended class recently, he doesn't know any of the answers. Assuming that Richard guesses on all eight questions, find the indicated probabilities. (Round your answers to three decimal places.)

(a) What is the probability that he will answer all questions correctly? 0.000015 Correct: Your answer is correct. (b) What is the probability that he will answer all questions incorrectly? 0.1001 Correct: Your answer is correct. (c) What is the probability that he will answer at least one of the questions correctly? Compute this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table. 0.8999 Correct: Your answer is correct. They should be equal, but may differ slightly due to rounding error (d) What is the probability that Richard will answer at least half the questions correctly? 0.1138 Correct: Your answer is correct.

Which of the following are continuous variables, and which are discrete?

(a) number of traffic fatalities per year in the state of Florida discrete (b) distance a golf ball travels after being hit with a driver continuous (c) time required to drive from home to college on any given day continuous (d) number of ships in Pearl Harbor on any given day discrete (e) your weight before breakfast each morning continuous

The figures below show histograms of several binomial distributions with n = 6 trials. Match the given probability of success with the best graph.

(a) p = 0.30 goes with graph II (b) p = 0.50 goes with graph I . (c) p = 0.65 goes with graph III . (d) p = 0.90 goes with graph IV . (e) In general, when the probability of success p is close to 0.5, would you say that the graph is more symmetrical or more skewed? symmetrical In general, when the probability of success p is close to 1, would you say that the graph is skewed to the right or to the left? What about when p is close to 0? When p is close to 1 it will be skewed left. When p is close to 0 it will be skewed right.

Which of the following are continuous variables, and which are discrete?

(a) speed of an airplane continuous (b) age of a college professor chosen at random continuous (c) number of books in the college bookstore discrete (d) weight of a football player chosen at random continuous (e) number of lightning strikes in Rocky Mountain National Park on a given day discrete

Consider each distribution. Determine if it is a valid probability distribution or not, and explain your answer.

(a) x 0 1 2 P(x) 0.23 0.64 0.13 Yes. The probabilities sum to 1. (b) x 0 1 2 P(x) 0.23 0.64 0.18 No. The probabilities do not sum to 1.

In a binomial experiment, is it possible for the probability of success to change from one trial to the next? Explain.

No. A binomial experiment requires that the probability of success be the same for each trial.

Consider the probability distribution of a random variable x. Is the expected value of the distribution necessarily one of the possible values of x?

No. The expected value can be a value different from the exact value of x.

Consider two binomial distributions, with n trials each. The first distribution has a higher probability of success on each trial than the second. How does the expected value of the first distribution compare to that of the second?

The expected value is higher for the first distribution.

What does it mean to say that the trials of an experiment are independent?

The outcome of one trial does not affect the probability of success on any other trial

What does the random variable for a binomial experiment of n trials measure?

The random variable measures the number of successes out of n trials.

For a binomial experiment, how many outcomes are possible for each trial? What are the possible outcomes?

two; success or failure

What does the expected value of a binomial distribution with n trials tell you?

the average number of successes