ES9 Additional Exercises

Chapter 5

1. "How women define holiday shopping," a USA Snapshot® (12-9-94) reported that 50% said "a pleasure," 22% said "a chore," 19% said "no big deal," and 8% said "a nightmare." The percentages do not sum to 100% due to round-off error.

a. What is the variable involved, and what are the possible values?

2. 
   Why is this variable not a random variable?
   

2. "Kids who smoke," a USA Snapshot (4-25-94), reports the percentage of children in each age group who smoke.

12 1.7

14 8.9

16 25.2

Is this a probability distribution? Explain why or why not.

3. A USA Snapshot® (11-1-94) titled "How many telephones we have" reported that 1% have none, 11% have one, 31% have two, and 57% have three or more. Let x equal the number of phones per home, and replace the category "three or more" with exactly "three."

a. Find the mean and standard deviation for the random variable x.

2. 
   Explain the effect of replacing the category "three or more" with "three" had on the distribution of x, the mean, and the standard deviation.
   

4. a. Use the probability distribution shown below and describe in your own words how the mean of the variable x is found.

x 1 2 3 4

P(x) 0.1 0.2 0.3 0.4

b. Find the mean of x.

c. Find the deviation from the mean for each x-value.

d. Find the value of each "squared deviation from the mean."

e. Recalling your answer to (a), find the mean of the variable "squared deviation."

f. "Variance" was the name given to the "mean of the squared deviations." Explain how formula (5.2) expresses the variance as a mean.

5. In California, 30% of the people have a certain blood type. What is the probability that exactly 5 out of a randomly selected group of 14 Californians will have that blood type? (Find the answer by using a table.)

6. On the average, 1 out of every 10 boards purchased by a cabinet manufacturer is unusable for building cabinets. What is the probability that 8, 9, or 10 of a set of 11 such boards are usable? (Find the answer by using a table.)

7. A local polling organization maintains that 90% of the eligible voters have never heard of John Anderson, who was a presidential candidate in 1980. If this is so, what is the probability that in a randomly selected sample of 12 eligible voters, 2 or fewer have heard of John Anderson?

8. A basketball player has a history of making 80% of the foul shots taken during games. What is the probability that he will miss three of the next five foul shots he takes?

9. According to an article in the February 1991 issue of Reader's Digest, Americans face a 1 in 20 chance of acquiring an infection while hospitalized. If the records of 15 randomly selected hospitalized patients are examined, find the probability that

a. none of the 15 acquired an infection while hospitalized.

2. 
   1 or more of the 15 acquired an infection while hospitalized.
   

10. An article in the Omaha World-Herald (12-1-94) stated that only about 60% of the individuals needing a bone marrow transplant find a suitable donor when they turn to registries of unrelated donors. In a group of 10 individuals needing a bone marrow transplant,

a. what is the probability that all 10 will find a suitable donor among the registries of unrelated donors?

b. what is the probability that exactly 8 will find a suitable donor among the registries of unrelated donors?

c. what is the probability that at least 8 will find a suitable donor among the registries of unrelated donors?

d. what is the probability that no more than 5 will find a suitable donor among the registries of unrelated donors?

11. Colorado Rockies baseball player Larry Walker’s league-leading batting average reached .344 after 415 times at bat during the 1998 season (ratio of hits to at bats). Suppose Walker has five official times at bat during his next game. Assuming no extenuating circumstances and that the binomial model will produce reasonable approximations, what is the probability that Walker:

a. gets less than two hits?

b. gets more than three hits?

c. goes five-for-five (all hits)?

12. According to Financial Executive (July/August 1993) disability causes 48% of all mortgage foreclosures. Given that 20 mortgage foreclosures are audited by a large lending institution,

a. find the probability that 5 or fewer of the foreclosures are due to a disability.

b. find the probability that at least 3 are due to a disability.

13. Seventy-five percent of the foreign-made autos sold in the United States in 1984 are now falling apart.

a. Determine the probability distribution of x, the number of these autos that are falling apart in a random sample of five cars.

b. Draw a histogram of the distribution.

2. 
   Calculate the mean and standard deviation of this distribution.
   

14. A 1998 survey conducted by Fortune revealed that the Marriott International workforce was composed of 50.3% minorities. A further subdivision revealed 6.0% Asian, 24.2% black, and 19.6% Hispanic. Source: Fortune, “The Diversity Elite”, August 3, 1998, p. 114.

Find the mean and standard deviation of all samples of 25 randomly selected employees of the Marriott International workforce for each of the three minority groups. Present your statistics in a table.

15. A USA Snapshot® titled "Stress does not love company" (11-3-94) answered the question "How people say they prefer to spend stressful times." Forty-eight percent responded "alone," 29% responded "with family," 18% responded "with friends," and 5% responded "other/don't know." Ten individuals are randomly selected and asked the question "How do you prefer to spend stressful times?"

a. What is the probability that two or fewer will respond by saying "alone"?

2. 
   Explain why this question can be answered using binomial probabilities.
   

16. For years, the manager of a certain company had sole responsibility for making decisions with regards to company policy. This manager has a history of making the correct decision with a probability of p. Recently company policy has changed, and now all decisions are to be made by majority rule of a three-person committee.

a. Each member makes a decision independently, and each has a probability of p of making the correct decision. What is the probability that the committee's majority decision will be correct?

b. If p = 0.1, what is the probability that the committee makes the correct decision?

c. If p = 0.8, what is the probability that the committee makes the correct decision?

d. For what values of p is the committee more likely to make the correct decision by majority rule than the former manager?

e. For what values (there are three) of p is the probability of a correct decision the same for the manager and for the committee? Justify your answer.

17. Suppose one member of the committee in Exercise 5.16 always makes the decision by rolling a die. If the die roll results in an even number, they vote for the proposal, and if an odd number occurs, they vote against it. The other two members still decide independently and have a probability of p of making the correct decision.

a. What is the probability that the committee's majority decision will be correct?

b. If p = 0.1, what is the probability that the committee makes the correct decision?

c. If p = 0.8, what is the probability that the committee makes the correct decision?

d. For what value of p is the committee more likely to make the correct decision by majority rule than the former manager?

e. For what values of p is the probability of a correct decision the same for the manager and for the committee? Justify your answer.

f. Why is the answer to (e) different than the answer to Exercise 5.16 (e)?