## MATH 533 Week 2 Homework Problems (Summer 2019)

**1.** **Question:** Many primary care doctors feel overworked and burdened by potential lawsuits. In fact, a group of researchers reported that 6464% of all general practice physicians do not recommend medicine as a career. Let x represent the number of sampled general practice physicians who do not recommend medicine as a career. Complete parts a through d.

- a. Explain why x is approximately a binomial random variable.
- b. Use the researchers’ report to estimate p for the binomial random variable.
- c. Consider a random sample of 28 general practice physicians. Use p from part b to find the mean and standard deviation of x, the number who do not recommend medicine as a career.
- d. For the sample of part c, find the probability that at least one general practice physician does not recommend medicine as a career.

**2. Question 3.36: **Problem at major companies. The organization Development Journal (Summer 2006) reported on the results of a survey of human resource officers (HROs)at major employers located in a southeastern city. The focus of the study was employee behavior, namely absenteeism, promptness to work, and turnover. The study found that 55% of the HROs had problems with employee absenteeism; also, 41% had problems with turnover. Suppose that 22% of the HROs had problems wit both absenteeism and turnover. Use this information to find the probability that an HRO selected from the group surveyed had problems with either employee absenteeism or employee turnover.

**3. Question** **3.42: **Stock market pari9tcipation and IQ. The Journal of Finance (December 2011) published a study of whether the decision to invest in the stock market is dependent on IQ. Information on a sample of 158,044 adults living on Finland formed the database for the study. An IQ score (from a low score of 1 to a high score of 9) was determined for each Finnish citizen as well as whether or not the citizen invested in the stock market. The next table gives the number of Finnish citizens in each IQ score/investment category. Suppose one of the 158,044 citizens is selected at random.

- What is the probability that the Finnish citizen invests in the stock market?
- What is the probability that the Finnish citizen has an IQ score of 6 or higher?
- What is the probability that the Finnish citizen invests in the stock market and has an IQ score of 6 or higher?
- What is the probability that the Finnish citizen invests in the stock market or has an IQ score of 6 or higher?
- What is the probability that the Finnish citizen does not invest in the stock market?
- Are the events {Invest in the stock market} and {IQ score of 1} mutually exclusive?

**4. Question 3.47: **Reliability of gas station air gauges. Tire and automobile manufacturers and consumer safety experts all recommend that drivers maintain proper tire pressure in their cars. Consequently, many gas stations now provide air pumps and air gauges for their customers. In a Research Note (Nov. 2001), the National Highway Traffic Safety Administration studied the reliability of gas station air gauges. The next table gives the percentage of gas stations that provide air gauges that over report the pressure level in the tire.

**5. Question 3.50: **For two events, A and B, P(A) = A, P(B) = 0.4, and P(B) = 0.2 and P(A}B) = 0.6:

- Find P(A∩B)
- Find P(B|A)

**6. Question 3.51: **For two events, A and B, P(A) = A, P(B) = 0.4, and P(B) = 0.2 and P(A}B) = 0.6:

- Find P(A∩B)
- Find P(B|A)
- Find P(A U B)

**7. Question 3.59: **World’s largest public companies. Forbes (Apr. 20, 2011) conducted a survey of the 20 largest public companies in the world. Of these 20 companies, 4 were banking or investment companies based in the United States. A total of 9 U.S companies were on the top 20 list. Suppose we select one of these 20 companies at random. Given that the company is based in the United States. What is the probability that it is a baking or investment company?

**8. Question 4.21: **NHTSA crash tests. Refer to the NHTSA crash test of new care models. Exercise 4.3 (p 188). A summary of the driver side star ratings for the 98 cars in the file is reproduced I the accompanying Minitab printout. Assume that one of the 98 cars is selected at random and let x equal the number of stars in the car’s driver-side star rating.

**9. Question 4.36: Expected lotto winnings. **Most states offer weekly lotteries to generate revenue for the state. Despite the long odds of winning, residents continues to gamble on the lottery of each week. In SIA, Chapter 3 (p.128), you learned that the chance of winning Florida’s Pick-6 Lotto game is 1 in approximately 23 million. Suppose you buy a $1 Lotto ticket in anticipation of winning the $7 million grand prize. Calculate your expected net winnings. Interpret result.

**10. Question 4.44: **If x is a binomial random variable, use Table I in appendix to find the following probabilities:

- P(x = 2) for n = 10, p = .4
- P(x ≤ 5) for n = 15, p = .6
- P(x > 1) for n = 5, p = .1
- P(x < 10) for n = 25, p = .7
- P(x ≥ 10) for n = 15, p = .9
- P(x = 2) for n = 20, p = .2

**11. Question 4.44: **If x is a binomial random variable, calculate µ, σ^{2}, and σ for each of the following:

- n = 25, p = .5
- n = 80, p = .2
- n = 100, p = .6
- n = 70, p = .9
- n = 60, p = .8
- n = 1000, p = .04

**12. Question 4.50: Physicians’ opinions on a career in medicine. **Many primary care doctors feel overworked and burdend by potential lawsuits. In fact, the Physicians’ Foundation reported that 60% of all general practice physicians in the United States do not recommend medicine as a career (Reuters, Nov. 18, 2008). Let x represent the number of sampled general practice physicians who do not recommend medicine as a career.

## STATUS

- Explain why x is approximately a binomial random variable.
- Use the Physicians’ Foundation report to estimate p for the binomial random variable.
- Consider a random sample of 25 general practice physicians. Use p form part b to find theh mean and standard deviation of x, the number who do not recommend medicine as a career.
- For the sample of part c, find the probability that at least one general practice physician does not recommend medicine as a career.

**13. Question 4.55: Bridge inspection ratings. **According to the National Bridge Inspection Standard (NBIS), public bridges over 20 feet in length must be inspected and rated every 2 years. The NBIS rating scale ranges from 0 (poorest rating) to 9 (highest rating). University of Colorado engineers used a probabilistic model to forecast the inspection ratings of all major bridges in Denver (Journal of Performance of Constructed Facilities, Feb. 2005). For the year 2020, the engineers forecast that 9% of all major Denver bridges will have ratings of 4 or below.

- Use the forecast to find the probability that in a random sample of 10 major Dever bridges, at least 3 will have an inspection rating of 4 or below in 2020.
- Suppose that you actually observes 3 or more of the sample of 10 bridges with inspection ratings of 4 or below in 2020. What interface can you make? Why?