# Mathematics

You are required to show all of your work/formulas

7.1 Given a normal distribution with m=100 and s=10, if you select a sample of n=25, what is the probability that X is

a. less than 95?

b. between 95 and 97.5?

c. above 102.2?

d. There is a 65% chance that X is above what value?

7.17 The Chartered Financial Analyst (CFA) Institute reported that 49% of its U.S. members indicate that lack of ethical culture within financial firms has contributed the most to the lack of trust in the financial industry. (Source: “Global Market Sentiment Sur-vey 2014,” cfa.is/Psk5O3.) Suppose that you select a sample of 100 CFA members.

a. What is the probability that the sample percentage indicating that lack of ethical culture within financial firms has contrib-uted the most to the lack of trust in the financial industry will be between 48% and 53%?

b. The probability is 90% that the sample percentage will be contained within what symmetrical limits of the population percentage?

c. The probability is 95% that the sample percentage will be contained within what symmetrical limits of the population percentage?

d. Suppose you selected a sample of 400 CFA members. How does this change your answers in (a) through (c)?

8.1 If X=85, s=8, and n=64, construct a 95% confidence interval estimate for the population mean, m.

8.9 A bottled water distributor wants to estimate the amount of water contained in 1-gallon bottles purchased from a nationally known water bottling company. The water bottling company’s specifications state that the standard deviation of the amount of water is equal to 0.02 gallon. A random sample of 50 bottles is selected, and the sample mean amount of water per 1-gallon bottle is 0.995 gallon

a. Construct a 99% confidence interval estimate for the popula-tion mean amount of water included in a 1-gallon bottle.

b. On the basis of these results, do you think that the distributor has a right to complain to the water bottling company? Why?

c. Must you assume that the population amount of water per bottle is normally distributed here? Explain.

d. Construct a 95% confidence interval estimate. How does this change your answer to (b)?

8.11 If X=75, S=24, and n=36, and assuming that the population is normally distributed, construct a 95% confidence interval estimate for the population mean, m.

8.17 The U.S. Department of Transportation requires tire manu-facturers to provide tire performance information on the sidewall of a tire to better inform prospective customers as they make purchasing decisions. One very important measure of tire perfor-mance is the tread wear index, which indicates the tire’s resistance to tread wear compared with a tire graded with a base of 100.A tire with a grade of 200 should last twice as long, on average, as a tire graded with a base of 100. A consumer organization wants to estimate the actual tread wear index of a brand name of tires that claims “graded 200” on the sidewall of the tire. A random sample of n=18 indicates a sample mean tread wear index of 195.3 and a sample standard deviation of 21.4.

a. Assuming that the population of tread wear indexes is normally distributed, construct a 95% confidence interval estimate for the population mean tread wear index for tires produced by this manufacturer under this brand name.

b. Do you think that the consumer organization should accuse the manufacturer of producing tires that do not meet the performance information provided on the sidewall of the tire? Explain.

c. Explain why an observed tread wear index of 210 for a particu-lar tire is not unusual, even though it is outside the confidence interval developed in (a).

8.33 What are the global trends that technology CEOs believe will transform their business? According to a PwC white paper, 105 of 117 technology CEOs from around the world responded that technological advances will transform their business and 42 responded that resource scarcity and climate change will trans-form their business. (Source: Fit for the Future: 17th Annual Global CEO Survey, available at pwc.to/PRQZYr.)

a. Construct a 95% confidence interval estimate for the popula-tion proportion of tech CEOs who indicate technological ad-vances as one of the global trends that will transform their business.

b. Construct a 95% confidence interval estimate for the popula-tion proportion of tech CEOs who indicate resource scarcity and climate change as one of the global trends that will trans-form their business.

c. Interpret the intervals in (a) and (b).

8.37 If you want to be 95% confident of estimating the popula-tion proportion to within a sampling error of {0.02 and there is historical evidence that the population proportion is approximately 0.40, what sample size is needed?

8.41 If the inspection division of a county weights and measures department wants to estimate the mean amount of soft-drink fill in 2-liter bottles to within {0.01 liter with 95% confidence and also assumes that the standard deviation is 0.05 liter, what sample size is needed?