1. Given a normal distribution with µ = 100 and σ = 10, if you select a sample of n = 25, what is the probability that X-bar is:
a. less than 95?
Calculate the z-score that corresponds to
b. between 95 and 97.5?
2. You plan to conduct a marketing experiment in which students are to taste one of two different brands of soft drink. Their task is to correctly identify the brand tasted. You select a random sample of 200 students and assume that the students have no ability to distinguish between the two brands.
a. What is the probability that the sample will have between 50% and 60% of the identifications correct?
b. The probability is 90% that the sample percentage is contained within what symmetrical limits of the population percentage?
c. What is the probability that the sample percentage of correct identifications is greater than 65%?
d. Which is more likely to occur-more than 60% correct identifications in the sample of 200 or more than 55% correct identifications in a sample of 1,000? Explain.
2. If = 85, σ = 8, and n = 64 construct a 95% confidence interval estimate of the population mean, µ.
3. Assuming that the population is normally distributed, construct a 95% confidence interval estimate for the population mean for each of the following samples:
Sample A: 1, 1, 1, 1, 8, 8, 8, 8
Sample B: 1, 2, 3, 4, 5, 6, 7, 8
Explain why these two samples produce different confidence intervals even though they have the same mean and range.
4. The U.S. Department of Transportation requires tire manufacturers to provide tire performance information on the sidewall of the tire to better inform prospective customers when making purchasing decisions. One very important measure of tire performance is the tread wear index, which indicates the tire's resistance to tread wear compared with a tire graded with a base of 100. This means that 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 of 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.
5. In a survey of 1,200 social media users, 76% said it is okay to friend co-workers, but 56% said it is not okay to friend your boss. (Data extracted from "Facebook Etiquette at Work," USA Today, March 24, 2010, p. 1B.)
a. Construct a 95% confidence interval estimate for the population proportion of social media users who would say it is okay to friend co-workers.
b. Construct a 95% confidence interval estimate for the population proportion of social media users who would say it is not okay to friend their boss.
6. If you want to be 99% confident of estimating the population mean to within a sampling error of + or - 20 and the standard deviation is assumed to be 100, what sample size is required?
7. If the manager of a paint supply store wants to estimate the mean amount of paint in a 1-gallon can to within ±0.004 gallon with 95% confidence and also assumes that the standard deviation is 0.02 gallon, what sample size is needed?