Examples Involving Confidence Intervals

Examples involving Confidence Intervals

1.The U.S. Department of Agriculture wants to determine the average number of eggs that children under 14 years of age consume each year. A random sample of 900 such children is obtained. In this sample, the average number of eggs consumed was = 86, and the population standard deviation is

σ = 16. Find a 99% confidence interval for the mean. Answer: (84.6267, 87.3733)

2.A department store decides to examine the proportion of shoppers interested in a boutique format for the store’s basement. Of the 300 shoppers surveyed, 195 think it would be a good idea.

a)Find a 95% confidence interval for the proportion of all shoppers who favor this idea.

Answer: (.5960, .7040)

b)Given your interval in part (a), can you conclude that the majority of all shopperswould approve of the boutique format? Why or why not? Answer: With 95% confidence, yes.

3.A farmer wants to determine whether different types of feed can influence the mean number of eggs that hens lay per month. In a random sample of 100 hens that ate feed 1, the average number of eggs per month was = 15.2 and the variance was 4. In a random sample of 100 hens that ate feed 2, the average number of eggs per month was = 14 and the variance was 4. Construct a 95% confidence interval for the population mean difference. Answer: (.6456, 1.7544)

4.A TV executive is interested in determining whether the proportion of people who watch a late-night talk show is higher with the regular host or a guest host. In a random sample of 400 people, 175 watch the show when the regular host is on. In a separate random sample of 500 people, 185 watch the show when a guest host is on. Calculate a 95% confidence interval for the difference in the population proportions. Answer: (.0030, .1320) 99% C.I. Answer: (-.0172, .1522)

5.The life in hours of a 75-watt light bulb is known to be normally distributed with a standard deviation of 25 hours. A random sample of 20 bulbs has a mean life of =1014 hours. Construct a 95% confidence interval for the mean life. Answer: (1003.0433, 1024.9567)

6.A large university is considering giving each faculty member a telephone answering machine rather than having calls forwarded to a secretary when the professor is not in his or her office. A budget committee wants to estimate the average amount of time that secretaries spend handling phone calls for absent faculty members. Because getting the information is relatively expensive, the sample size must be kept small. A random sample of 16 secretaries is observed. The results are = 36 minutes per day and s 2 = 320. Assuming the population distribution is normal, find a 95% confidence interval for the mean number of minutes per day that secretaries handle phone calls for absent professors.

Answer: (26.4699, 45.5301)

7. Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation .75.

a)How large a sample size is necessary to estimate the true average porosity if the width of the 95% interval is to be .40? Answer: 55

b)What sample size is necessary to estimate μ to within .2 with 99% confidence? Answer: 94