PROBLEMS ON CONFIDENCE INTERVAL



Problems on Confidence Interval for Mean:

1. IF sample mean = 85, ( = 8, and n = 64, set up a 95% confidence interval estimate for the population mean µ.

2. The quality control manager at a light bulb factory needs to estimate the mean life of a large shipment of life bulbs. The process standard deviation is known to be 100 hours. A random sample of 64 light bulbs indicated a sample mean life of 350 hours.

a. Obtain the standard error of the mean.

b. Set up a 95% confidence interval estimate of the true population mean life of light bulbs in this shipment

c. Do you think that the manufacture has the right to state that the light bulb last an average of 400 hours?

3. It is known from the manufacturer’s specifications that the standard deviation of the amount of paint is equal to 0.02 gallon. A random sample of 50 cans is selected, and the sample mean amount of paint per 1-gallon can is 0.995 gallon.

a. Obtain the standard error of the mean.

b. Set up 95% confidence interval estimate of the true population mean amount of paint included in a 1-gallon.

4. The bottling plant has informed the inspection division that the s.d. for 2- liter bottles is 0.05 liter. A random sample of 100 2-liter bottles obtained from this bottling plant indicates a sample mean of 1.99 liters. Set up a 95% confidence interval estimate for true population mean amount of soft drink in each bottle.

5. If sample mean = 75, s = 24, and n = 36, and assuming that the population is normally distributed, ser up a 95% confidence interval estimate for the population mean µ.

6. Set up a 95% confidence interval estimate for the population mean, based on each of the following sets of data separately, assuming that the population is normally distributed:

Set 1: 1 1 1 1 8 8 8 8

Set 2: 1 2 3 4 5 6 7 8

7. A manufacture of computer paper has a production process that operates continuously throughout an entire production shift. The paper is expected to have an average length of 11 inches and the standard deviation is known to be 0.01 inch. Suppose a random sample of 100 sheets and the average paper length is found to be 10.998 inches. Set up 95% and 99% confidence interval estimate of the population average paper length.

8. The customer service department of a local gas utility wants to estimate the average length of time between the entry of the service request and the connection of service. A random sample of 15 houses is selected from the records available during the past year. The results records in number of days are as follows

86 78 96 73 99 78 72 104 114 137 126 117 114 103 86

Set up 95% and 99% confidence interval estimate of the population average waiting time in the past year.

9. A stationary shop would like to estimate the average retail value of greeting cards that it has in its inventory. A random sample of 20 greeting cards indicated an average value of Rs.167 and a standard deviation of Rs.32. Set up a 95% confidence interval estimate of all greeting cards that are in its inventory.

10. Population consists of the Fortune 500 Companies (Fortune Web Site), as ranked by Revenues. You are trying to find out the average Revenues for the companies on the list. The population standard deviation is $ 15,056.37. A random sample of 64 companies obtained a sample mean of $ 10,672.87. Give a 90% confidence interval estimate for the average Revenues.

11. A race car driver tested his car for time from 0 to 60 mph, and in 20 testes obtained an average of 4.85 seconds with a standard deviation of 1.47 seconds. Calculate a 95% confidence interval for the time from 0 to 60 mph.

12. Quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired length of the insulation is 12 feet. It is known that the standard deviation in the cutting length is 0.15 feet. A sample of 70 cut sheets yields a mean length of 12.14 feet. This sample will be used to obtain a 90%, 95% and 99% confidence interval estimate for the mean length cut by machine. Obtain and interpret this confidence interval.

13. A rational consumer is smart enough to conduct statistical analysis. He was worried about DDC’S average filling of milk in a packet labeled. 500 ml net. He randomly sampled 10 such different packets and found the sample mean of 495 ml with a sample standard deviation (s) of 9. Set up a 95% confidence interval of the population mean µ.

14. A major department store chain is interested in estimating the average amount its credit card customers spent on their first visit to the chain’s new store in the mall. Fifteen credit card accounts were randomly sampled and analyzed with the following results, sample mean= $50.50 and s2 = 400. Construct a 95% confidence interval for the average amount its credit card customers spent on their first visit to the chain’s new store in the mall, assuming that the amount spent follows a normal distribution.

Confidence Interval for Proportion:

1. If n = 200 and X = 50, set up a 95% confidence interval estimate of the population proportion.

2. A random sample of 500 apples was taken from a large consignment and 60 were found be bad. Obtain the 95% confidence limits for the proportion of bad apples in the consignments.

3. A random sample of 100 consumers is obtained, and it is found that 34 people in the sample are users of foreign made products the rest are users of domestic products. Obtain 95% confidence interval estimate for the domestic product users.

4. The telephone company wants to estimate the proportion of households that would purchases an additional telephone line if it were made available at a substantially reduced installation cost. A random sample of 500 households is selected. The results indicate that 135 of the households would purchase the additional telephone line at a reduced installation cost. Construct a 99% confidence interval estimate of the population proportion of households that would purchase the additional telephone line.

5. In recent survey conducted by the society for human resource management, 453 of 853 personnel officials indicated that job seekers sometimes falsify past salaries.

a. Set up a 99% confidence interval estimate for the population proportion of personnel officials who believe that job seekers sometimes falsify past salaries. Which interval is wider?

b. Set up a 95% confidence interval estimate for the population proportion of personnel officials who believe that job seekers sometimes falsify past salaries.

c. Which interval is wider?

6. In a survey of 763 women who had started their own businesses, 229 said that they launched their businesses for greater freedom. Only 99 indicated that a desire to make more money drove them to start their businesses.

a. Set up 90% confidence interval estimate of the population proportion of women who start new businesses to gain freedom.

b. Set up 90% confidence interval estimate of the population proportion of women who start new businesses to earn more money.

7. When a sample of 70 retail executives was surveyed regarding the poor November performance of the retail industry, 42 executive believed that decreased sale were due to unseasonably warm temperatures, resulting in consumer’s delaying purchase of cold-weather items.

a. Estimate the standard error of the proportion of retail executive who blames warm weather for low sales.

b. Construct 95 percent confidence limit for this proportion and interpret the limit

8. In an attempt to study the problem of using cell phones while driving. A survey of drivers who use cell phones determined that 46% of the respondents reported having had to swerve and 10% knew someone who had had a crash while talking on a cell phone. Suppose the survey was based on 500 respondents.

a. Construct a 95% confidence interval estimate for the proportion of drivers who reported having had to swerve.

b. Construct a 95% confidence interval estimate for the proportion of drivers who knew someone who had had a crash while talking on a cell phone.

9. In a survey by walker information, workers were asked if they felt comfortable reporting misconduct by their fellow workers. Whereas 1344 answered yes and 1456 answered no. Construct a 95% confidence interval estimate for the proportion of all workers who fell comfortable reporting misconduct by their fellow workers.

10. A survey of working women in North America was conducted by the Clinique uite of Estee lauder cosmetics. Of the 1000 women surveyed, 55% believed that companies should hold positions for those on maternity leave for six months or less and 45% felt that they should hold those positions for more than six months. Construct a 95% confidence interval estimate for the proportion of all working women in North America who believe that companies should hold positions for those on maternity leave for more than six months.

11. Mr. Sharma is a professional statistician and currently teaches Business Statistics at college X. he surveyed 100 statisticians and found that 90 of them were in favor of computer - aided statistical analysis in BBA.

a. Estimate the standard error of proportion.

b. Set up a 95% confidence interval estimate of the population proportion of the statisticians who are in favor of computer – aided statistical analysis in BBA.

12. The librarian at the K.U. sampled 45 textbooks at the university and determined that, of these 45 textbooks, 60% had been marked up in price more than 50% over wholesale cost. Give a 95% confidence interval for the proportion of the books marked up more than 50%.

13. A marketing research firm wants to estimate the share that foreign companies have in the American market for certain products. A random sample of 100 consumers is obtained, and it is found that 34 people in the sample are users of foreign made products the rest are users of domestic products. Give a 95% confidence interval estimate for the share of foreign products in this market.

14. After an extensive advertising campaign, the manger of a company wants to estimate the proportion of potential customers that recognize a new product. She samples 120 potential consumers and finds that 54 recognize this product. She uses this sample information to obtain a 95% confidence interval estimate. Determine her estimate in the question.

15. A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. Use a 90%, 95% 99% confidence interval to estimate the true proportion of students who receive financial aid. Also interpret these intervals.

16. Dr. Khatri noted economist, surveyed 150 households in a particular place and found that 63 of them were unable to bear the school fee of their children. Set up a 95% confidence interval estimate for the true proportion of households who cannot actually bear the school fee of their children

17. The European Union Executive Commission conducted a study of 6543 European adults. Of those surveyed, 56% said that the euro single currency would promote economic growth and 73% knew the correct date of the changeover.

18. Obtain the value of standard error of the proportion.

19. Construct a 95% confidence interval estimate for the proportion of European adults who believed that the euro would promote economic growth.

20. Construct a 90% confidence interval estimate for the proportion of European adults who knew the correct date of the currency changeover.

Determination of sample size

1. The university is considering raising tuition to improve school facilities, and they want to determine what percentages of students favor the increase. The university has decided to within 2 percent of the true value. How large a sample is needed to guarantee this accuracy regardless of the true percentage?

2. An important proposal must be voted on, and a politician wants to find the proportion of people who are in favor of the proposal. Find the sample size needed to estimate the true proportion to within +/- 0.05 at the 95 percent confidence level. Assume you have no strong feeling about what the proportion is. How would your sample size changes if you believe about 75% of the people favor the proposal? How would it change if only about 25 percent favor the proposal?

3. A speed-reading course guarantees a certain reading rate increase within 2 days. The teacher knows a few people will not be able to achieve this increase, so before stating the guaranteed percentage of people who achieve the reading rate increase, he wants to be 95% confident that the percentage has been estimated to within +/- 5 percent of the true value. What is the most conservative sample size needed for this problems.

4. If the population standard deviation is 78, find the sample size necessary to estimate the true mean within 50 points for a confidence level of 95%.

5. Given a population with a standard deviation of 8.6, what size sample is needed to estimate the mean of the population within +/- 0.5 with 99 percent confidence?

6. The management of southern textiles has recently come under fire regarding the supposedly determined effects on health caused by its manufacturing process. A social scientist has advanced a theory that the employees who die from natural causes exhibit remarkable consistency in their life-span the upper and lower limits of their life-span differ by no more than 550 weeks. For a confidence level of 95 percent, how large a sample should be examined to find the average life-span of these employees within +/- 30 weeks.

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