Exam 8: Confidence Interval Estimation

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% confidence interval to estimate the true proportion of students who receive financial aid.

SCENARIO 8-12-A You are the quality control manager of a water bottles company. One of the biggest complaints in the past years has been the breakage and, hence, the concern on the durability of the connector between the lid and the bottle which many users use as a handle for the bottles. To collect evidence before implementing any modification to the production process, your department has subjected 50 water bottles to a durability test and the following data on the number of times the handles have been used to lift the bottles before they break are contained in the file SCENARIO 8-12-A.XLSX. 495 499 502 500 491 498 498 495 488 516 513 486 504 503 493 504 489 500 495 499 501 507 511 496 486 497 510 504 493 482 511 502 520 514 486 514 500 505 512 500 504 498 503 514 474 489 488 506 517 490 Assume that and the number of times the handles have been used to lift the bottles before they break follows a normal distribution. -Referring to Scenario 8-12-A, the sampling error of the 95% confidence interval for the mean number of times the handles have been used to lift the bottles before they break is _____.

The t distribution is used to develop a confidence interval estimate of the population mean when the population standard deviation is unknown.

SCENARIO 8-12-A You are the quality control manager of a water bottles company. One of the biggest complaints in the past years has been the breakage and, hence, the concern on the durability of the connector between the lid and the bottle which many users use as a handle for the bottles. To collect evidence before implementing any modification to the production process, your department has subjected 50 water bottles to a durability test and the following data on the number of times the handles have been used to lift the bottles before they break are contained in the file SCENARIO 8-12-A.XLSX. 495 499 502 500 491 498 498 495 488 516 513 486 504 503 493 504 489 500 495 499 501 507 511 496 486 497 510 504 493 482 511 502 520 514 486 514 500 505 512 500 504 498 503 514 474 489 488 506 517 490 Assume that and the number of times the handles have been used to lift the bottles before they break follows a normal distribution. -Referring to Scenario 8-12-A, it is possible that the mean of the population will fall between the computed lower and upper limit of the 95% confidence interval for the mean number of times the handles have been used to lift the bottles before they break.

The sample mean is a point estimate of the population mean.

SCENARIO 8-10 A sales and marketing management magazine conducted a survey on salespeople cheating on their expense reports and other unethical conduct. In the survey on 200 managers, 58% of the managers have caught salespeople cheating on an expense report, 50% have caught salespeople working a second job on company time, 22% have caught salespeople listing a "strip bar" as a restaurant on an expense report, and 19% have caught salespeople giving a kickback to a customer. -Referring to Scenario 8-10, determine the sample size needed to estimate the proportion of managers who have caught salespeople working a second job on company time to within $\pm 0.02$ with 95% confidence.

SCENARIO 8-10 A sales and marketing management magazine conducted a survey on salespeople cheating on their expense reports and other unethical conduct. In the survey on 200 managers, 58% of the managers have caught salespeople cheating on an expense report, 50% have caught salespeople working a second job on company time, 22% have caught salespeople listing a "strip bar" as a restaurant on an expense report, and 19% have caught salespeople giving a kickback to a customer. -Referring to Scenario 8-10, we are 95% confident that the population mean number of managers who have caught salespeople cheating on an expense report is between 0.5116 to 0.6484.

A university system enrolling hundreds of thousands of students is considering a change in the way students pay for their education. Currently, the students pay $400 per credit hour. The university system administrators are contemplating charging each student a set fee of $7,000 per quarter, regardless of how many credit hours each takes. To see if this proposal would be economically feasible, the administrators would like to know how many credit hours, on the average, each student takes per quarter. A random sample of 250 students yields a mean of 14.1 credit hours per quarter and a standard deviation of 2.3 credit hours per quarter. Suppose the administration wanted to estimate the mean to within 0.1 hours at 95% reliability and assumed that the sample standard deviation provided a good estimate for the population standard deviation. How large a total sample would they need to take?

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77, 11.90, 11.64, 11.84, 12.13, 11.99, and 11.77. -Referring to Scenario 8-4, the critical value for a 99% confidence interval for this sample is __________.

SCENARIO 8-10 A sales and marketing management magazine conducted a survey on salespeople cheating on their expense reports and other unethical conduct. In the survey on 200 managers, 58% of the managers have caught salespeople cheating on an expense report, 50% have caught salespeople working a second job on company time, 22% have caught salespeople listing a "strip bar" as a restaurant on an expense report, and 19% have caught salespeople giving a kickback to a customer. -Referring to Scenario 8-10, the sampling error of a 95% confidence interval estimate of the population proportion of managers who have caught salespeople working a second job on company time is __________.

A quality control engineer is interested in estimating the proportion of defective items coming off a production line. In a sample of 300 items, 27 are defective. A 90% confidence interval for the proportion of defectives from this production line would go from __________ to __________.

A prison official wants to estimate the proportion of cases of recidivism. Examining the records of 250 convicts, the official determines that there are 65 cases of recidivism. A confidence interval will be obtained for the proportion of cases of recidivism. Part of this calculation includes the estimated standard error of the sample proportion. For this sample, the estimated standard error is __________.

The width of a confidence interval estimate for a proportion will be

A race car driver tested his car for time from 0 to 60 mph, and for 20 tests obtained a mean of 4.85 seconds with a standard deviation of 1.47 seconds. A 95% confidence interval for the 0 to 60 mean time is 4.52 seconds to 5.18 seconds.

For a t distribution with 12 degrees of freedom, the area between - 2.6810 and 2.1788 is 0.980.

SCENARIO 8-8 The president of a university would like to estimate the proportion of the student population that owns a personal computer. In a sample of 500 students, 417 own a personal computer. -Referring to Scenario 8-8, we are 99% confident that between 79.11% and 87.69% of the student population own a personal computer.

SCENARIO 8-2 A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired mean 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 99% confidence interval for the mean length cut by machine. -Referring to Scenario 8-2, the confidence interval indicates that the machine is not working properly.

SCENARIO 8-9 A university wanted to find out the percentage of students who felt comfortable reporting cheating by their fellow students. A surveyed of 2,800 students was conducted and the students were asked if they felt comfortable reporting cheating by their fellow students. The results were 1,344 answered "Yes" and 1,456 answered "no". -Referring to Scenario 8-9, a 95% confidence interval for the proportion of the student population who feel comfortable reporting cheating by their fellow students is narrower than a 99% confidence interval.

The standardized normal distribution is used to develop a confidence interval estimate of the population proportion regardless of whether the population standard deviation is known.

The difference between the sample mean and the population mean is called the sampling error.