Exam 8: Confidence Interval Estimation

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, a 90% confidence interval calculated from the same data would be narrower than a 99% confidence interval.

SCENARIO 8-13-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 bottle to 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 Scenario8-13-DataA.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 the population standard deviation is 5 and the number of times the handles have been used to lift the bottles before they break follows a normal distribution. -Referring to Scenario 8-13-A, 95% of the sample means 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 confidence interval obtained will always correctly estimate the population parameter.

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, construct a 95% confidence interval estimate of the population proportion of managers who have caught salespeople giving a kickback to a customer.

SCENARIO 8-3 To become an actuary, it is necessary to pass a series of 10 exams, including the most important one, an exam in probability and statistics. An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program. They take a sample of 8 actuarial students in this program and determine that their scores are: 2, 5, 8, 8, 7, 6, 5, and 7. This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program. -Referring to Scenario 8-3, it is possible that the confidence interval obtained will not contain the mean score for all actuarial students in the special study program.

SCENARIO 8-12-B 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-B.XLSX. 493 506 515 491 500 505 517 510 506 503 503 491 495 496 496 505 493 486 504 483 514 494 497 501 493 490 510 494 494 495 494 486 495 506 506 507 502 498 510 501 500 505 492 486 501 496 501 521 510 498 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-B, 95% of the sample means 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.

SCENARIO 8-6 After an extensive advertising campaign, the manager 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 that goes from 0.36 to 0.54. -Referring to Scenario 8-6, 95% of the time, the proportion of people that recognize the product will fall between 0.36 and 0.54.

The t distribution is used to construct confidence intervals for the population mean when the population standard deviation is unknown.

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

SCENARIO 8-13-B 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 bottle to 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 Scenario8-13-DataB.XLSX. 493 506 515 491 500 505 517 510 506 503 503 491 495 496 496 505 493 486 504 483 514 494 497 501 493 490 510 494 494 495 494 486 495 506 506 507 502 498 510 501 500 505 492 486 501 496 501 521 510 498 Assume that the population standard deviation is 5 and the number of times the handles have been used to lift the bottles before they break follows a normal distribution. -Referring to Scenario 8-13-B, 95% of all confidence intervals constructed similarly to this one with a sample size of 50 will contain the mean of the population.

SCENARIO 8-7 A hotel chain wants to estimate the mean number of rooms rented daily in a given month. The population of rooms rented daily is assumed to be normally distributed for each month with a standard deviation of 240 rooms. During February, a sample of 25 days has a sample mean of 370 rooms. -Referring to Scenario 8-7, the parameter of interest is the proportion of the rooms rented daily in a given month.

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. If the dean wanted to estimate the proportion of All students receiving financial aid to within 3% with 99% reliability, how many students would need To be sampled?

When determining the sample size necessary for estimating the true population mean, which factor is not considered when sampling with replacement?

A major department store chain is interested in estimating the mean 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: $\bar { X } = \$ 50.50 \text { and } S = 20$ . Assuming the Distribution of the amount spent on their first visit is normal, what is the shape of the sampling Distribution of the sample mean that will be used to create the desired confidence interval for ??

SCENARIO 8-5 A sample of salary offers (in thousands of dollars) given to management majors is: 48, 51, 46, 52, 47, 48, 47, 50, 51, and 59. Using this data to obtain a 95% confidence interval resulted in an interval from 47.19 to 52.61. -Referring to Scenario 8-5, 95% of the time, the sample mean salary offer to management majors will fall between 47.19 and 52.61.

The standardized normal distribution is used to develop a confidence interval estimate of the population proportion when the sample size is sufficiently large.

Other things being equal, as the confidence level for a confidence interval increases, the width of the interval increases.

The t distribution approaches the standardized normal distribution when the number of degrees of freedom increases.

SCENARIO 8-7 A hotel chain wants to estimate the mean number of rooms rented daily in a given month. The population of rooms rented daily is assumed to be normally distributed for each month with a standard deviation of 240 rooms. During February, a sample of 25 days has a sample mean of 370 rooms. -Referring to Scenario 8-7, the parameter of interest is the mean number of rooms rented daily in a given month.

For a given data set, the confidence interval will be wider for 95% confidence than for 90% confidence.