Exam 8: Confidence Interval Estimation

A population parameter is used to estimate a confidence interval.

SCENARIO 8-11 A poll was conducted by the marketing department of a video game company to determine the popularity of a new game that was targeted to be launched in three months. Telephone interviews with 1,500 young adults were conducted which revealed that 49% said they would purchase the new game. The margin of error was $\pm 3$ percentage points. -Referring to Scenario 8-11, the standard error is 3%.

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, the lower and upper bounds of the 95% confidence interval for the mean number of times the handles have been used to lift the bottles before they break are ____ and _____.

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, the critical value used to construct the 95% confidence interval for the mean number of times the handles have been used to lift the bottles before they break is _____.

SCENARIO 8-11 A poll was conducted by the marketing department of a video game company to determine the popularity of a new game that was targeted to be launched in three months. Telephone interviews with 1,500 young adults were conducted which revealed that 49% said they would purchase the new game. The margin of error was $\pm 3$ percentage points. -Referring to Scenario 8-11, you are 99% confidence that the percentage of the targeted young adults who will purchase the new game is somewhere between 46% and 52%.

The head librarian at the Library of Congress has asked her assistant for an interval estimate of the mean number of books checked out each day. The assistant provides the following interval estimate: From 740 to 920 books per day. If the head librarian knows that the population standard deviation is 150 books checked out per day, and she asked her assistant to use 25 days of data to construct the Interval estimate, what confidence level can she attach to the interval estimate?

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, the critical value for a 99% confidence interval for this sample is __________.

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, 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.

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, 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 _____.

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, it is possible that the 99% confidence interval calculated from the data will not contain the proportion of the student population who feel comfortable reporting cheating by their fellow students.

The upper bound of a 90% confidence interval estimate of the population mean is 34.32.

Holding the width of a confidence interval fixed, increasing the level of confidence can be achieved with a lower sample size.

An economist is interested in studying the incomes of consumers in a particular country. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in a Mean income of $15,000. What total sample size would the economist need to use for a 95% Confidence interval if the width of the interval should not be more than $100?

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, a 90% confidence interval for the mean of the population from the same sample would be wider than the 95% confidence interval for the mean number of times the handles have been used to lift the bottles before they break.

Suppose a 95% confidence interval for μ has been constructed. If it is decided to take a larger sample and to decrease the confidence level of the interval, then the resulting interval width would ) (Assume that the sample statistics gathered would not change very much for the new sample.)

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 between 51.16% and 64.84% of managers in the population have caught salespeople cheating on an expense report.

When determining the sample size for a proportion for a given level of confidence and sampling error, the closer to 0.50 that π is estimated to be, the sample size required __________.

The t distribution allows the calculation of confidence intervals for means for small samples when the population variance is not known, regardless of the shape of the distribution in the population.

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, a 95% confidence interval for the proportion of the student population who own a personal computer is narrower than a 99% confidence interval.

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, it is possible that the 99% confidence interval calculated from the data will not contain the population mean number of rooms rented daily in a given month.