Exam 8: Confidence Interval Estimation

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, what is the needed sample size to obtain a 95% confidence interval in estimating the percentage of the targeted young adults who will purchase the new game to within ±5% if you do not have the information on the 49% in the interviews who said that they would purchase the new game?

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, it is possible that the mean of the population is not between 47.19 and 52.61.

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 sample means will fall between 47.19 and 52.61.

The sampling error can either be positive or negative.

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, we are 99% confident that the total number of the student population who feel comfortable reporting cheating by their fellow students is between 0.4557 and 0.5043.

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, a 99% 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.

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, 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 turns out to be (1,000, 2,100). To make more useful inferences from the data, it is desired to reduce the width of the confidence interval. Which of the Following will result in a reduced interval width?

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

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 99% confidence interval will contain 99% of the student population who own a personal computer.

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

As an aid to the establishment of personnel requirements, the director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each. For this sample, $\bar { X } = 396 \text { and } S = 100$ . Using the sample standard deviation as an estimate for the population standard deviation, what size sample should the director choose if she wishes to estimate the mean number of admissions per 24-hour period to within 1 admission with 99% reliability,?

Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is put into a fund called an endowment, and the college Spends only the interest earned by the fund. A recent survey of 8 private colleges in the United States Revealed the following endowments (in millions of dollars): 60.2, 47.0, 235.1, 490.0, 122.6, 177.5, 95)4, and 220.0. Summary statistics yield $\bar { X } = 180.975 \text { and } S = 143.042$ . Calculate a 95% Confidence interval for the mean endowment of all the private colleges in the United States assuming A normal distribution for the endowments. a) $\$ 180.975 \pm \$ 94.066$ b) $\$ 180.975 \pm \$ 99.123$ c) $\$ 180.975 \pm \$ 116.621$ d) $\$ 180.975 \pm \$ 119.586$

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 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 8-12-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 not 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, it is possible that the true proportion of people that recognize the product is not between 0.36 and 0.54.

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, it is possible that the 99% confidence interval calculated from the data will not contain the sample proportion of students who own a personal computer.

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 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 8-7 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, we are 99% confident that the average number of rooms rented daily in a given month is somewhere between 246.36 and 493.64.

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, what is the needed sample size to obtain a 90% confidence interval estimate of the percentage of the targeted young adults who will purchase the new game by allowing the same level of margin of error?