Exam 9: Fundamentals of Hypothesis Testing: One-Sample Tests

A Type II error is committed when

SCENARIO 9-11-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 Scenario9-11-DataA.XLSX. 1495 1499 1502 1500 1491 1498 1498 1495 1488 1516 1513 1486 1504 1503 1493 1504 1489 1500 1495 1499 1501 1507 1511 1496 1486 1497 1510 1504 1493 1482 1511 1502 1520 1514 1486 1514 1500 1505 1512 1500 1504 1498 1503 1514 1474 1489 1488 1506 1517 1490 Assume that the number of times the handles have been used to lift the bottles before they break follows a normal distribution. You want to test to see if there is enough evidence that the mean number of times the handles have been used to lift the bottles before they break is more than 1500. -Referring to Scenario 9-11-A, the appropriate hypotheses are: a) $H _ { 0 } : \mu = 1500$ versus $H _ { 1 } : \mu \neq 1500$ b) $H _ { 0 } : \mu \geq 1500$ versus $H _ { 1 } : \mu < 1500$ c) $H _ { 0 } : \mu \leq 1500$ versus $H _ { 1 } : \mu > 1500$ d) $H _ { 0 } : \mu \neq 1500$ versus $H _ { 1 } : \mu = 1500$

A pizza chain is considering opening a new store in an area that currently does not have any such stores. The chain will open if there is evidence that more than 5,000 of the 20,000 households in The area have a favorable view of its brain. It conducts a telephone poll of 300 randomly selected Households in the area and finds that 96 have a favorable view. The decision on the hypothesis Test using a 5% level of significance is: a) to reject $H _ { 0 }$ in favor of $H _ { 1 }$ . b) to accept $H _ { 0 }$ in favor of $H _ { 1 }$ . c) to fail to reject $H _ { 0 }$ in favor of $H _ { 1 }$ . d) we cannot tell what the decision should be from the information given.

A pizza chain is considering opening a new store in an area that currently does not have any such stores. The chain will open if there is evidence that more than 5,000 of the 20,000 households in The area have a favorable view of its brain. It conducts a telephone poll of 300 randomly selected Households in the area and finds that 96 have a favorable view. The value of the test statistic in This problem is approximately equal to:

SCENARIO 9-11-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 Scenario9-11-DataA.XLSX. 1495 1499 1502 1500 1491 1498 1498 1495 1488 1516 1513 1486 1504 1503 1493 1504 1489 1500 1495 1499 1501 1507 1511 1496 1486 1497 1510 1504 1493 1482 1511 1502 1520 1514 1486 1514 1500 1505 1512 1500 1504 1498 1503 1514 1474 1489 1488 1506 1517 1490 Assume that the number of times the handles have been used to lift the bottles before they break follows a normal distribution. You want to test to see if there is enough evidence that the mean number of times the handles have been used to lift the bottles before they break is more than 1500. -Referring to Scenario 9-11-A, the p-value of the test is ________.

SCENARIO 9-1 Microsoft Excel was used on a set of data involving the number of defective items found in a random sample of 46 cases of light bulbs produced during a morning shift at a plant. A manager wants to know if the mean number of defective bulbs per case is greater than 20 during the morning shift. She will make her decision using a test with a level of significance of 0.10. The following information was extracted from the Microsoft Excel output for the sample of 46 cases: $n = 46$ ; Arithmetic Mean $= 28.00$ ; Standard Deviation $= 25.92$ ; Standard Error $= 3.82$ ; Null Hypothesis: $H _ { 0 } : \mu \leq 20 ; \alpha = 0.10 ; \mathrm { df } = 45 ; T$ Test Statistic $= 2.09$ ; One-Tail Test Upper Critical Value $= 1.3006 ; p$ -value $= 0.021 ;$ Decision $=$ Reject. -The statement of the null hypothesis always contains an equality.

A pizza chain is considering opening a new store in an area that currently does not have any such stores. The chain will open if there is evidence that more than 5,000 of the 20,000 households in The area have a favorable view of its brain. It conducts a telephone poll of 300 randomly selected Households in the area and finds that 96 have a favorable view. The pizza chain's conclusion from The hypothesis test using a 5% level of significance is:

SCENARIO 9-1 Microsoft Excel was used on a set of data involving the number of defective items found in a random sample of 46 cases of light bulbs produced during a morning shift at a plant. A manager wants to know if the mean number of defective bulbs per case is greater than 20 during the morning shift. She will make her decision using a test with a level of significance of 0.10. The following information was extracted from the Microsoft Excel output for the sample of 46 cases: $n = 46$ ; Arithmetic Mean $= 28.00$ ; Standard Deviation $= 25.92$ ; Standard Error $= 3.82$ ; Null Hypothesis: $H _ { 0 } : \mu \leq 20 ; \alpha = 0.10 ; \mathrm { df } = 45 ; T$ Test Statistic $= 2.09$ ; One-Tail Test Upper Critical Value $= 1.3006 ; p$ -value $= 0.021 ;$ Decision $=$ Reject. -The test statistic measures how close the computed sample statistic has come to the hypothesized population parameter.

For a given sample size, the probability of committing a Type II error will increase when the probability of committing a Type I error is reduced.

SCENARIO 9-1 Microsoft Excel was used on a set of data involving the number of defective items found in a random sample of 46 cases of light bulbs produced during a morning shift at a plant. A manager wants to know if the mean number of defective bulbs per case is greater than 20 during the morning shift. She will make her decision using a test with a level of significance of 0.10. The following information was extracted from the Microsoft Excel output for the sample of 46 cases: $n = 46$ ; Arithmetic Mean $= 28.00$ ; Standard Deviation $= 25.92$ ; Standard Error $= 3.82$ ; Null Hypothesis: $H _ { 0 } : \mu \leq 20 ; \alpha = 0.10 ; \mathrm { df } = 45 ; T$ Test Statistic $= 2.09$ ; One-Tail Test Upper Critical Value $= 1.3006 ; p$ -value $= 0.021 ;$ Decision $=$ Reject. -"Is the intended sample size large enough to achieve the desired power of the test for the level of significance chosen?" should be among the questions asked when performing a hypothesis test

The marketing manager for an automobile manufacturer is interested in determining the proportion of new compact-car owners who would have purchased a GPS navigation system if it Had been available for an additional cost of $300. The manager believes from previous Information that the proportion is 0.30. Suppose that a survey of 200 new compact-car owners is Selected and 79 indicate that they would have purchased the GPS navigation system. If you were To conduct a test to determine whether there is evidence that the proportion is different from 0.30 And decided not to reject the null hypothesis, what conclusion could you reach?

SCENARIO 9-1 Microsoft Excel was used on a set of data involving the number of defective items found in a random sample of 46 cases of light bulbs produced during a morning shift at a plant. A manager wants to know if the mean number of defective bulbs per case is greater than 20 during the morning shift. She will make her decision using a test with a level of significance of 0.10. The following information was extracted from the Microsoft Excel output for the sample of 46 cases: $n = 46$ ; Arithmetic Mean $= 28.00$ ; Standard Deviation $= 25.92$ ; Standard Error $= 3.82$ ; Null Hypothesis: $H _ { 0 } : \mu \leq 20 ; \alpha = 0.10 ; \mathrm { df } = 45 ; T$ Test Statistic $= 2.09$ ; One-Tail Test Upper Critical Value $= 1.3006 ; p$ -value $= 0.021 ;$ Decision $=$ Reject. -A sample is used to obtain a 95% confidence interval for the mean of a population. The confidence interval goes from 15 to 19. If the same sample had been used to test the null hypothesis that the mean of the population is equal to 18 versus the alternative hypothesis that the mean of the population differs from 18, the null hypothesis could be rejected at a level of significance of 0.05.

SCENARIO 9-3 An appliance manufacturer claims to have developed a compact microwave oven that consumes a mean of no more than 250 W. From previous studies, it is believed that power consumption for microwave ovens is normally distributed with a population standard deviation of 15 W. A consumer group has decided to try to discover if the claim appears true. They take a sample of 20 microwave ovens and find that they consume a mean of 257.3 W. -Referring to Scenario 9-3, the consumer group can conclude that there is enough evidence that the manufacturer's claim is not true when allowing for a 5% probability of committing a Type I error.

SCENARIO 9-1 Microsoft Excel was used on a set of data involving the number of defective items found in a random sample of 46 cases of light bulbs produced during a morning shift at a plant. A manager wants to know if the mean number of defective bulbs per case is greater than 20 during the morning shift. She will make her decision using a test with a level of significance of 0.10. The following information was extracted from the Microsoft Excel output for the sample of 46 cases: $n = 46$ ; Arithmetic Mean $= 28.00$ ; Standard Deviation $= 25.92$ ; Standard Error $= 3.82$ ; Null Hypothesis: $H _ { 0 } : \mu \leq 20 ; \alpha = 0.10 ; \mathrm { df } = 45 ; T$ Test Statistic $= 2.09$ ; One-Tail Test Upper Critical Value $= 1.3006 ; p$ -value $= 0.021 ;$ Decision $=$ Reject. -Referring to Scenario 9-1, the manager can conclude that there is sufficient evidence to show that the mean number of defective bulbs per case is greater than 20 during the morning shift with no more than a 1% probability of incorrectly rejecting the true null hypothesis.

SCENARIO 9-7 A major home improvement store conducted its biggest brand recognition campaign in the company's history. A series of new television advertisements featuring well-known entertainers and sports figures were launched. A key metric for the success of television advertisements is the proportion of viewers who "like the ads a lot". A study of 1,189 adults who viewed the ads reported that 230 indicated that they "like the ads a lot." The percentage of a typical television advertisement receiving the "like the ads a lot" score is believed to be 22%. Company officials wanted to know if there is evidence that the series of television advertisements are less successful than the typical ad (i.e. if there is evidence that the population proportion of "like the ads a lot" for the company's ads is less than 0.22) at a 0.01 level of significance. -Referring to Scenario 9-7, state the alternative hypothesis for this study.

SCENARIO 9-11-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 Scenario9-11-DataA.XLSX. 1495 1499 1502 1500 1491 1498 1498 1495 1488 1516 1513 1486 1504 1503 1493 1504 1489 1500 1495 1499 1501 1507 1511 1496 1486 1497 1510 1504 1493 1482 1511 1502 1520 1514 1486 1514 1500 1505 1512 1500 1504 1498 1503 1514 1474 1489 1488 1506 1517 1490 Assume that the number of times the handles have been used to lift the bottles before they break follows a normal distribution. You want to test to see if there is enough evidence that the mean number of times the handles have been used to lift the bottles before they break is more than 1500. -Referring to Scenario 9-11-A, the value of the test statistic is ________.

SCENARIO 9-9 The president of a university claimed that the entering class this year appeared to be larger than the entering class from previous years but their mean SAT score is lower than previous years. He took a sample of 20 of this year's entering students and found that their mean SAT score is 1,501 with a standard deviation of 53. The university's record indicates that the mean SAT score for entering students from previous years is 1,520. He wants to find out if his claim is supported by the evidence at a 5% level of significance. -Referring to Scenario 9-9, the evidence proves beyond a doubt that the mean SAT score of the entering class this year is lower than previous years.

If an economist wishes to determine whether there is evidence that mean family income in a community equals $50,000

SCENARIO 9-11-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 Scenario9-11-DataB.XLSX. 1493 1506 1515 1491 1500 1505 1517 1510 1506 1503 1503 1491 1495 1496 1496 1505 1493 1486 1504 1483 1514 1494 1497 1501 1493 1490 1510 1494 1494 1495 1494 1486 1495 1506 1506 1507 1502 1498 1510 1501 1500 1505 1492 1486 1501 1496 1501 1521 1510 1498 Assume that the number of times the handles have been used to lift the bottles before they break follows a normal distribution. You want to test to see if there is enough evidence that the mean number of times the handles have been used to lift the bottles before they break is more than 1500. -Referring to Scenario 9-11-B, you can conclude that there is not enough evidence that the mean number of times the handles have been used to lift the bottles before they break is more than 1500 when allowing for a 1% probability of committing a Type I error.

The owner of a local nightclub has recently surveyed a random sample of n = 250 customers of the club. She would now like to determine whether or not the mean age of her customers is Greater than 30. If so, she plans to alter the entertainment to appeal to an older crowd. If not, no Entertainment changes will be made. Suppose she found that the sample mean was 30.45 years And the sample standard deviation was 5 years. If she wants to have a level of significance at 0)01, what decision should she make? a) Reject $H _ { 0 }$ . b) Reject $H _ { 1 }$ . c) Do not reject $H _ { 0 }$ . d) We cannot tell what her decision should be from the information given.