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

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, 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.

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, the value of $\beta$ is 0.90.

The symbol for the power of a statistical test is a) $\alpha$ . b) $1 - \alpha$ . c) $\beta$ . d) $1 - \beta$ .

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, state the null hypothesis for this study.

SCENARIO 9-4 A drug company is considering marketing a new local anesthetic. The effective time of the anesthetic the drug company is currently producing has a normal distribution with a mean of 7.4 minutes with a standard deviation of 1.2 minutes. The chemistry of the new anesthetic is such that the effective time should be normally distributed with the same standard deviation, but the mean effective time may be lower. If it is lower, the drug company will market the new anesthetic; otherwise, they will continue to produce the older one. A sample of size 36 results in a sample mean of 7.1. A hypothesis test will be done to help make the decision. -Referring to Scenario 9-4, the value of the test statistic is ________.

SCENARIO 9-10 A manufacturer produces light bulbs that have a mean life of at least 500 hours when the production process is working properly. Based on past experience, the population standard deviation is 50 hours and the light bulb life is normally distributed. The operations manager stops the production process if there is evidence that the population mean light bulb life is below 500 hours. -Referring to Scenario 9-10, if you select a sample of 100 light bulbs and are willing to have a level of significance of 0.01, the probability of the operations manager not stopping the process if the population mean bulb life is 510 hours 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. -Referring to Scenario 9-1, the lowest level of significance at which the null hypothesis can be rejected is ______.

SCENARIO 9-6 The quality control engineer for a furniture manufacturer is interested in the mean amount of force necessary to produce cracks in stressed oak furniture. She performs a two-tail test of the null hypothesis that the mean for the stressed oak furniture is 650. The calculated value of the Z test statistic is a positive number that leads to a p-value of 0.080 for the test. -Referring to Scenario 9-6, suppose the engineer had decided that the alternative hypothesis to test was that the mean was greater than 650. What would be the p-value of this one-tail test?

A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim against the alternative that the actual proportion of doctors who recommend aspirin Is less than 0.90, a random sample of 100 doctors was selected. Suppose you reject the null Hypothesis. What conclusion can you reach?

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, which of the following best describes the Type II error?

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 no more than 1500 when allowing for a 1% probability of committing a Type I error.

The value that separates a rejection region from a non-rejection region is called the _______.

SCENARIO 9-6 The quality control engineer for a furniture manufacturer is interested in the mean amount of force necessary to produce cracks in stressed oak furniture. She performs a two-tail test of the null hypothesis that the mean for the stressed oak furniture is 650. The calculated value of the Z test statistic is a positive number that leads to a p-value of 0.080 for the test. -Referring to Scenario 9-6, if the test is performed with a level of significance of 0.10, the null hypothesis would be rejected.

SCENARIO 9-10 A manufacturer produces light bulbs that have a mean life of at least 500 hours when the production process is working properly. Based on past experience, the population standard deviation is 50 hours and the light bulb life is normally distributed. The operations manager stops the production process if there is evidence that the population mean light bulb life is below 500 hours. -Referring to Scenario 9-10, if you select a sample of 100 light bulbs and are willing to have a level of significance of 0.01, the probability of the operations manager failing to stop the process if the population mean bulb life is 490 hours is _____.

If a researcher rejects a true null hypothesis, she has made a _______error.

SCENARIO 9-2 A student claims that he can correctly identify whether a person is a business major or an agriculture major by the way the person dresses. Suppose in actuality that if someone is a business major, he can correctly identify that person as a business major 87% of the time. When a person is an agriculture major, the student will incorrectly identify that person as a business major 16% of the time. Presented with one person and asked to identify the major of this person (who is either a business or an agriculture major), he considers this to be a hypothesis test with the null hypothesis being that the person is a business major and the alternative that the person is an agriculture major. -Referring to Scenario 9-2, what is the value of $\alpha$ ?

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 evidence proves beyond a doubt that the mean number of defective bulbs per case is greater than 20 during the morning shift.

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, 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 no more than 1500 when allowing for a 1% probability of committing a Type I error.

A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim against the alternative that the actual proportion of doctors who recommend aspirin Is less than 0.90, a random sample of 100 doctors was selected. Suppose that the test statistic is - 2)20. Can you conclude that H0 should be rejected at the (a) $\alpha = 0.10 , \text { (b) } \alpha = 0.05 \text {, and (c) } \alpha$ = 0)01 level of Type I error?

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, the parameter of interest is