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

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 null hypothesis would be rejected if a 5%probability of committing a Type I error is allowed.

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 9-3 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 appropriate hypotheses to determine if the manufacturer's claim appears reasonable are: a) $H _ { 0 } : \mu = 250$ versus $H _ { 1 } : \mu \neq 250$ b) $H _ { 0 } : \mu \geq 250$ versus $H _ { 1 } : \mu < 250$ c) $H _ { 0 } : \mu \leq 250$ versus $H _ { 1 } : \mu > 250$ d) $H _ { 0 } : \mu \geq 257.3$ versus $H _ { 1 } : \mu < 257.3$

How many tissues should the Kimberly Clark Corporation package of Kleenex contain?Researchers determined that 60 tissues is the mean number of tissues used during a cold.Suppose a random sample of 100 Kleenex users yielded the following data on the number of tissues used during a cold: $\bar { X } = 52$ , S = 22.Suppose the test statistic does fall in the rejection region at $\alpha$ = 0.05.Which of the following decision is correct? a) At $\alpha = 0.05$ , you do not reject $H _ { 0 }$ . b) At $\alpha = 0.05$ , you reject $H _ { 0 }$ . c) At $\alpha = 0.05$ , you do not reject $H _ { 0 }$ . d) At $\alpha = 0.10$ , you do not reject $H _ { 0 }$ .

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 pounds.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 0.05, the engineer can conclude that the mean amount of force necessary to produce cracks in stressed oak furniture is 650.

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 9-7 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 at a 0.01 level of significance. -Referring to Scenario 9-7, state the null hypothesis for this study.

SCENARIO 9-5 A bank tests the null hypothesis that the mean age of the bank's mortgage holders is less than or equal to 45 years, versus an alternative that the mean age is greater than 45 years.They take a sample and calculate a p-value of 0.0202. -Referring to Scenario 9-5, the bank can conclude there is insufficient evidence to conclude that the mean age is greater than 45 years at a significance level of $\alpha$ = 0.01.

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

The test statistic measures how close the computed sample statistic has come to the hypothesized population parameter.

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, what critical value should the president use to determine the rejection region?

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 shewants to have a level of significance at 0.01, what decision should she make?

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 highest level of significance at which the null hypothesis cannot be rejected is .

The larger the p-value, the more likely you are to reject the 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 at a 0.01 level of significance. -Referring to Scenario 9-7, the null hypothesis will be rejected if the test statistic is

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 parameter of interest is

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.

The symbol for the level of significance of a statistical test 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 0.05, the probability of the operations manager not stopping the process when the process is in fact working properly is in fact below 500 hours 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 .