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

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 alternative hypothesis for this study.

"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

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.

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 numberof tissues used during a cold: $\bar { X } = 52$ , S = 22.Give the null and alternative hypotheses to determine if the number of tissues used during a cold is less than 60. a) $H _ { 0 } : \mu \leq 60$ and $H _ { 1 } : \mu > 60$ . b) $H _ { 0 } : \mu \geq 60$ and $H _ { 1 } : \mu < 60$ . c) $H _ { 0 } : \bar { X } \geq 60$ and $H _ { 1 } : \bar { X } < 60$ . d) $H _ { 0 } : \bar { X } = 52$ and $H _ { 1 } : \bar { X } \neq 52$ .

If a researcher does not reject a true null hypothesis, she has made a decision.

If the Type I error ( $\alpha$ ) for a given test is to be decreased, then for a fixed sample size n

With a test being robust, it does not lose power when the shape of the population distribution departs from a normal distribution.

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 null hypothesis will be rejected at 1%level of significance.

For a given level of significance, if the sample size is increased but the summary statistics remain the same, the probability of committing a Type I error will increase.

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 incorrectly stopping the process if the population mean bulb life is 510 hours is .

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 null hypothesis would be rejected at a significance level of $\alpha$ = 0.01.

Which of the following would be an appropriate null hypothesis?

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 20 versus thealternative hypothesis that the mean of the population differs from 20, the null hypothesis could be rejected at a level of significance of 0.10.

You know that the probability of committing a Type II error ( $\beta$ ) is 5%, you can tell that the power of the test 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 president can conclude that there is sufficient evidence to show that the mean SAT score of the entering class this year is lower than previous years with no more than a 10% probability of incorrectly rejecting the true null hypothesis.

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, if the same sample was used to test the opposite one-tail test, what would be that test's p-value?

In instances in which there is insufficient evidence to reject the null hypothesis, you must make it clear that this has proven that the null hypothesis is true.

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.If she wants to have a level of significance at 0.01, what rejection region should she use? a) Reject $H _ { 0 }$ if $t < - 2.3263$ . b) Reject $H _ { 0 }$ if $t < - 2.5758$ . c) Reject $H _ { 0 }$ if $t > 2.3263$ . d) Reject $H _ { 0 }$ if $t > 2.5758$ . .

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?

The power of a statistical test is