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

If the p-value is less than α in a two-tailed test

$\text { TABLE 9-1 }$ Microsoft Excel was used on a set of data involving the number of parasites found on 46 Monarch butterflies captured in Pismo Beach State Park. A biologist wantsto know if the mean number of parasites per butterfly is over 20 . She will make her decision usinga test witha level of significance of $0.10$ . The following informationwas extracted from the Microsoft Excel outputfor the sample of 46 Monarchbutterflies: n=46; Arithmetic Mean =28.00; Standard Deviation =25.92; Standard Error =3.82; Null Hypothesis: :\mu\leq20.000;\alpha=0.10;df=45;T Test Statistic =2.09; One-Tailed Test Upper Critical Value =1.3006; p-value =0.021; Decision = Reject. -Referring to Table 9-1, the biologist can conclude that there is sufficient evidence to show that the average number of parasites per Monarch butterfly in Pismo Beach State Park is over 20 with no more than a 5% probability of incorrectly rejecting the true null hypothesis.

In testing a hypothesis, statements for the null and alternative hypotheses as well as the selection of the level of significance should precede the collection and examination of the data.

TABLE 9-8 One of the biggest issues facing e-retailers is the ability to turn browsers into buyers. This is measured by the conversion rate, the percentage of browsers who buy something in their visit to a site. The conversion rate for a company's web site was 10.1% The web site at the company was redesigned in an attempt to increase its conversion rates. Samples of 200 browsers at the redesigned site were selected. Suppose that 24 browsers made a purchase. The company officials would like to know if there is evidence of an increase in conversion rate at the 5% level of significance. -Referring to Table 9-8, what critical value should the company officials use to determine the rejection region?

TABLE 9- 1 Microsoft Excel was used on a set of data involving the number of parasites found on 46 Monarch butterflies captured in Pismo Beach State Park. A biologist wants to know if the mean number of parasites per butterfly is over 20. 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 Monarch butterflies: n = 46; Arithmetic Mean = 28.00; Standard Deviation = 25.92; Standard Error = 3.82; Null Hypothesis: H₀: µ c 20.000; a = 0.10; df = 45; T Test Statistic = 2.09; One- Tailed Test Upper Critical Value = 1.3006; p- value = 0.021; Decision = Reject. -Referring to Table 9-1, the evidence proves beyond a doubt that the mean number of parasites on butterflies in Pismo Beach State Park is over 20.

It is not considered an unethical practice to not allow potential respondents be permitted to self-select for a study or be purposely selected.

A major videocassette rental 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 are equipped with videocassette recorders (VCRs). It conducts a telephone poll of 300 randomly selected households in the area and finds that 96 have VCRs. The rental chain's conclusion from the hypothesis test using a 3% level of significance is

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 we conclude that H₀ should be rejected at the (a) α = 0.10, (b) α = 0.05, and (c) α = 0.01 level of Type I error?

TABLE 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 an average 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 normal 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 Table 9-4, what is the power of the test if the average effective time of the anesthetic is 7.0 using a 0.10 level of significance?

Which of the following would be an appropriate alternative hypothesis?

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 over 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. What is the p-value associated with the test statistic?

TABLE 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 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 Table 9-2, what is the value of ??

TABLE 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-tailed 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 Table 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.

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 over 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 be 99% confident in her decision, what decision should she make?

The larger is the p-value, the more likely one is to reject the null hypothesis.

TABLE 9- 1 Microsoft Excel was used on a set of data involving the number of parasites found on 46 Monarch butterflies captured in Pismo Beach State Park. A biologist wants to know if the mean number of parasites per butterfly is over 20. 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 Monarch butterflies: n=46; Arithmetic Mean =28.00; Standard Deviation =25.92; Standard Error =3.82; Null Hypothesis: :\mu\leq20.000;\alpha=0.10;df=45;T Test Statistic =2.09; One-Tailed Test Upper Critical Value =1.3006; p-value =0.021; Decision = Reject. -Referring to Table 9-1, the power of the test is ______if the mean number of parasites per butterfly on Monarch butterflies in Pismo Beach State Park is 18 using a 0.1 level of significance and assuming that the population standard deviation is 25.92.

TABLE 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 an average 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 normal 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 Table 9-4, what is the power of the test if the average effective time of the anesthetic is 7.5 using a 0.10 level of significance?

TABLE 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 an average 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 normal 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 Table 9-4, the appropriate hypotheses are

TABLE 9-3 An appliance manufacturer claims to have developed a compact microwave oven that consumes an average of no more than 250 W. From previous studies, it is believed that power consumption for microwave ovens is normally distributed with a 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 an average of 257.3 W. -Referring to Table 9-3, what is the power of the test if the average power consumption of all such microwave ovens is in fact 248 W using a 0.05 level of significance?

The power of a statistical test is