Question

7.48 The Graded Naming Test and sociocultural differences: Researchers often use z tests to compare their samples to known population norms. The Graded Naming Test (GNT) asks respondents to name objects in a set of 30 black-and-white drawings. The test, often used to detect brain damage, starts with easy words like kangaroo and gets progressively more difficult, ending with words like sextant. The GNT population norm for adults in England is 20.4. Roberts (2003) wondered whether a sample of Canadian adults had different scores than adults in England. If they were different, the English norms would not be valid for use in Canada. The mean for 30 Canadian adults was 17.5. For the purposes of this exercise, assume that the standard deviation of the adults in England is 3.2. Conduct all six steps of a z test. Be sure to label all six steps. Some words on the GNT are more commonly used in England. For example, a mitre, the headpiece worn by bishops, is worn by the archbishop of Canterbury in public ceremonies in England. No Canadian participant correctly responded to this item, whereas 55% of English adults correctly responded. Explain why we should be cautious about applying norms to people different from those on whom the test was normed. When we conduct a one-tailed test instead of a two-tailed test, there are small changes in steps 2 and 4 of hypothesis testing. (Note: For this example, assume that those from populations other than the one on which it was normed will score lower, on average. That is, hypothesize that the Canadians will have a lower mean.) Conduct steps 2, 4, and 6 of hypothesis testing for a one-tailed test. Under which circumstance—a one-tailed or a two-tailed test—is it easier to reject the null hypothesis? Explain.

Answer 1

A)

Here, in this example, we are applying the study on a sample that is different from the population. The sample is from Canada whereas the population is from England. This accounts for difference in their culture, traditions etc.

B)

2) Specify the desired level of significance and the sample size

Suppose that a = 0.05 and n = 30 are chosen for this test

4) Determine the critical values

For a = 0.05 the critical Z values are -1.645 (because its a lower tailed test as we need to test if Canadians have a lower mean)

6) Is the test statistic in the rejection region?

Z=x-//​n=17.5-20.4/3.5/30=-4.54. Hence reject the null hypothesis.

C)

One-tailed tests make it easier to reject the null hypothesis when the alternative is true. A large sample, two-sided, 0.05 level t test puts a probability of 0.025 in each tail. It needs a t statistic of less than -1.96 to reject the null hypothesis of no difference in means. A one-sided test puts all of the probability into a single tail. It rejects the hypothesis for values of t less than -1.645.