Question

An investigator thinks that people under the age of forty have vocabularies that are different than those of people over sixty years of age. The investigator administers a vocabulary test to a group of 31 younger subjects and to a group of 31 older subjects. Higher scores reflect better performance. The mean score for younger subjects was 14.0 and the standard deviation of younger subject's scores was 5.0. The mean score for older subjects was 20.0 and the standard deviation of older subject's scores was 6.0. Does this experiment provide evidence for the investigator's theory?

a) Provide, in words, the null and alternative hypotheses. (4marks)
b) Provide the decision rule for rejecting the null hypothesis, including the critical value(s) for the appropriate statistic. (6marks)
c) Using an alpha level of .05, test the null hypothesis. As part of this test, please compare the actual value for the appropriate statistic against the critical value(s) for the appropriate statistic. (5marks)
d) State your conclusion regarding the results from this test in language that a friend of yours with no knowledge of statistics could understand.

Answer 1

In this experiment, the investigator tests the theory that people under the age of forty have vocabularies different from those over sixty years of age. The process involves setting up null and alternative hypotheses, establishing a decision rule, and conducting a hypothesis test using the appropriate statistical test and an alpha level of 0.05. Based on the test results, a conclusion is drawn regarding the investigator's theory.

Step-by-step explanation:

a) Null Hypothesis: The vocabularies of people under the age of forty are the same as those of people over sixty years of age.

Alternative Hypothesis: The vocabularies of people under the age of forty are different from those of people over sixty years of age.

b) Decision Rule: To reject the null hypothesis, we will compare the test statistic, in this case, the difference in means, against the critical value(s) for the appropriate statistic. For this experiment, we can use a t-test to compare the means of the two groups.

c) Test the Null Hypothesis: Using an alpha level of 0.05, we perform a two-sample t-test to compare the mean scores of the younger subjects and the older subjects. The calculated test statistic is (mean of younger group - mean of older group) / (standard deviation of the difference in means). We compare this value to the critical value obtained from the t-distribution table or a statistical software.

d) Conclusion: Based on the results of the t-test, we compare the calculated test statistic to the critical value. If the calculated test statistic falls in the rejection region, we reject the null hypothesis. In this case, if the calculated test statistic falls outside the critical value, we have evidence to support the investigator's theory that the vocabularies of people under the age of forty are different from those of people over sixty years of age.

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