Example problem using the Z-test in the process to test statistical hypotheses for a research problem

Research Problem: We randomly select a group of 9 subjects from a population with a mean IQ of 100 and standard deviation of 15 (, ).
We give the subjects intensive "Get Smart" training and then administer an IQ test. The sample mean IQ is 113 and the sample standard deviation is 10. Did the training result in a significant increase in IQ score?

The research question for this experiment is - Does training subjects with the Get Smart training program, increase their IQ significantly over the average IQ for the general population? We will use the six step process to test statistical hypotheses for this research problem.

  1. State null hypothesis and alternative hypothesis:

  2. Set the alpha level:

  3. Calculate the value of the proper statistic:

    Since this problem involves comparing a single group's mean with the population mean and the standard deviation for the population is known, the proper statistical test to use is the Z-test.

    Z = 2.6

  4. State the rule for rejecting the null hypothesis:

    We need to find the value of Z that will only be exceeded 5% of the time since we have set our alpha level at .05. Since the Z score is normally distributed (or has the Z distribution), we can find this 5% level by looking at the table in Appendix A in the textbook. We look for .45 in column 2 (area from the mean to Z) since that point would have 5% of the scores at or higher than it. The associated Z-score would be 1.64 (or 1.65).

    Our rejection rule then would be: Reject H0 if .

  5. Decision: Reject H0, p < .05, one-tailed.

    Our decision rule said reject H0 if the Z value is equal to or greater than 1.64. Our Z value was 2.6 and 2.6 is greater than 1.64 so we reject H0. We also add to the decision the alpha level (p < .05) and the tailedness of the test (one-tailed).

  6. Statement of results: The average IQ of the group taking the Get Smart training program is significantly higher than that of the general population.

    If we reject the null hypothesis, we accept the alternative hypothesis. The statement of results then states the alternative hypothesis which is the research question stated in the affirmative manner.

We mentioned that we use the Z-test to compare the mean of a sample with the population mean when the population standard deviation is known. We will now turn to the statistic to use when the standard deviation of the population is not know, the one-sample t-test.

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