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Notes on Topic 9:
Z-Tests and T-Tests:
One Sample Hypothesis Tests

    Hypothesis Testing: Z-Test

    Example Experiment

    Prenatal exposure to alcohol on birthweight in rats. The researcher's sample has n=16 rat pups. We assume that

    1. The population of birthweights is normally distributed.
    2. The population has a mean birthweight of 18 grams.
    3. The population has a standard deviation of 4 grams.

    Here are the four steps involved in the statistical hypothesis test:

    1. State the Hypotheses:
      • Null hypothesis: No effect for alcohol consumption on birth weight. Their weight will be 18 grams. In symbols:
      • Alternative Hypothesis: Alcohol will effect birth weight. The weight will not be 18 grams. In symbols:
    2. (Classical Approach)
      Set the decision criteria:
      • Specify alpha, the significance level. We specify:
      • Determine the critical value of Z. We do this for the choosen significance level. For a non-directional test the critical value of Z is the value that has alpha-percent of the area more extreme than Z. For alpha=.05 we look up a Z that has .025 of the distribution beyond it. This is a Z of +1.96 and -1.96.

      (Contemporary Approach)
      The computer calculates the exact probability of the result of the experiment.

    3. Gather Data:
      Two experimenters carried out the experiment. They got the following two samples of results:
      Experiment 1 Experiment 2
      Sample Mean = 13 Sample Mean = 16.5

    4. Evaluate Null Hypothesis:
      For each experiment we calculate the standard error of the mean, then calculate Z for each experiment. Classically, we would then look up the P value for the obtained Z. Contemporary practice is to have the computer determine the P for the obtained Z. Then we make a decision.
      • Determine the standard error of the mean. The standard error is calculated by the formula:

        For these data the value is 4/sqrt(16) = 1.

      • Calculate the Test-Statistic. To determine how unusual the mean of the sample we will get is, we will use the Z formula to calculate Z for our sample mean under the assumption that the null hypothesis is true. The Z formula is:

        Note that the population mean is 18 under the null hypothesis, and the standard error is 1, as we just calculated. We then can calculate Z by using the obtained sample mean.

      • Make a Decision:
        • Classical Approach: We look the obtained Z up in the Z table to find P, and compare it to the Critical Z value. If the obtained P is less than alpha, we reject the null hypothesis.
        • Contemporary Approach: The computer calculates the P value. We report it and let the reader/listener decide.

          The P value that indicates how unusual the obtained sample's mean is, under the null hypothesis of no effect. In other words, it indicates how often we would obtain the results by chance alone. Using this P value we decide whether to reject or retain the null hypothesis.

          Here's what happens for each experiment:

          Experiment 1 Experiment 2
          Sample Mean = 13
          Z = (13-18)/1 = -5.0
          p < .0000
          Reject Ho
          Sample Mean = 16.5
          Z = (16.5-18)/1 = -1.5
          p = .1339
          Retain Ho

          Here is ViSta's report for these two experiments:

          ViSta Applet
          Report for Univariate Analysis of Experiment 1 Data.

          ViSta Applet
          Report for Univariate Analysis of Experiment 2 Data.

    Next Topic: The T-Test