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Notes on Topic 8:
Hypothesis Testing

Directional (one-tailed) Techniques


    We wish to answer the same basic question that was asked with two-tailed testing:

    "How likely are we to get this sample if the null hypothesis is true?"
    but we test a directional null hypothesis, as stated below:

    We have the same basic four steps of hypothesis testing:

    1. State the Hypotheses:
      • Null hypothesis: Alcohol consumption does not decrease birth weight. Their weight will not be less than 18 grams (i.e., it will be equal to or greater than 18 grams). In symbols:
      • Alternative Hypothesis: Alcohol will decrease birth weight. The weight will be less than 18 grams. In symbols:
    2. Define the decision method:
      • (Classic Approach: Define Decision Criterion)

      • Determine the standard error of the mean (standard deviation of the distribution of sample means) for samples of size 16. The standard error is calculated by the formula:

        The value is 4/sqrt(16) = 1.

      • 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. All we need to calculate Z is a sample mean.

        When we get the data we will calculate Z and then look it up in the Z table to see how unusual the obtained sample's mean is, if the null hypothesis Ho is true, using a one-tailed probabililty.

    3. Gather Data:
      The two experimenters got these different sets of data:

      Experiment 1 Experiment 2
      Sample Mean = 13 Sample Mean = 16.5

    4. Evaluate Null Hypothesis:
      We calculate Z for each experiment, and then look up the P value for the obtained Z, and make a decision. Here's what happens for each experiment:
      Experiment 1 Experiment 2
      Sample Mean = 13
      Z = (13-18)/1 = -5.0
      p < .0000
      Reject Ho
      ViSta Applet
      Sample Mean = 16.5
      Z = (16.5-18)/1 = -1.5
      p = .0670
      Do Not Reject Ho
      ViSta Applet

      ViSta's Report for Univariate Analysis of Experiment 1 Data.

      ViSta's Report for Univariate Analysis of Experiment 2 Data.