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

    Hypothesis Tesitng: T-Test Example

    Example: Eyespot Data

    Many accounts suggest that many species of animals find direct stares from another animal aversive. Some moths have developed eye-spot patterns on their wings or bodys to ward off predators. An experiment was done using 16 moth-eating birds. These birds were tested in a two-chambered box that they were free to roam in from side to side. One chamber had two eye-spot patterns painted on the wall. The other chamber had plain walls. Each bird was left in the chamber for 60 minutes, and the amount of time spent in the plain chamber was recorded.

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

    1. State the Hypotheses: The null hypothesis is that there is no effect for eye-spots painted on the wall. Birds should spend half their time -- 30 minutes -- in each room. The alternative hypothesis is that there is an effect. In symbols the null and alternative hypotheses are:
    2. Set the decision criteria: We arbitrarily decide on an alpha level

      We also note that there are 15 degrees of freedom:

      The table tells us that the critical region consists of t values less than -2.131 and greater than +2.131.

    3. Gather Data:

      The mean is 35 minutes in the plain side, and the sample variance is 81.

    4. Evaluate Null Hypothesis:
      The formula for the estimated standard error is:

      For these data, the estimated standard error is Sqrt(81/16) = 2.25

      The formulat for T is:

      Thus, for these data, T = (35-30)/2.25 = 2.22

      Since 2.22 is in the critical region, we reject the null hypothesis that the presence of eye-spot patterns does not influence behavior.

    ViSta Applet
    Report for the Eyespots Data.

  1. Two more ViSta examples
    1. Z-Test: In-class survey SAT Math and Verbal data (pop mean=460, pop stdv=100). ViSta Data

      Report for the SAT Verbal Variable
      Report for the SAT Math Variable

    2. T-Test: Newcomb Lightspeed ViSta Data

      These data are from the first experiment to determine the speed of light. The best modern measurements correspond to a passage time of 33.02 in this experiment. The population standard deviation is unknown.

      Report for the Newcomb Lightspeed Data