Ed230A

Introduction to Research Design and Statistics

Testing Means (Normal Distribution)


The Standard Normal Distribution

The normal distribution is used to test differences in means when samples are large (n > 30) or when the population variance is known. The test itself is known as a Z-test.

Z-test for Single Samples

Sampling Distribution of a Mean

Hypotheses

  • 2-tail -    H0: μ1 = value          H1: μ1 ≠ value
  • 1-tail -    H0: μ1 <= value        H1: μ1 > value
  • 1-tail -    H0: μ1 >= value        H1: μ1 < value

    Example

    The mean of a sample of 49 seniors enrolled in a vocational education program was 53.4 on a measure of mechanical ability. The state norms for this test has a mean of 50 and a variance of 196.

    The principal of vocational school wants to test the hypothesis that the seniors from her school are superior in mechanical ability to seniors in general throughout the state.

    Would the results have been significant if run as a two-tail test?

    Z-test for Independent Groups (Between-Subjects)

    Sampling Distribution of the Difference between Means

    Hypotheses

  • 2-tail -    H0: μ1 = μ2          H1: μ1 ≠ μ2
  • 1-tail -    H0: μ1 <= μ2        H1: μ1 > μ2
  • 1-tail -    H0: μ1 >= μ2        H1: μ1 < μ2

    Z-test Formulas

  • Standard error of differences between means

  • Z-test

    Looking at two groups

    In this section we will use the zdemo2 (available from ATS) command to look at several different combinations of means and standard deviation for two groups. The first example will have different means, 50 and 60, but equal standard deviations, sd = 10.

    zdemo2 50 10 60 10

    Next, there means will stay the same, 50 and 60, but the standard deviation of the second group will be increased to 20.

    zdemo2 50 10 60 20

    Finally, the both means will be set to 50 while the standard deviations will remain at 10 and 20.

    zdemo2 50 10 50 20

    Z-test Example

    In a study of the effects of special class placement on the achievement of educably mentally retarded (EMR) children, 72 EMR children were randomly assigned to two groups: 36 to special education classes and 36 to regular mainstream classes.

    The following results were obtained:

    Special Ed
    Group
    Regular Ed
    Group
    n3636
    Mean50.447
    Variance2525

    Is there a significant difference between the two groups?


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    Phil Ender, 30Jun98