Z-tests

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.

H₀: μ₁ <= 50 H₁: μ₁ > 50

First, compute the standard deviation from the variance:
σ = SQRT(σ²) = SQRT(196) = 14

Next, compute the standard error of the mean
se = σ/SQRT(n) = 14/SQRT(49) = 14/7 = 2

Compute the Z-test:
Z = (Xbar - μ)/se = (53.4 - 50)/2 = +1.7

The critical value for Z at alpha = 0.05 is +1.645

Thus, the principal concluded that the seniors at her school scored significantly higher that the state average.

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
n 36 36
Mean 50.4 47
Variance 25 25

	Special Ed Group	Regular Ed Group
n	36	36
Mean	50.4	47
Variance	25	25

Is there a significant difference between the two groups?

H₀: μ₁ = μ₂ H₁: μ₁ ≠ μ₂

First, compute the standard deviations from the variance:
σ = SQRT(σ²) = SQRT(25) = 5 for each group

Next, computer the standard errors for each group:
se = σ/SQRT(n) = 5/SQRT(36) = 5/6 = 0.833 for each group

Compute the standard error of the differences in means:
SQRT(se²₁ + se²₂) = SQRT(.833² + .833²) = 1.18

Compute the Z-test:
Z = (mean₁ - mean₂)/se_dif = (50.4 - 47)/1.18 = +2.88

The critical value of Z is ±2.58

Thus, it is concluded that the children in the special education classes scored higher than those in the regular classes and that the difference was significant at alpha = 0.01. That is, the samples come from two different populations which have different population means.

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Phil Ender, 30Jun98