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Paired-Sample Z-Test: A Statistical Method for Comparing Two Related Samples, Schemes and Mind Maps of Experimental Design

University of Wales, SwanseaExperimental Design

The paired-sample z-test is a statistical method used to compare the means of two related samples, where each subject in one group is paired with a subject in the other group. This test assumes that the sample forms two treatment groups, the null hypothesis states that the average of the pair differences is not significantly different than zero, and the population distribution is arbitrary. The test involves computing the test statistic, determining the rejection region, and calculating the p-value. A step-by-step guide to performing the paired-sample z-test, including an example problem.

Typology: Schemes and Mind Maps

2021/2022

Uploaded on 09/27/2022

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Paired-sample z-test
1. Assumptions
•Experimental Design: The sample forms two treatment groups,
where each subject in one group is paired with a subject in the other
group.
•Null Hypothesis: The average of the pair differences is not signifi-
cantly different than zero.
•Population Distribution: Arbitrary.
•Sample Size: The sample size of each treatment group is greater than
or equal to 30.
2. Inputs for the paired-sample z-test
•Sample size: n
•Sample mean: ¯
dof the differences, where di=x1i−x2i,
x1i∈Treatment Group 1; x2i∈Treatment Group 2.
•Sample standard deviation: sd(SD of the differences)
•Standard error of mean: SEmean =sd
√n
•Null hypothesis value: 0
•The level of the test: α
3. Five Steps for Performing the Test of Hypothesis
1. State null and alternative hypotheses:
H0:µ= 0, H1:µ6= 0
2. Compute test statistic:
z=¯
d−µ
SEmean
,
assuming the null hypothesis value 0 for µ.
1
pf3

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Download Paired-Sample Z-Test: A Statistical Method for Comparing Two Related Samples and more Schemes and Mind Maps Experimental Design in PDF only on Docsity!

Paired-sample z-test

1. Assumptions

  • Experimental Design: The sample forms two treatment groups, where each subject in one group is paired with a subject in the other group.
  • Null Hypothesis: The average of the pair differences is not signifi- cantly different than zero.
  • Population Distribution: Arbitrary.
  • Sample Size: The sample size of each treatment group is greater than or equal to 30.

2. Inputs for the paired-sample z-test

  • Sample size: n
  • Sample mean: d¯ of the differences, where di = x 1 i − x 2 i, x 1 i ∈ Treatment Group 1; x 2 i ∈ Treatment Group 2.
  • Sample standard deviation: sd (SD of the differences)
  • Standard error of mean: SEmean =

sd √ n

  • Null hypothesis value: 0
  • The level of the test: α

3. Five Steps for Performing the Test of Hypothesis

  1. State null and alternative hypotheses:

H 0 : μ = 0, H 1 : μ 6 = 0

  1. Compute test statistic:

z =

d¯ − μ SEmean

assuming the null hypothesis value 0 for μ.

  1. Compute 100(1 − α)% confidence interval I for z.
  2. If z ∈ I, accept H 0 ; if z /∈ I, reject H 1 and accept H 0.
  3. Compute p-value.

4. Discussion

For the difference variable di = x 1 i − x 2 i,, the central limit theorem insures that d¯ ∼ N (μx, σ^2 x/

n). The paired two-sample z-test reduces to a one- sample z-test on the differences di.

4. A Sample Problem

Freedman, Pisani, and Purves, p. p. 476: A legislative committee wants to see if there is a significance difference in tax revenue between the proposed new tax law and the existing tax law. The committee has a staff member randomly choose 100 representative tax returns. For the ith return, it com- putes the tax x 2 i using the proposed new tax law, and compares it to the tax x 1 i paid under the existing law. The staff member then computes the differences di = x 2 i − x 1 i and tests whether there is a significant difference between the proposed new law and the existing law. Here are the summary statistics:

n = 100 d¯ = − 219 sd = 725 c = 0 α = 0. 05

SEmean =

sx √ n

The five steps of the z-test:

  1. State the null and alternative hypotheses:

H 0 = 0, H 1 6 = 0

  1. Compute the test statistic:

z =

x¯ − μ SEmean