Two Sample Test - Basic Statistics for Sociology - Lecture Slides, Slides of Statistics for Psychologists

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Chapter 9

Hypothesis Testing:

Two Sample Test for Means and

Proportions

Introduction:

• The two sample test is similar to the one sample test, except that we are now testing for differences between two populations rather than a sample and a population. There are three types of two sample tests:
• Hypothesis Testing with Sample Means (Large Samples)
• Hypothesis Testing with Sample Means (Small Samples)
• Hypothesis Testing with Sample Proportions (Large Samples)

Null Hypothesis:

• The H 0 is that the populations are the same.

H 0 : μ 1 = μ 2

• If the difference between the sample statistics is large enough, or, if a difference of this size is unlikely , assuming that the H 0 is true, we will reject the H 0 and conclude there is a difference between the populations.

Null Hypothesis (cont.)

• The H 0 is a statement of “no difference”
• The 0.05 level will continue to be our indicator of a

significant difference

• We change the sample statistics to a Z score, place

the Z score on the sampling distribution and use Appendix A to determine the probability of getting a difference that large if the H 0 is true.

Formula for Hypothesis Testing with Sample

Means (Large Samples)

( ) Χ− Χ = Χ−Χ σ Z^12

Explanation of formula:

• The numerator is the difference in sample means.
• The denominator is the “pooled estimate” of the standard error for both samples.
• The pooled estimate is calculated by using the sample information in the following formula:

Χ 1 − Χ 2

σ Χ − Χ

1 1 2 1

−

−

Χ − Χ = n

s

n

s σ

Example: Hypothesis Testing in the Two Sample Case

• Text P. 244 Problem 9.5 b (Email messages):
  • Middle class families average 8.7 email messages and working class families average 5.7 messages.
  • The middle class families seem to use email more but is the difference significant?

Problem Information:

E-Mail Messages

Sample 1 (M.Class) Sample 2 (W.Class)

= 8.7 = 5. S 1 = 0.3 S 2 = 1. n 1 = 89 n 2 = 55

Χ (^1) Χ 2

Step 2 State the Null Hypothesis

• H 0 : μ 1 = μ 2
  • The Null asserts there is no significant difference between the populations.
• H 1 : μ 1 ≠ μ 2
  • The research hypothesis contradicts the H 0 and asserts there is a significant difference between the populations.

Step 3 Select the Sampling Distribution and Establish the Critical Region

• Sampling Distribution = Z distribution
• Alpha (α) = 0.
• Z (critical) = ± 1.

Step 5 Make a Decision

• The obtained test statistic (Z = 19.74) falls in the Critical Region so reject the null hypothesis.
• The difference between the sample means is so large that we can conclude (at α = 0.05) that a difference exists between the populations represented by the samples.
• The difference between the email usage of middle class and working class families is significant (Z=19.74, α=.05)

Two-tailed Hypothesis Test:

When α = .05, then .025 of the area is distributed on either side of the curve in area (C ) The .95 in the middle section represents no significant difference between the two populations. The cut-off between the middle section and +/- .025 is represented by a Z-value of +/- 1.96.

Z= -1.

c

Z = +1.

c Z=19. I

Significance Vs. Importance

• As long as we work with random samples, we

must conduct a test of significance.

• Significance is not the same thing as

importance.

• Differences that are otherwise trivial or uninteresting may be significant.

Significance Vs. Importance

• When working with large samples, even small

differences may be significant.

• The value of the test statistic (step 4) is an inverse function of n.
• The larger the n, the greater the value of the test statistic, the more likely it will fall in the critical region (region of rejection) and be declared significant.