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Week 9:Dependent t-test

Paired Samples t- test for two dependent samples

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Dependent Samples When have two dependent or related

samples. • Same group measured twice (Time 1 vs.

Time 2; Pretest and Posttest).• Samples are matched on some variable. Each score in one sample is paired with

a specific score in the other sample. Such data are correlated data.

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Examples of Research Questions: Is there a significant difference

students’ mathematics achievement when taught through traditional methods and hands-on problem-solving method?

IV = method taught (values = traditional [baseline], hands-on problem-solving)

DV = mathematics achievement (score, continuous)

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Examples of Research Questions:

Is there a significant difference in morbidly obese students’ pre-exercise weight and post-exercise weight? Rather than comparing the means of the

pre and post, we compare the pre and post scores for each individual.

IV: Time (pre or post)DV: Weight (Value = pounds, continuous)

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An investigator for NASA examines the effect of cabin temperature on reaction time. A random sample of 10 astronauts and pilots is selected. Each person’s reaction time to an emergency light is measured in a simulator where the cabin temperature is maintained at 70 degrees F and again the next day at 95 degrees F.

IV: Temperature (values = 70F or 95F)DV: Reaction Time (Value = seconds,

continuous)

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Is there a significant difference between husband and wife’s annual income?

IV: Spouse (values = husband, wife)DV: Annual income (Value = dollars, continuous)

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Steps in hypothesis testing:

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Step 1:State the hypotheses

Null hypothesis: or Ho: µD ≥ 0 or Ho: µD ≤ 0

Alternative hypothesis: or or

* Subscript D indicates difference.

01: H D01: H D

01: H D

00: H D

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Step 2: Set Criterion for Rejecting HO

1) Compute degrees of freedomdf = n – 1 whereby n = number of

pairs2) Set alpha level3) Locate critical value(s)

Table C. 3 (page 638 of text) – same as in an Independent t - test

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Step 3: Compute test statistic

Whereby:D = after-before

= Sample Standard Deviation

of difference (D) scores, divided

by

S D

Dt

xx 12

S D

n

S D

n

DD

Sum of individual differences

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Example Computation:

Before After D = after - before

5 6 1 8 9 1 4 5 1 3 6 3 7 7 0 8 10 2

S D

Dt

8203111D

42.6

03.1

nSS D

D

09.342.

3.1

S D

Dt

3.16

8

n

DD Standard

deviation of the differences

Number of pairs

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Step 4: Compare Test Statistic to Criterion

Use t distribution in the appendix to find the critical values (given alpha level, df, and directionality of the test).

In this example, df = n-1= 6-1 = 5

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Step 5: Make decision Use t distribution in the

appendix to find the critical values (given alpha level, df, and directionality of the test).

The graph on the right shows an example of two-tailed test with the c.v. equal to ± 2.776.

For our example, use Table C.3 on page 638 to find out the critical value(s). With alpha = 0.05 and df = 5, the critical values are ± 2.571 (two-tailed test).