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Session__10

# Session__10 - Review of Paired T-tests A paired t-test is...

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Review of Paired T-tests

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A paired t-test is used when we have observations for the same group of people at two time points. Each person gets a “difference score.” Observation Observation Time 1 Time 1 Time 2 Time 2 Difference Difference 1 1 x x 11 11 x x 21 21 X X d1 d1 =X =X 11 11 -X -X 21 21 2 2 x x 12 12 x x 22 i i x x 1i 1i x x 2i 2i n n x x 1n 1n x x 2n 2n X X d2 d2 =X =X 12 12 -X -X 22 22 X X di di =X =X 1i 1i -X -X 2i 2i X X dn dn =X =X 1n 1n -X -X 2n 2n
The null hypothesis is: H 0 : There is no difference between the means. H 0 : µ1 = µ2// µ D = 0 (D i = X 1 i - X 2 i ) The alternative hypothesis is: H 1 : There is a difference between the means. H 1 : µ1 ≠ µ2// µ D ≠ 0 ( D i = X 1 i - X 2 i ) We use the paired t-test to see if the means are different between scores at two different points in time.

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Note: Research Questions Research Questions Hypothesis Hypothesis No Difference No Difference Any Difference Any Difference Pop 1 Pop 1 Pop 2 Pop 2 Pop 1 < Pop 2 Pop 1 < Pop 2 Pop 1 Pop 1 Pop 2 Pop 2 Pop 1 > Pop 2 Pop 1 > Pop 2 H H 0 μ μ D = 0 = 0 μ μ D 0 0 μ μ D 0 0 H H 1 μ μ D 0 0 μ μ D < 0 < 0 μ μ D > 0 > 0 Be careful with identifying hypotheses for directional tests. You believe that μ 2 has the higher mean. You believe that μ 1 has the higher mean.
Once again, the paired t-test has certain assumptions. 1. Sample of paired observations is randomly drawn. 2. Both sets are normally distributed with same variance. 3. The established pairs of observations are correlated.

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Formulae for paired t-test 1 2 1 2 d d X X X X X X t s s - - = = SE of Mean Difference d d X X s s n = 1 df v n = = - 2 SD of Difference 1 d d X X S n = - Because we only have one group of people, the degrees of freedom is n -1.
Example 1 You work in Human Resources. You want to see if there is a difference in test scores following a new training program. You collect the following test score data: Name Before (1) After (2) Sam 85 94 Tamika 94 87 Brian 78 79 Mike 87 88 Using a two-tailed test with alpha = 0.05, was the training effective?

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1. State the null and alternative hypotheses.
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