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The t Test for Two Independent Samples PowerPoint PPT Presentation

The t Test for Two Independent Samples Compare means of two groups Experimental—treatment versus control Existing groups—males versus females Notation—subscripts indicate group M 1 , s 1 , n 1 M 2 , s 2 , n 2 Null and alternative hypotheses

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The t Test for Two Independent Samples

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The t test for two independent samples l.jpg

The t Test for Two Independent Samples

  • Compare means of two groups

    • Experimental—treatment versus control

    • Existing groups—males versus females

  • Notation—subscripts indicate group

    • M1, s1, n1 M2, s2, n2

  • Null and alternative hypotheses

    • translates into

    • translates into


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  • Criteria for use

    • Dependent variable is quantitative, interval/ratio

    • Independent variable between-subjects

    • Independent variable has two levels

  • t-test

    • Basic form

    • One sample


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Two sample

  • Difference between sample means M1 - M2

    • Population parameter

  • Sampling distribution of the difference

    • Difference between M1 and M2 drawn from population


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Standard error of the difference

  • Population variance known

    • Sum of

  • Estimate from samples

  • Differences more variable than scores


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Variability of mean differences

  • Randomly generated set of 1000 means

    • Μ= 50, σM = 10

    • Take difference between pairs


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S2pooled Pooled Variance

  • Homogeneity of variance

    • Assume two samples come from populations with equal σ2’s

    • Two estimates of σ2 — and

  • Weighted average


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df = df1 + df2 = (n1-1) + (n2-1) = n1 + n2 - 2

t-test


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Hypothesis testing

  • Two-tailed

    • H0: µ1 = µ2, µ1 - µ2 = 0

    • H1: µ1 ≠ µ2, µ1 - µ2 ≠ 0

  • One-tailed

    • H0: µ1 ≥ µ2, µ1 - µ2 ≥ 0

    • H1: µ1 < µ2, µ1 - µ2 < 0

  • Determine α

  • Critical value of t

    • df = n1 + n2 - 2


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Assumptions

  • Random and independent samples

  • Normality

  • Homogeneity of variance

    • SPSS—test for equality of variances, unequal variances t test

    • t-test is robust


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H0: µ1 = µ2, µ1 - µ2 = 0

H1: µ1 ≠ µ2, µ1 - µ2 ≠ 0

df = n1 + n2 - 2 =10 + 7 – 2 = 15

=.05

t(15) = 2.131

Example 1


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  • t(15) = –2.325, p < .05 (precise p = 0.0345)


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df = n1 + n2 - 2 = 15 + 15 – 2 = 28

=.05, t(28) = 2.049

Example 2


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  • t(28)= –.947, p > .05


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Confidence Interval for the Difference

  • Example 1

    • -3.257 - (2.131*1.401) < µ1 - µ2 < -3.257 + (2.131*1.401) = -6.243 < µ1 - µ2 < -0.272

  • Example 2

    • -0.867 - (1.701*5.221) < µ1 - µ2 < -0.867 + (1.701*5.221) = -9.748 < µ1 - µ2 < 8.014

  • Includes 0 retain H0


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SPSS

  • Analyze

    • Compare Means

      • Independent-Samples T Test

  • Dependent variable(s)—Test Variable(s)

  • Independent variable—Grouping Variable

    • Define Groups

    • Cut point value

  • Output

    • Levene’s Test for Equality of Variances

    • t Tests

      • Equal variances assumed

      • Equal variances not assumed


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Output Example 1


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Effect size

  • Cohen’s d =

    • Example 1 Cohen’s d

    • Example 2 Cohen’s d

  • r2 or η2

    • G = grand mean


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Factors Influencing t–test and Effect Size

  • Mean difference M1 – M2

    • Larger difference, larger t

    • Larger difference, larger r2 and Cohen’s d


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  • Example 1, subtract 1 from first group, add 2 to second group

    • M1 – M2 increases from –3.257 to –6.257

    • unaffected t increases from –2.325 to –4.466

    • r2increases from


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  • Magnitude of sample variances

    • As sample variances increase:

    • t decreases

    • Cohen’s d and r2 decreases

      • SSExplainedunchanged, SSErrorand SSTotal increases, S2pooled increases


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  • Sample size

    • Larger sample smaller t affects

    • No effect on Cohen’s d, minimal effect on r2

    • First example increase n1from 10 to 30 and n2 from 7 to 21


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