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The t Test for Two Independent Samples PowerPoint Presentation

The t Test for Two Independent Samples

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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

- Criteria for use
- Dependent variable is quantitative, interval/ratio
- Independent variable between-subjects
- Independent variable has two levels

- t-test
- Basic form
- One sample

Two sample

- Difference between sample means M1 - M2
- Population parameter

- Sampling distribution of the difference
- Difference between M1 and M2 drawn from population

Standard error of the difference

- Population variance known
- Sum of

- Estimate from samples
- Differences more variable than scores

Variability of mean differences

- Randomly generated set of 1000 means
- Μ= 50, σM = 10
- Take difference between pairs

S2pooled Pooled Variance

- Homogeneity of variance
- Assume two samples come from populations with equal σ2’s
- Two estimates of σ2 — and

- Weighted average

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

t-testHypothesis 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

Assumptions

- Random and independent samples
- Normality
- Homogeneity of variance
- SPSS—test for equality of variances, unequal variances t test
- t-test is robust

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

- t(28)= –.947, p > .05

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

SPSS

- Analyze
- Compare Means
- Independent-Samples T Test

- Compare Means
- 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

Effect size

- Cohen’s d =
- Example 1 Cohen’s d
- Example 2 Cohen’s d

- r2 or η2
- G = grand mean

Factors Influencing t–test and Effect Size

- Mean difference M1 – M2
- Larger difference, larger t
- Larger difference, larger r2 and Cohen’s d

- 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

- Magnitude of sample variances group
- As sample variances increase:
- t decreases
- Cohen’s d and r2 decreases
- SSExplainedunchanged, SSErrorand SSTotal increases, S2pooled increases

- Sample size group
- 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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