the t test for two independent samples l.
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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 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
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
slide2
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
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
Standard error of the difference
  • Population variance known
    • Sum of
  • Estimate from samples
  • Differences more variable than scores
variability of mean differences
Variability of mean differences
  • Randomly generated set of 1000 means
    • Μ= 50, σM = 10
    • Take difference between pairs
s 2 pooled pooled variance
S2pooled Pooled Variance
  • Homogeneity of variance
    • Assume two samples come from populations with equal σ2’s
    • Two estimates of σ2 — and
  • Weighted average
hypothesis testing
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
assumptions
Assumptions
  • Random and independent samples
  • Normality
  • Homogeneity of variance
    • SPSS—test for equality of variances, unequal variances t test
    • t-test is robust
example 1
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
confidence interval for the difference
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
slide15
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
effect size
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
Factors Influencing t–test and Effect Size
  • Mean difference M1 – M2
    • Larger difference, larger t
    • Larger difference, larger r2 and Cohen’s d
slide20
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
slide21
Magnitude of sample variances
    • As sample variances increase:
    • t decreases
    • Cohen’s d and r2 decreases
      • SSExplainedunchanged, SSErrorand SSTotal increases, S2pooled increases
slide22
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