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t Test for Two Independent Samples
t test for two independent samples
 Basic Assumptions
 Independent samples are not paired with other
observations
 Null hypothesis states that there is no difference
between the means of the groups
 Or
 H0: µ1 - µ2 ≤ 0
t test for two independent samples
 Basic Assumptions
 Alternate hypothesis
 H1: µ1 - µ2 > 0
 Two other possible alternate hypotheses
 Directional less than
 H1: µ1 - µ2 < 0
 Or
 Nondirectional
 H1: µ1 - µ2 ≠ 0
t ratio
(X1 – X2) – (µ1 - µ2)hyp
 t=
sx1 – x2
Calculation steps for t ratio for two
independent means
Phase I
1. Assign a value to n1
2. Sum all X1 scores
3. Find mean for X1
4. Square each X1 score
5. Sum all squared X1 scores
6. Solve for SS1
Repeat for X2
Calculation steps for t ratio for two
independent means
Phase II
7. Calculate pooled variance using formula p 290
SS1 + SS2
s2p = n1 + n2 – 2
8. Calculate standard error p 291
9. Substitute numbers to get t ratio
Pooled variance estimate
 The pooled variance represents the mean of the
variances for the two samples
 Estimated standard error uses calculated pooled
variance
p-value
 The p-value indicates the degree of rarity of the
observed test result when combined with all
potentially more deviant test results.
 Smaller p-values tend to discredit the null hypothesis
and support the research hypothesis.
Significance??
 Statistical significance between pairs of sample
means implies only that the null hypothesis is probably
false, and not whether it’s false because of a large or
small difference between the population means.
Confidence intervals
 Confidence intervals for µ1 - µ2 specify ranges of values
that, in the long run, include the unknown effect
(difference between population means) a certain
percent of the time.
X1 – X2 ± (tconf )(sx1 – x2 )
But wait ……. there is more!!
Significance??
 Statistical significance between pairs of sample
means implies only that the null hypothesis is probably
false, and not whether it’s false because of a large or
small difference between the population means.
Effect size: Cohen’s d
__mean difference_
 d = standard deviation =
X1 – X2
√ s2 p
Effect size: Cohen’s d
 Interpreting d
 Effect size is small if d is less than 0.2
 Effect size is medium if d is in the vicinity of 0.5
 Effect size is large if d is more than 0.8
Assumptions when using t ratio
 Both underlying populations are normally distributed
 Both populations have equal variances
 If these are not met you might try:
 Increasing sample size
 Equate sample sizes
 Use a less sensitive (yet more complex) t test
 Use a less sensitive test such as Mann-Whitney U test