t Test for Two Independent Samples

t test for two independent samplesBasic Assumptions

Independent samples are not paired with other observations

Null hypothesis states that there is no difference between the means of the groups

OrH0: µ1 - µ2 ≤ 0

t test for two independent samplesBasic Assumptions

Alternate hypothesisH1: µ1 - µ2 > 0

Two other possible alternate hypothesesDirectional less thanH1: µ1 - µ2 < 0

OrNondirectionalH1: µ1 - µ2 ≠ 0

t ratio

(X1 – X2) – (µ1 - µ2)hyp

t = sx1 – x2

Calculation steps for t ratio for two independent means

Phase I1. 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 II7.Calculate pooled variance using formula p

290 SS1 + SS2

s2p = n1 + n2 – 2

8.Calculate standard error p 2919.Substitute numbers to get t ratio

Pooled variance estimateThe pooled variance represents the mean of

the variances for the two samplesEstimated standard error uses calculated

pooled variance

p-valueThe 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 intervalsConfidence 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_ X1 – X2

d = standard deviation = √ s2p

Effect size: Cohen’s dInterpreting d

Effect size is small if d is less than 0.2Effect size is medium if d is in the vicinity of

0.5Effect size is large if d is more than 0.8

Assumptions when using t ratioBoth underlying populations are normally

distributedBoth populations have equal variances

If these are not met you might try:Increasing sample sizeEquate sample sizesUse a less sensitive (yet more complex) t testUse a less sensitive test such as Mann-Whitney

U test