Goal: To compare the responses to two treatments or to compare the characteristics of two populations.

* Have a separate, independent sample from each population

e.g. randomized comparative experiment, comparing groups (M vs. F)

Ask yourself: means or proportions, one or two distinct populations?

1. Graphical comparisons

2. Numerical comparisons

3. Inference - do the groups differ more than we would expect by chance alone?

Need to know how the statistic behaves in repeated samples

Example Medical researcher wants to compare a new treatment to the old treatment. Randomly assigns 10 patients to each treatment. She observes how many in each group exhibit satisfactory recovery. She finds 7 patients in the new treatment and 5 patients in the old treatment recover.

2. Graphical Summary

new old treatment

Are you convinced this difference in recovery rates is not just due to chance?

Parameter:

Statistic:

Null Hypothesis:

Alternative Hypothesis:

Overall recovery rate:

Are the recovering patients distributed among the two groups randomly?

Sampling Distribution of ₁ - ₂ when p₁=p₂

No. of

samples

Shape?

Center?

Variability?

Assume H₀ is true,

Pooled estimate:

Standard error:

Confidence Interval for p₁-p₂:

Technical assumptions: SRS, normal approximation to Binomial

Rule of thumb: number of successes and failures in each sample > 5

Example Want to study the long term effects of preschool programs for poor children. Follow two groups of Californian children since early childhood: control group of 61 did not attend preschool. Another groups of 62 attended preschool as 3 and 4 year olds. Response variable is need for social service as adults: 49 of the control sample and 38 of the preschool sample needed social services.

- Does this data satisfy the technical assumptions?

- Test for a difference in the need for social services between these two groups

- How large is the difference? Estimate at the 95% confidence level.