Hypothesis Testing for Two Proportions

Overview

This lecture reviews how to check conditions for hypothesis testing involving two proportions using an apnea (infant therapy) study as an example.

The three main conditions are: randomness, large sample size, and independence.
Randomness: Both samples must be randomly selected (random infants in therapy and placebo groups).

Calculate pooled sample proportion: ( \hat{p} = (X_1 + X_2)/(N_1 + N_2) ).
(X) refers to the number of "successes" (e.g., infants with death or disability).
In the example: (X_1 = 377), (X_2 = 431), (N_1 = 937), (N_2 = 932).
Pooled proportion: ( \hat{p} = 808 / 1869 = 0.43 ).

Multiply ( \hat{p} ) by each group's sample size to find expected successes: e.g., (0.43 \times 937 \approx 403).
Calculate expected failures: sample size minus expected successes (e.g., (937 - 403 = 534)).
Both expected successes and failures in each group must be greater than 10.

Independence between groups: Therapy and placebo groups consist of different infants; one group does not affect the other.
Independence within groups: Each baby's breathing outcome is independent of other babies; assumption holds unless stated otherwise.

Significance Level (α) — The probability of a Type I error; default is 0.05 if not specified.
Pooled Proportion ((\hat{p})) — Combined proportion of "successes" from both groups: ( (X_1 + X_2)/(N_1 + N_2) ).
Success — The outcome of interest (e.g., infants suffering death or disability).
Random Sample — Each subject is selected by chance.
Independence — One subject’s outcome does not affect another’s, both between and within groups.

Practice checking all three conditions (randomness, large sample, independence) on sample problems.
Use pooled proportion to calculate expected successes and failures for both groups.
Prepare for calculations if all conditions are satisfied.