12. Two-sample Confidence Intervals

Topics and Notes

1. Confidence interval of the difference of two independent population means
• a. Normal confidence intervals
•   ■ Both sample sizes are large (bigger than 30 by convention)
•   ■ Normal populations with known variances. In this case, no restrictions on sample sizes
• b. small sample t-confidence interval
•   ■ Both populations must be normal
•   ■ Both population variance are unknown but equal
•   ■ Require estimating pool variance by combining two random samples.
$$s^2_{pool}=\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2 - 2}$$
2. Confidence interval of the difference of two independent population proportions
• a. Assumptions
•   ■ n₁p̂₁ ≥ 5 and   ■ n₁(1-p̂₁) ≥ 5, AND
•   ■ n₂p̂₂ ≥ 5 and   ■ n₂(1-p̂₂) ≥ 5.
• b. Sampling distribution of (p̂₁ - p̂₁) is normal.
• c. The margin of error
$$E=Z_{\alpha/2}\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$$
3. Lecture Note
• Two-sample confidence intervals HTML PDF