11. Two-sample Tests

Topics and Notes

1. Normal test of two independent population means
• a. Assumptions
•   ■ Both sample sizes are large, OR
•   ■ If one of the sample size is small, both populations are normal and both variances are given.
• b. Sampling distribution of the difference of two sample means
•   ■ $\mu_1-\mu_2$ is approximated by $\bar{x}_1 - \bar{x}_2$
•   ■ Variance of ($\bar{x}_1 - \bar{x}_2$) is $s^2_{\bar{x}_1-\bar{x}_2} = \sqrt{s_1^2/n_1 + s_2^2/n_2}$
•   ■ Null hypothesis H₀: μ₁ - μ₂ = d₀
•   ■ Test statistic is $TS = \frac{(\bar{x}_1-\bar{x}_2)-d_0}{\sqrt{s_1^2/n_1 + s_2^2/n_2}}$
• c. 6-step procedure: both critical and p-value methods.
2. t-test of two independent population means
• a. Assumptions: normal population, unknown but equal variances
•   ■ Both populations are normal
•   ■ Population variances are unknown (i.e., need to be estimated from data)
•   ■ Both populations are equal (This assumption is only for this class)
• b. Test statistic
•   ■ Variance of pooled samples $s^2_{\text{pool}} = \frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}$
•   ■ Null hypothesis H₀: μ₁ - μ₂ = d₀
•   ■ Test statistic $TS = \frac{(\bar{x}_1-\bar{x}_2)-d_0}{s_{\text{pool}}/\sqrt{1/n_1 + 1/n_2}} \rightarrow N(0,1)$
• b. pooled sample variance and the 6-step procedure.
3. Lecture Note
• Two-sample tests of the population mean HTML PDF

Topics and Notes

1. Practice exercises #11 WEB LINK
2. [Optional]Read sections 11.1 of Navidi's textbook
3. Two-sample Tests for Population Means INTERACTIVE APPS
4. Flow-chart of two-sample testing procedures: FLOW CHART

Weekly Assignments

1. There is no assignment for this week due to midterm exam #3.