Hypothesis Testing for Two Independent Means

Comparing Two Independent Means

• We compare the Olympic sponsorship (response) between USA and Tonga (two populations).
• We compare the effectiveness (response) between Drug A and Drug B (two treatments).

How to identify this type of situation

1. There are two samples: {Sample 1, Sample 2}
2. Each sample is randomly drawn from different, non-overlapping populations.

• Sample 1 is drawn from Population 1 with $\mu_1$ and $\sigma_1$
• Sample 2 is drawn from Population 2 with $\mu_2$ and $\sigma_2$

1. The two populations are independent (there is no matching of individuals or observations in the two samples)

1. Let $i=1,2$. Each of the two samples will consist of:

• Sample mean $\overline{x}_i$
• Sample standard deviation $s_i$
• $σ_1\$and $σ_2$ (population standard deviations) are unknown and, instead, $s_1$ and$\ s_2$ (sample standard deviations) are used.
• Sample size $n_i$
• It's okay if sample sizes differ as they do not need to be equal.
• Central Limit Theorem applies:
• If both population distributions are normal, then the sample sizes do not need to be large.
• If both population distributions are not normal, then the sample sizes need to be large.
• When in doubt, you should have sufficiently large sample sizes.

Hypothesis Test Steps: Comparing Two Independent Means

• Step 1: State the hypotheses
• Is this a one-sided or two-sided test?
• To make inferences about the difference in the population means $\mu_1-\mu_2$ (two-sided test), the hypotheses are:

• You can also test if one population mean is greater/less than the other population mean at either direction (one-sided test), depending on the question:

• Step 2: Note the significance level $\alpha$
• You may find the critical value, depending on $\alpha,\ df,\$ and # of sides
• Step 3: Locate the relevant variables and run the appropriate test
• Find $\overline{x}_1,\ \overline{x}_2,\ s_1,\ s_2,\ n_1,\ n_2$
• To Pool or Not to Pool?
• If we can assume $σ_1\ =\ σ_2$$\rightarrow$ pooled variance t-test
• Otherwise, we assume $σ_1\ \ne\ σ_2$ $\rightarrow$ unequal variance t-test
• Step 4: Find the p-value
• The p-value is based on your test statistic and # of sides
• If p-value $<\alpha\ \rightarrow$ Reject $H_o$
• If p-value $>\alpha\ \rightarrow$ Fail to reject $H_o$
• You can also compare the critical value with the test statistic
• If $\left|t\right|>\left|CV\right|\rightarrow$ Reject $H_o$
• If $\left|t\right|<\left|CV\right|\rightarrow$ Fail to reject $H_o$
• Step 5: Draw your conclusion
• If you reject $H_o$, you conclude that there is evidence for $H_a$. Example: "There is evidence that the means differ."
• If you fail to reject $H_o$, you conclude that there is no evidence for $H_a$. Example: "There is no evidence that the means differ."