Hypothesis Testing with Two Samples: Study Notes

Chapter 8: Hypothesis Testing with Two Samples

8.1 Testing the Difference Between Means (Independent Samples, Known Standard Deviations)

This section introduces hypothesis testing for comparing the means of two independent populations when the population standard deviations are known. It covers the identification of independent and dependent samples, the formulation of hypotheses, and the execution of the two-sample z-test.

• Independent Samples: Two samples are independent if the selection of one sample does not influence the selection of the other. For example, comparing test scores from two different groups of people.

• Dependent Samples (Paired or Matched Samples): Two samples are dependent if each member of one sample corresponds to a member of the other sample, such as before-and-after measurements on the same subjects.

Example: Measuring triglyceride levels in the same patients before and after treatment produces dependent samples. Measuring scores from different groups (e.g., males vs. females) produces independent samples.

8.1.1 Two-Sample Hypothesis Test: Concepts

A two-sample hypothesis test compares two parameters (usually means or proportions) from two populations. The null hypothesis typically states that there is no difference between the population parameters.

• Null Hypothesis (H0): $H_0: \mu_1 = \mu_2$ (no difference in means)

• Alternative Hypothesis (Ha): $H_a: \mu_1 \neq \mu_2$, $\mu_1 > \mu_2$, or $\mu_1 < \mu_2$ (depending on the claim)

8.1.2 Conditions for the Two-Sample z-Test

The two-sample z-test for the difference between means is appropriate when:

• The population standard deviations ($\sigma_1$ and $\sigma_2$) are known.

• The samples are randomly selected and independent.

• The populations are normally distributed, or each sample size is at least 30 (Central Limit Theorem applies).

8.1.3 Sampling Distribution and Test Statistic

When the above conditions are met, the sampling distribution of the difference between sample means ($\bar{x}_1 - \bar{x}_2$) is normal with:

• Mean: $\mu_{\bar{x}_1 - \bar{x}_2} = \mu_1 - \mu_2$

• Standard Error: $\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$

The test statistic is:

$z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$

Usually, the null hypothesis assumes $\mu_1 - \mu_2 = 0$.

8.1.4 Steps for the Two-Sample z-Test

1. State the null and alternative hypotheses.

2. Check that all conditions for the z-test are met.

3. Calculate the test statistic using the formula above.

4. Determine the critical value(s) and rejection region(s) based on the significance level ($\alpha$) and the type of test (two-tailed, left-tailed, or right-tailed).

5. Compare the test statistic to the rejection region and make a decision (reject or fail to reject $H_0$).

8.1.5 Example: Credit Card Debt Comparison

Scenario: A group claims there is a difference in mean credit card debts between Montana and Pennsylvania. Two independent random samples of 250 people from each state are taken, with known population standard deviations.

• Step 1: State hypotheses: $H_0: \mu_1 = \mu_2$, $H_a: \mu_1 \neq \mu_2$

• Step 2: Check conditions: Standard deviations known, samples random and independent, sample sizes large ($n \geq 30$)

• Step 3: Calculate $z$ using the formula above

• Step 4: Find critical values for $\alpha = 0.05$ (two-tailed test)

• Step 5: Compare $z$ to rejection regions; if $z$ is not in the rejection region, fail to reject $H_0$

Conclusion: If $z$ is not in the rejection region, there is not enough evidence to support the claim of a difference in means.

8.1.6 Example: Comparing Rents Using Technology

Scenario: A researcher claims that the average rent in Seattle is less than in Santa Ana. Independent samples are taken, and population standard deviations are known.

• Set up hypotheses: $H_0: \mu_1 = \mu_2$, $H_a: \mu_1 < \mu_2$

• Use technology (e.g., TI-84 calculator) to compute the test statistic and p-value

• Compare the test statistic to the critical value for a left-tailed test at $\alpha = 0.01$

• If the test statistic is not in the rejection region, fail to reject $H_0$

Conclusion: There is not enough evidence at the 1% significance level to support the claim that Seattle rents are lower.

8.1.7 Summary Table: Independent vs. Dependent Samples

Type of Samples	Description	Example
Independent	Samples are unrelated; selection of one does not affect the other	Comparing test scores of males vs. females
Dependent (Paired)	Samples are related; each member of one sample corresponds to a member of the other	Before-and-after measurements on the same subjects

8.1.8 Key Formulas

• Standard Error for Difference of Means:

• $SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$

• z-Test Statistic:

• $z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{SE}$

Additional info: In practice, if population standard deviations are unknown, a two-sample t-test is used instead. The z-test is less common in real-world applications but is important for understanding the theory of hypothesis testing with large samples and known variances.