Chapter 9: Inferences from Two Samples

This chapter focuses on statistical methods for comparing two populations using sample data. It covers hypothesis testing and confidence intervals for two proportions, two means (independent and dependent samples), and two variances. These techniques are essential for determining whether observed differences between groups are statistically significant.

Two Proportions – Hypothesis Tests

Hypothesis tests for two proportions are used to compare the proportions of a certain characteristic in two independent populations.

• Null Hypothesis (H0): (the population proportions are equal)

• Test Statistic:

• and are the sample proportions

• is the pooled sample proportion

• are the number of successes in each sample; are the sample sizes

Example: In a randomized controlled trial in Kenya, insecticide-treated bednets were tested to reduce malaria. Among 343 infants using bednets, 15 developed malaria. Among 329 infants not using bednets, 27 developed malaria. Use a 0.01 significance level to test if the incidence of malaria is lower for infants using bednets.

Critical Value and p-Value Methods

• Critical Value Method: Compare the test statistic to the critical value from the standard normal distribution.

• p-Value Method: Calculate the p-value and compare it to the significance level .

Two Proportions – Confidence Intervals

Confidence intervals estimate the difference between two population proportions.

• Confidence Interval Formula:

where

• If the confidence interval does not include 0, there is evidence that and are different.

• Conclusions from confidence intervals may differ from those of hypothesis tests.

Example: In a World Series, the Houston Astros were ordered to keep the roof closed for some games and open for others. Construct a 90% confidence interval for the difference between the two proportions of wins.

Two Means – Hypothesis Tests (Independent Samples)

These tests compare the means of two independent populations. The approach depends on whether population variances are known or assumed equal.

• Independent Samples: Samples from one population are not matched with those from the other.

• Dependent Samples: Samples are paired (matched pairs).

Possible Scenarios

• Both variances unknown, not assumed equal

• Both variances known

• Both variances unknown, assumed equal

Test Statistic (for independent samples, variances not assumed equal):

• are sample means; are sample variances; are sample sizes

• Degrees of Freedom: Use the smaller of and

Example: Compare mean FICO scores for borrowers of high- and low-interest mortgages to test if interest rate affects FICO score.

Critical Value and p-Value Methods

• Use the t-distribution to find critical values or p-values.

Two Means – Confidence Intervals (Independent Samples)

Estimate the difference between two population means using independent samples.

• Confidence Interval Formula:

where

• If the confidence interval does not include 0, there is evidence of a difference between the means.

Example: Construct a 95% confidence interval for the difference in mean FICO scores between two groups of borrowers.

Two Means – Hypothesis Tests (Dependent Samples)

Used when samples are paired (matched pairs), such as before-and-after measurements or matched subjects.

• Test Statistic:

• = individual difference between paired values

• = mean of the differences

• = standard deviation of the differences

• = number of pairs

Example: Compare hospital admissions from motor vehicle crashes on two different Fridays to test for a difference.

Two Means – Confidence Intervals (Dependent Samples)

Estimate the mean of the differences from dependent samples.

• Confidence Interval Formula:

Two Variances – Hypothesis Tests

Tests for comparing the variances (or standard deviations) of two populations, often using the F-distribution.

• F-Distribution: The ratio follows an F-distribution if both populations are normal and have equal variances.

• Test Statistic:

• Degrees of freedom: ,

• Shape depends on two different degrees of freedom

• Cannot be negative; not symmetric

Interpreting an F-Test

• If and are close, is close to 1 (evidence for equal variances)

• If and are far apart, is far from 1 (evidence for different variances)

Example: Compare the standard deviations of braking distances for two types of cars to test if they have the same variance.

Groupwork Example

The following data are self-reported heights and measured heights (in inches) for males aged 18-21:

• Are the two samples independent or dependent?

• Construct a 95% confidence interval estimate of the mean difference between reported and measured heights. Does the CI include zero? What does that suggest?

Additional info: Since each reported height is paired with a measured height for the same individual, these are dependent samples (matched pairs). If the confidence interval for the mean difference includes zero, it suggests no significant difference between reported and measured heights.