Inferences from Two Samples

Overview

This chapter introduces statistical methods for making inferences about two populations using sample data. The focus is on comparing proportions and means from two independent samples, including hypothesis testing and confidence interval estimation. These techniques are essential for determining whether observed differences between groups are statistically significant.

9-1 Two Proportions

Key Concepts

• Objective: Test claims about two population proportions and construct confidence intervals for their difference.

• Methods apply to proportions, probabilities, or decimal equivalents of percentages.

Notation for Two Proportions

• Population 1:

  • p1: population proportion

  • n1: size of first sample

  • x1: number of successes in first sample

  • \( \hat{p}_1 = \frac{x_1}{n_1} \): sample proportion

  • \( \hat{q}_1 = 1 - \hat{p}_1 \): complement of sample proportion

• Population 2: Notations \( p_2, n_2, x_2, \hat{p}_2, \hat{q}_2 \) apply similarly.

Pooled Sample Proportion

• The pooled sample proportion combines the two sample proportions into one:

Requirements for Inference

• Samples must be simple random samples.

• Samples must be independent (no natural pairing or matching).

• Each sample must have at least 5 successes and 5 failures: , , , .

Test Statistic for Two Proportions

To test :

Where under .

Confidence Interval for

The confidence interval estimate is:

Where the margin of error is:

Equivalent Methods

• The P-value method and critical value method are equivalent for hypothesis testing.

• The confidence interval method is not equivalent to the P-value or critical value methods.

Hypothesis Tests

• Null hypothesis:

• Estimates and are combined to form the pooled sample proportion .

Example: Comparing Success Rates of E-Cigarettes and Nicotine Replacement

• Data:

  • E-cigarette group:

  • Nicotine replacement group:

• Hypotheses:

• Significance level:

• Pooled proportion:

• Complement:

• Test statistic:

• P-value: For two-tailed test,

• Conclusion: Since , reject . There is a significant difference in success rates.

• Confidence interval:

  • Interval does not contain 0, supporting the conclusion of a significant difference.

Example Application: Comparing treatment effectiveness in medical studies, such as smoking cessation methods.

9-2 Two Means: Independent Samples

Key Concepts

• Objective: Test hypotheses about two independent population means and construct confidence intervals for their difference.

Independent vs. Dependent Samples

• Independent samples: Sample values from one population are not related to those from the other population.

• Dependent samples (matched pairs): Sample values are paired based on some inherent relationship.

Notation for Two Means

• Population 1:

  • : population mean

  • : population standard deviation

  • : sample size

  • : sample mean

  • : sample standard deviation

• Population 2: Notations apply similarly.

Requirements for Inference

• Population standard deviations and are unknown and not assumed equal.

• Samples are independent and simple random samples.

• Either both sample sizes are large () or both samples come from populations with normal distributions.

Test Statistic for Two Means (Independent Samples)

To test :

(Often under )

Degrees of Freedom

• Simple estimate:

• More accurate formula (used by technology):

Confidence Interval for

The confidence interval estimate is:

Where and

Equivalent Methods

• The P-value method, critical value method, and confidence interval method are equivalent for two means (independent samples).

Example Application

• Comparing mean heights: Testing whether the mean height of U.S. Army male personnel in 1988 is less than in 2012.

• Hypotheses:

• Test statistic: Calculated using sample means and standard deviations.

• P-value: If greater than significance level, fail to reject .

• Confidence interval: If interval contains 0, no significant difference is indicated.

Example: Comparing average heights, weights, or test scores between two groups.

Summary Table: Key Formulas

Additional info: These methods are foundational for comparing groups in experimental and observational studies, such as clinical trials, educational interventions, and social science research.