Chapter 8: Estimation – Additional Topics

Overview

This chapter focuses on constructing confidence intervals for the difference between two population means (using both dependent and independent samples) and for the difference between two population proportions. These methods are essential for comparing groups in business and economics research.

Confidence Intervals for Differences in Means

Population Means: Dependent Samples

When comparing two related populations, such as measurements taken before and after a treatment on the same subjects, dependent (paired) samples are used. The goal is to estimate the confidence interval for the mean difference between the two populations.

• Paired or matched samples: Each observation in one sample is uniquely paired with an observation in the other sample.

• Repeated measures: The same subjects are measured twice (e.g., before and after treatment).

• Difference between paired values: for

• Assumptions: Both populations are normally distributed.

Key Formulas

• Mean of paired differences:

• Sample standard deviation of differences:

• Confidence interval for mean difference:

• Margin of error:

Example: Paired Samples

Six people participate in a weight loss program. Their weights before and after are recorded, and the differences are calculated. The mean difference and standard deviation are computed, and a 95% confidence interval is formed using the formulas above. If the interval contains zero, we cannot be 95% confident that the program causes weight loss.

Additional info: Calculations yield , , .

Population Means: Independent Samples

When comparing two unrelated groups, independent samples are used. The goal is to estimate the confidence interval for the difference between two population means.

• Independent samples: Data from different sources; selection from one population does not affect the other.

• Point estimate:

• Assumptions: Samples are randomly and independently drawn; populations are normally distributed.

Cases for Population Variances

Key Formulas

• Known variances:

• Unknown, assumed equal variances (pooled):

• Unknown, assumed unequal variances:

Example: Independent Samples (Pooled Variance)

Comparing CPU speeds for two processors:

Pooled variance , . Confidence interval: Mhz.

Confidence Intervals for Differences in Proportions

Population Proportions

When comparing proportions (e.g., the proportion of men and women with college degrees), the goal is to estimate the confidence interval for the difference between two population proportions.

• Assumptions: Both sample sizes are large (generally at least 40 observations each).

• Point estimate:

Key Formulas

• Standard error:

• Confidence interval:

Example: Two Population Proportions

Form a 90% confidence interval for the difference between the proportion of men and women with college degrees:

• Men:

• Women:

• Standard error:

• For 90% confidence,

• Confidence interval:

Since the interval does not contain zero, we are 90% confident that the proportions are not equal.

Summary Table: Confidence Interval Methods

Key Takeaways

• Confidence intervals allow us to estimate the range in which the true difference between population parameters lies, with a specified level of confidence.

• Choice of method depends on sample type (dependent/independent) and knowledge of population variances.

• For proportions, large sample sizes are required for normal approximation.

• Interpretation of intervals is crucial: if the interval contains zero, there may be no significant difference between groups.