unit 2

Unit Overview

• Unit Title: Inference for the Means of Two Populations

• Unit Focus: Understanding methods to compare the means of two populations using statistical inference techniques.

Outline of the Unit

• Key Topics:

  • Matched pairs t-procedures

  • Inference for equality of means in two populations:

    • When population variances are equal

    • When population variances are unequal

  • Assumptions:

    • Normality

    • Independence

Paired Data

• Definition: Data collected in pairs to analyze differences rather than individual observations.

• Common Situations:

  • Two different variables measured for each individual (e.g., comparing grades in different subjects).

  • Measurements taken at different times or conditions (e.g., blood pressure measurements before and after medication).

  • Similar individuals receiving different treatments for comparative analysis (e.g., twins on different diets).

Matched Pairs t-Procedures

• Purpose: Detect differences in responses to two treatments based on pairs of observations.

• Parameter of Interest: ( \mu_d ) - true mean of the differences of all pairs in the population.

• Assumptions:

  • Differences follow a normal distribution with mean ( \mu_d \) and standard deviation ( \sigma_d \).

  • Pairs represent a Simple Random Sample (SRS) from the population.

  • Observations are dependent within pairs.

• Methodology: Confidence intervals and hypothesis tests constructed similarly to one-sample tests, focusing instead on differences.

Confidence Intervals for ( \mu_d )

• Formula: ( \bar{x}d , \pm t{(n-1)} \cdot \frac{s_d}{\sqrt{n}} )

  • ( \bar{x}_d ): sample mean difference

  • ( t_{(n-1)} ): critical value from t-distribution

  • ( s_d ): sample standard deviation of differences

  • ( n ): number of pairs

Example of Confidence Interval Calculation

• Scenario: Testing whether premium gasoline yields better mileage.

• Sample Data: Mileage of 8 cars run on regular and premium gas with calculated differences.

• Interval Calculation:

  • If ( \bar{x}_d = 2 ), ( s_d = 2 ), and critical value ( t = 2.365 ):

  • Confidence Interval: ( 2 \pm 2.365 \cdot \frac{2}{\sqrt{8}} = (0.33, 3.67) )

Hypothesis Testing with Matched Pairs

• Test Statistic: ( t = \frac{\bar{x}d - \mu{d0}}{\frac{s_d}{\sqrt{n}}} )

  • Where ( \mu_{d0} ) is the hypothesized mean difference (usually 0).

• Example: Conducting a hypothesis test to evaluate whether premium gasoline is more effective.

• P-value Calculation: Determine based on the t-distribution corresponding to the calculated statistic.

Two-Sample t Procedures

• Independent vs. Paired Samples: Explain the difference between independent samples and matched pairs in terms of hypothesis tests and intervals.

• Assumptions for Independent Samples:

  • Normal distributions

  • Samples must be independent

Pooled vs. Unpooled Procedures

• Pooled Methods: Used under the assumption that population variances are equal.

• Unpooled Methods: Used when population variances are not assumed to be equal.

• Critical Rule: Calculate whether to assume equal variances by evaluating the ratio ( \frac{max(s_1, s_2)}{min(s_1, s_2)} )

• Confidence Interval for Equality of Means when variances are unequal:

  • ( (\bar{x}_1 - \bar{x}2) \pm t{critical} \cdot SE )

Conclusion

• Robustness of Procedures: Two-sample t procedures are more robust against violations of normality than one-sample procedures, especially with equal sample sizes.

• Practical Implications: Understanding these inference methods equips researchers to handle real-world data comparisons effectively.