DIS reading Agresti-etal-Indep-T-Test-Comparing-Two-Means summary

Section 10.2: Quantitative Response - Comparing Two Means

Overview

• This section discusses methods for comparing two groups based on a quantitative response variable by analyzing their means.

• The focus is on determining what the difference between sample means tells us about the difference between the population means.

Comparing Two Groups on a Quantitative Variable

• Graphical Analysis

  • Graphs, such as box plots, are effective in conveying information about comparative analysis of two groups.

  • Example: Side-by-side box plots illustrate differences in product ratings between two groups (Text & Graph vs. Text Only).

• Experimental Setup

  • In a study, one group read a text with a bar graph, while the other group read the same text without the graph.

  • The ratings (on a 9-point scale) were compared:

    • Group 1 (Text & Graph): Mean = 6.83, SD = 1.18, n = 30

    • Group 2 (Text Only): Mean = 6.13, SD = 1.43, n = 31

  • Difference in means = 6.83 - 6.13 = 0.7, indicating that the group with the graph rated the drug's effectiveness higher.

Sampling Distribution of the Difference Between Two Means

• Standard Error Calculation

  • The standard error (se) captures the variability of the sampling distribution of the difference between two means.

  • For large samples from two groups, the standard error formula is: se = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}

    • For example, given standard deviations for Group 1 and Group 2, se was calculated to be 0.335.

• Confidence Intervals

  • The confidence interval for the population mean differences is constructed as:

(\mu_1 - \mu_2) \pm 1.96(se)

• A 95% confidence interval from the above data was calculated as (0.03, 1.38) indicating the likely range of the difference in population means.

Significance Testing for Population Means

• General Overview

  • Testing the null hypothesis (H0: \mu_1 = \mu_2) involves using a t-test statistic: t = \frac{(x_1 - x_2) - 0}{se}

  • For significant findings, P-values dictate whether to reject H0 based on observed results.

• Two-Sided vs One-Sided Tests

  • Two-sided tests evaluate if there's a difference without favoritism towards which mean is larger, while one-sided tests anticipate a specific direction of difference.

Example Study: Cell Phone Use While Driving

• Experimental Design

  • A study investigating cell phone use and reaction times involved 64 students divided into a cell phone group and a control group.

  • Measured response times showed:

    • Cell Phone Group: Mean = 585 ms

    • Control Group: Mean = 533 ms

  • Observed difference = 51.6 ms, significant at P-value = 0.0110, leading to the conclusion that cell phone use negatively affects reaction times.

Summary of Key Concepts

• Confidence Interval for Difference of Means

  • When comparing groups, confidence intervals provide insight into the potential range of population mean differences.

  • Intervals containing only positive values indicate that the first group (with graph) rates the drug more favorably than the second group (text only).

• Considerations and Assumptions

  • The validity of the confidence interval and significance tests relies on the assumption of independent samples, approximate normality of distributions, and equality of standard deviations if using specific test methods.