12. Inference for Two Independent Means

Comparing Two Groups

Previously, we made inferences about a single mean from one group.

Now we extend to comparing two independent groups.

Research question: Is there a difference between the population means of two groups?

Same framework:

• CI: Point estimate ± margin of error
• Hypothesis testing with p-values

Data Structure for Two Means

When comparing two means:

• Response variable: Numerical
• Explanatory variable: Binary categorical (defines the two groups)

This is the same structure as two-proportion inference, but now the response is numerical instead of categorical.

Statistics and Parameters for Two Means

Statistic of interest: \(\bar{x}_1 - \bar{x}_2\) (difference in sample means)

Goal: Make inferences about \(\mu_1 - \mu_2\) (difference in population means)

The Key Insight

Like in two proportions under the null hypothesis (\(\mu_n = \mu_s\)):

• The groups have the same birth weight
• Whether a mother smokes or not doesn’t affect their outcome
• The outcome is independent of group assignment

Key idea: If group doesn’t matter, we can shuffle the birth weights between the groups

Permutation: Shuffling Response Values

A permutation randomly reassigns the response values (outcomes):

• Group assignments stay fixed (867 nonsmokers, 114 smokers)
• We randomly shuffle which weight goes with which mother
• This simulates “what if the outcome had nothing to do with group”

After shuffling, we calculate the difference in means

This gives us ONE value from the null distribution

Estimating the Difference in Means Using a Mathematical Model

Pooled Standard Deviation

When comparing two groups, we pool the variability to get a combined estimate of the common population SD.

\[s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}}\]

For birth weights:

\[s_p = \sqrt{\frac{(867-1) \cdot 1.23^2 + (114-1) \cdot 1.6^2}{867+114-2}} = 1.28\]

SE for Difference in Means

The standard error for the difference in sample means:

\[SE = s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}\]

For birth weights:

\[SE = 1.28 \times \sqrt{\frac{1}{867} + \frac{1}{114}} = 0.128\]

T-Statistic for Two Means

The T-statistic measures how far the observed difference is from the null value in SE units:

\[T = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{SE}\]

Degrees of freedom: \(df = n_1 + n_2 - 2 = 867 + 114 - 2 = 979\)

For birth weights:

\[T = \frac{0.59 - 0}{0.128} = 4.65\]

Conditions for Two-Sample T-Test

Independence:

• Random sample from each population ✓
• Observations independent within and between groups ✓

Normality:

• Both groups have n ≥ 30, no extreme outliers ✓

Equal variance:

• Similar spread in both groups (SDs within a factor of two) ✓

Conditions are met for using the t-distribution.

Bootstrap Confidence Interval for Difference in Means

• The method for calculating bootsrap confidence intervals for a difference in means is similar to the methods we used for a difference in proportions
• We create bootstrap samples from each group (resampling with replacement) and calculate a difference in means
• We do this 1,000 time and use the resulting sampling distribution to calculate a bootstrap percentile CI

Mathematical Model CI Formula for Difference in Means

When conditions are met:

\[\text{CI} = (\bar{x}_1 - \bar{x}_2) \pm t^*_{df} \times SE\]

Where:

• \(\bar{x}_1 - \bar{x}_2\) = observed difference in sample means
• \(t^*_{df}\) = critical value from t-distribution with df = \(n_1 + n_2 - 2\)
• \(SE = s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}\)

Technical Note: Equal Variance

What Affects Strength of Evidence?

For hypothesis testing, stronger evidence = smaller p-value = larger |T|

Recall: \(T = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{SE}\) where \(SE = s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}\)

Three factors affect |T| (and thus p-value):

1. Effect size: Larger \(|\bar{x}_1 - \bar{x}_2|\) → larger |T| → smaller p-value
2. Sample size: Larger \(n_1\) and \(n_2\) → smaller SE → larger |T| → smaller p-value
3. Variability: Smaller \(s_1\) and \(s_2\) → smaller \(s_p\) → smaller SE → larger |T| → smaller p-value

What Affects CI Width?

For confidence intervals: Width = \(2 \times t^* \times SE\)

Where \(SE = s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}\)

Three factors affect width:

1. Confidence level: Higher confidence → larger \(t^*\) → wider CI

2. Sample size: Larger \(n_1\) and \(n_2\) → smaller SE → narrower CI

3. Variability: Smaller \(s_1\) and \(s_2\) → smaller \(s_p\) → smaller SE → narrower CI

Summary

For comparing two independent means:

Conditions: Independence + Normality + Equal variance

References

• Introduction to Modern Statistics (2e) textbook by Mine Çetinkaya-Rundel and Johanna Hardin
• Sections 20.1 - 20.4