Topic 4.7 Notes – Constructing a Confidence Interval for the Difference Between Two Population Means

What a Two-Sample t-Interval Estimates

This interval estimates a difference in population means. The parameter is $\mu_1 - \mu_2$, and you have to name it in context.

If group 1 is students using Method A and group 2 is students using Method B, then the parameter might be “the population mean quiz score for students using Method A minus the population mean quiz score for students using Method B.”

A few things have to stay straight:

• The point estimate is the sample difference $\bar{x}_1 - \bar{x}_2$.
• Order matters. If you switch to $\bar{x}_2 - \bar{x}_1$, every sign flips, including the interval.
• This is for a quantitative response, like time, height, score, or amount.
• The groups must be independent.
• Population standard deviations are unknown, which is why this is a $t$-interval.
• This is not for proportions or categorical data.
• This is not for matched pairs. If the data are paired, you make one difference per pair and use a one-sample $t$-interval on those differences.

When This Procedure Is Appropriate

Before calculating anything, the setting has to match the method.

You need all of these:

• a quantitative response variable
• two groups or populations
• independent samples or independently assigned treatment groups
• a goal of estimating a difference in means
• unknown population standard deviations

Then check the inference conditions.

Conditions

• Randomization
  • You need two independent random samples, or a randomized experiment with two treatments.
  • Independence between groups comes from the study design.
• 10% condition
  • If sampling without replacement, check both groups.
  • $n_1 \le 0.10N_1$ and $n_2 \le 0.10N_2$
  • You do not need this for a randomized experiment.
• Sample data condition
  • Either both populations are approximately normal, or both sample sizes are at least 30.
  • If either sample is under 30, then both sample distributions should have no strong skewness or outliers.

That “both groups” point gets tested a lot. One nice-looking sample does not rescue the other one.

Checking normality in both groups

Building the Interval

Here’s the full structure for a two-sample $t$-interval for $\mu_1 - \mu_2$.

1. Parameter and procedure
   State that you are constructing a two-sample $t$-interval for $\mu_1 - \mu_2$.

2. Point estimate

   $$\bar{x}_1 - \bar{x}_2$$

3. Standard error

   $$SE_{\bar{x}_1-\bar{x}_2} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$

   The variances add because the samples are independent. You do not add standard deviations directly.

4. Critical value
   Use $t^*$ for your confidence level and appropriate degrees of freedom.

   • Technology finds df for you.
   • It will be between the smaller of $n_1-1$ and $n_2-1$, and $n_1+n_2-2$.
   • With a table, using the smaller df is conservative.

5. Margin of error

   $$ME = t^*\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$

6. Confidence interval

   $$(\bar{x}_1-\bar{x}_2)\pm ME$$

Quick example. If $\bar{x}_1=52$, $\bar{x}_2=47$, $s_1=8$, $s_2=6$, $n_1=25$, $n_2=25$, then the center is $52-47=5$. Everything is built around that 5, with the same subtraction order.

Calculator note: use 2-SampTInt and if it asks about pooling, choose No.

What the Interval Means

A correct interpretation sounds like this:

“We are 95% confident that the true difference in population mean waiting time, $\mu_1-\mu_2$, is between 1.2 and 4.8 minutes.”

A few meanings come from the interval itself:

• The interval estimates the population difference, not the sample difference.
• The center is $\bar{x}_1-\bar{x}_2$.
• A wider margin of error means a wider interval.
• If 0 is in the interval, a difference of 0 is plausible, so there is no clear evidence of a population mean difference at that confidence level.
• If 0 is not in the interval, a real difference in population means is plausible.
• If the whole interval is positive, population 1 likely has the larger mean.
• If the whole interval is negative, population 1 likely has the smaller mean.

Common Mistakes to Avoid

• Treating matched pairs like independent samples
• Naming the procedure but not defining the parameter in context
• Switching subtraction order halfway through
• Writing vague condition checks instead of using study details
• Checking the normality condition for only one sample
• Using $z$ instead of $t$
• Choosing pooled or assuming equal variances
• Interpreting the interval as about individual values instead of population means
• Forgetting the units