Topic 7.6 Notes – Confidence Intervals for the Difference of Two Means

What a Confidence Interval for μ₁ − μ₂ Does

Here, the parameter of interest is:

$$\mu_1 - \mu_2$$

That is the true difference between two population means.

From your samples, you calculate the point estimate:

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

So the entire interval is estimating one thing: the difference between populations.

If we repeated this sampling method many times, about C% of the intervals would capture the true value of μ₁ − μ₂.

How to Interpret 0 in the Interval

• If 0 is inside the interval → a difference of 0 is plausible → no convincing evidence of a difference.
• If the entire interval is positive → μ₁ > μ₂.
• If the entire interval is negative → μ₁ < μ₂.

On an FRQ, they love hiding the conclusion in the sign. Always ask yourself: does this interval include 0?

When to Use a Two-Sample t-Interval

This procedure is appropriate when:

• You are comparing one quantitative variable
• Across two independent groups
• Using two independent random samples OR a randomized experiment
• Population standard deviations are unknown (almost always true)

This is called a two-sample t-interval (not pooled).

Do not use it for:

• Matched pairs (use a one-sample t-interval on differences)
• Categorical data (that’s a two-proportion z-interval)

The key word is independent. If the same subjects appear in both groups, it is not this procedure.

Why the Sampling Distribution Works

You already know that for one mean, the sampling distribution of $\bar{x}$ is approximately normal under the right conditions.

Now we apply that idea to the difference:

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

If:

• Both populations are normal
• Both sample sizes are at least 30

then the sampling distribution of $\bar{x}_1 - \bar{x}_2$ is approximately normal.

Its:

• Mean = $\mu_1 - \mu_2$
• Standard deviation = $\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$

Since we don’t know σ₁ and σ₂, we estimate with s₁ and s₂ and use a t-distribution.

Conditions for Inference

You must check independence and normality.

Independence

• Two independent random samples OR randomized experiment with random assignment.
• If sampling without replacement:
  • $n_1 \le 10\%$ of population 1
  • $n_2 \le 10\%$ of population 2

If it’s a randomized experiment, the 10% condition is not required.

Nearly Normal Condition

We need the sampling distribution of $\bar{x}_1 - \bar{x}_2$ to be approximately normal.

That happens if:

• Both populations are normal
• Both $n_1 \ge 30$ and $n_2 \ge 30$

If samples are small, check boxplots or histograms for strong skewness or outliers in either group.

If one sample is small and heavily skewed, stop. The procedure is not appropriate.

The Formula and Pieces

The confidence interval is:

$$(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$

It follows the familiar structure:

point estimate ± (critical value × standard error)

Standard Error

$$\text{SE} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$

Margin of Error

$$\text{ME} = t^* \times \text{SE}$$

Degrees of freedom are handled by your calculator.

On a TI-84:

• STAT → TESTS → 2-SampTInt
• Choose Stats if given summary statistics
• Select not pooled

AP does not require pooling.

Example of Interpretation (In Context)

Suppose you compare mean weekly study hours of students in two different prep programs and get a 95% CI of (1.2, 3.8).

A correct interpretation:

We are 95% confident that the true difference in mean weekly study hours between students in Program 1 and Program 2 (μ₁ − μ₂) is between 1.2 and 3.8 hours.

That means Program 1 students study, on average, between 1.2 and 3.8 more hours per week.

Never say “there is a 95% probability the true difference is in this interval.” The parameter is fixed.

Common Mistakes to Avoid

• Mixing up the order of subtraction.
• Interpreting each mean separately instead of the difference.
• Forgetting to define μ₁ − μ₂ in context.
• Skipping the 10% condition for sampling.
• Using pooled procedures.

On FRQs, most lost points happen in the conditions or the context interpretation, not the math.