Topic 4.8 Notes – Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Means

What a Confidence Interval for μ₁ − μ₂ Means

This interval is about two population means, not about individual people and not about sample means.

• The parameter is $\mu_1-\mu_2$
• The point estimate is $\bar{x}_1-\bar{x}_2$

So if group 1 is students using Method A and group 2 is students using Method B, then the interval estimates:

$$\mu_1-\mu_2=\text{true mean score for Method A} - \text{true mean score for Method B}$$

The interval gives plausible values for that true mean difference.

A full AP Stats interpretation has to name:

• both groups or populations
• the quantitative variable
• the population means
• the subtraction order
• the units

Example wording:

“We are 95% confident that the interval from 2.1 to 5.4 contains the true difference in population mean exam score, $\mu_1-\mu_2$, for Method A minus Method B, measured in points.”

That confidence level has a specific meaning. If you repeated the random sampling or random assignment process many times and built an interval each time, about 95% of those intervals would capture the fixed value of $\mu_1-\mu_2$.

One interval you already computed either contains the parameter or it does not. The 95% is about the method’s long-run success rate, not the probability for this one finished interval.

Order of Subtraction and What the Signs Mean

This is one of the easiest places to lose points. The sign only means something after you know which group is first.

For an interval for $\mu_1-\mu_2$:

• all positive values mean $\mu_1>\mu_2$
• all negative values mean $\mu_1<\mu_2$
• $0$ means $\mu_1=\mu_2$

If you reverse the order, every sign flips. An interval $(a,b)$ for $\mu_1-\mu_2$ becomes $( -b,-a)$ for $\mu_2-\mu_1$.

Example:

• for $\mu_1-\mu_2$, interval is $(-3.2,-0.8)$
• for $\mu_2-\mu_1$, interval is $(0.8,3.2)$

A negative interval does not mean “no difference.” It means the first group’s mean is lower.

Using the Interval to Justify a Claim

Values inside the interval are plausible. Values outside are not supported at that confidence level. The usual benchmark is $0$, because $0$ means equal population means.

Step by step

1. State the interval in context.
   Example: “We are 90% confident that the true difference in mean battery life, Brand X minus Brand Y, is between $-0.4$ and $1.1$ hours.”

2. Check whether $0$ is in the interval.

3. Write the conclusion in evidence language.

   • If $0$ is not in the interval, there is convincing evidence the means differ.
   • If the whole interval is above $0$, there is convincing evidence that $\mu_1>\mu_2$.
   • If the whole interval is below $0$, there is convincing evidence that $\mu_1<\mu_2$.
   • If $0$ is in the interval, there is not sufficient evidence that the means differ.
   • If $0$ is an endpoint, count it as included.

This also works for claims about another value $d$, not just $0$. If $d$ is inside the interval, it’s plausible. If it’s outside, the interval gives evidence against that claim.

What Conclusions You Can and Cannot Make

When the interval contains $0$, say:

• “There is not convincing evidence of a difference.”

Do not say:

• “The means are equal.”
• “There is no difference.”

When the interval excludes $0$, say:

• “There is convincing evidence” or “sufficient evidence.”

Do not say:

• “proved”
• “definitely”

A tiny interval like $(0.1,0.3)$ can show a statistically significant difference, but that difference might be too small to matter in real life.

There’s also a direct link to hypothesis tests:

$$\alpha = 1 - \frac{C}{100}$$

So a 95% confidence interval matches a two-sided test at $\alpha=0.05$.

• $0$ outside the interval ↔ reject $H_0:\mu_1-\mu_2=0$
• $0$ inside the interval ↔ fail to reject $H_0$

Your conclusion also depends on how data were collected:

• random samples let you generalize to populations
• random assignment lets you make cause-and-effect claims about treatments
• observational studies do not justify causation, even if $0$ is outside the interval

Common Mistakes to Avoid

• Treating the interval as if it describes individual observations
• Forgetting subtraction order and reversing the direction
• Writing about $\bar{x}_1-\bar{x}_2$ when the interval estimates $\mu_1-\mu_2$
• Saying “there is 95% probability the true difference is in the interval”
• Saying 95% of observations are in the interval
• Saying “no difference” when $0$ is included
• Ignoring rounding when $0$ is very close to an endpoint
• Making causal claims from observational data
• Forgetting that the two-sample $t$-interval conditions must be met