7.7 Justifying a Claim About the Difference of Two Means Based on a Confidence Interval

A confidence interval for a difference of population means estimates $\mu 1 \mu 2$. To justify a claim, check whether 0 is inside the interval and explain the result in context.

If 0 is inside the interval, the data do not provide sufficient evidence that the population means differ. If 0 is outside the interval, the interval supports a claim that the means differ in the direction shown by the subtraction order.

Why This Matters for the AP Statistics Exam

This topic is a classic AP Statistics communication task. The exam expects you to reference the samples, name the populations, use the correct parameter order, and connect the interval to a claim without overstating what the data prove.

A statistical claim is a statement about a population parameter, such as a mean $\mu$ or proportion $p$. With two populations, claims often concern the difference in parameters, such as $\mu_1 - \mu_2 = 0$ versus $\mu_1 - \mu_2 \neq 0$.

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Making a Conclusion

When given a confidence interval and making a conclusion, there is a preset template that we can use that helps us to be sure to include all of the vocabulary necessary to score well on the AP Statistics exam.

Template

Our template should go as follows:

• "Based on these samples, we are ___% confident that the true difference in population means (Group A − Group B) is from ___ to ___ (context of problem).
• "In repeated random sampling with the same sample size, approximately C% of confidence intervals created will capture the difference of population means (include context).

As with any AP Statistics Free Response question, it is always imperative that your answer includes context of the problem given. After all, a comprehensive interpretation for a confidence interval for the difference of two population means should include a reference to the samples taken and details about the populations they represent.

When testing for the difference in two populations, it is also important to use that interval to test the claim that the two population means differ. If 0 is included in your interval, we do not have sufficient evidence that the population means differ. If after completing your interval you do not have 0 in your interval, the results provide evidence that the population means differ in the direction indicated by the interval.

Finishing Our Apples to Apples Example

Let's finish our example with the green apples and red apples.

Suppose our confidence interval for the difference in mean apple weights was (0.408, 0.592), using green minus red as the subtraction order. Our conclusion should go as follows:

"Based on these samples of green apples (n1 = __) and red apples (n2 = __), we are 95% confident that the true difference in population mean weights (green − red) is between 0.408 and 0.592 units. Since 0 is not included in our interval, these data provide strong evidence that μ_green > μ_red (the population mean weight of green apples exceeds that of red apples)."

Notice that because 0 was not in our interval, the interval suggests a positive difference (green − red), which supports the conclusion above.

Relating Confidence Intervals and Sample Sizes

When all other things remain the same, the width of the confidence interval for the difference of two means tends to decrease as the sample sizes increase**.** That’s right: the margin of error is half the width of the interval. Margin of error = (critical value) × (standard error); interval width = 2 × margin of error. The margin of error decreases as sample sizes increase, approximately at a 1/√n rate. For two samples, SE = sqrt(s1^2/n1 + s2^2/n2); increasing n1 and/or n2 reduces SE and narrows the interval.

Recall that the confidence interval represents the range of values within which the true population parameter is likely to fall, with a certain level of confidence. In the case of the difference of two means, the confidence interval is calculated by taking the difference between the two sample means and adding and subtracting the margin of error. The margin of error is calculated based on the standard error of the difference between the means and the desired level of confidence.

Increasing the sample size allows for a more precise estimate of the population parameter, which results in a narrower confidence interval and a smaller margin of error.