AP Statistics Inference for the Difference of Two Means Lesson (Free PDF)

AP Statistics math lesson: Inference for the Difference of Two Means – Maths PDF

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This two means lesson for AP Statistics covers inference for the difference of two means (Common Core AP STAT 7.6, AP STAT 7.7, AP STAT 7.8, AP STAT 7.9, HSS-IC.B.5), with a free PDF to print.

To compare two means from independent groups, use a two-sample t procedure with standard error sqrt(s1^2/n1 + s2^2/n2). The interval is the difference in sample means plus or minus t* times that standard error. Check independence, the 10 percent condition, and normality or large samples. For matched data, use a paired t procedure on the differences.

AP Statistics · Common Core State Standards: AP STAT 7.6, AP STAT 7.7, AP STAT 7.8, AP STAT 7.9, HSS-IC.B.5

Key facts in this two means lesson

Key facts

Parameter: mu1 – mu2; statistic: x-bar1 – x-bar2.
SE = sqrt(s1^2/n1 + s2^2/n2); df from the calculator, or min(n1 – 1, n2 – 1) conservatively.
Interval: (x-bar1 – x-bar2) +/- t* SE; test: t = (x-bar1 – x-bar2)/SE with H0: mu1 – mu2 = 0.
Conditions: independent random samples or random assignment; 10%; each group normal or n >= 30.
Same subjects measured twice or matched pairs: use a one-sample t on the differences.

What do we compare with two means?

Two-sample procedures compare the means \(\mu_1\) and \(\mu_2\) of two populations, or the mean responses of two treatments. The statistic is \(\overline{x}_1-\overline{x}_2\). The groups must be independent: different people in each group, with no pairing.

Scores in two independent classes taught with different methods.

Example: two classes are taught with different methods. The difference in sample means is \(78.4-72.1=6.3\) points.

How do you build a two-sample t interval?

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

The degrees of freedom come from the calculator (or the smaller of \(n_1-1\) and \(n_2-1\) as a safe conservative choice). Conditions: independent random samples or random assignment; 10% condition for each sample; and each group approximately normal or large (at least 30). Here both groups have at least 30 students.

Boxplots of both classes: no outliers and roughly symmetric.

\(SE=\sqrt{\frac{9.2^{2}}{35}+\frac{11.5^{2}}{40}}=2.393\), \(df\approx 72.5\), \(t^{*}=1.993\), \(ME=4.769\). The 95% interval is (1.53, 11.07).

95% interval for mu_A – mu_B: (1.53, 11.07); 0 is not included.

We are 95% confident that the interval from 1.53 to 11.07 points captures the true difference in mean scores between the populations taught with Method A and Method B. Since the whole interval is positive, Method A seems to give higher mean scores.

How do you test H0: mu1 = mu2?

State \(H_0:\mu_1-\mu_2=0\) against \(H_a:\mu_1-\mu_2\neq 0\) (or a one-sided alternative). The test statistic is:

\[t=\frac{(\overline{x}_1-\overline{x}_2)-0}{\sqrt{\frac{s_1^{2}}{n_1}+\frac{s_2^{2}}{n_2}}}\]

Here \(t=\frac{6.3}{2.393}=2.63\), and the two-sided p-value is \(0.0104\).

Two-sided p-value for t = 2.63: 0.0104.

Since 0.0104 is less than 0.05, we reject \(H_0\). There is convincing evidence of a difference in mean scores, and the interval tells us the size.

When is a paired procedure needed?

Choose the procedure by the design.

If the same students take both methods, or are matched in pairs, compute the differences and use a one-sample t procedure on them. Mixing up the two designs is the most common error.

Two means lesson: questions and answers

How do I know whether to use a two-sample or a paired procedure?

Ask whether the two sets of measurements come from different individuals (two-sample) or from the same or matched individuals (paired). Pairing creates dependence, so the differences are analyzed as one sample.

How are the degrees of freedom found?

The calculator uses a formula that gives a non-integer df based on both standard deviations and sample sizes. If you work by hand, use the smaller of n1 – 1 and n2 – 1, which is conservative.

Why do the variances add under the square root?

The two samples are independent, so the variance of the difference is the sum of the variances. Standard deviations never add directly.

What does it mean when the interval for mu1 – mu2 contains 0?

It means 0 is a plausible value for the difference, so there is not convincing evidence that the two population means differ.

Do I need equal standard deviations?

No. The two-sample t procedures in AP Statistics do not assume equal variances; each group keeps its own standard deviation in the standard error.

How to use this two means lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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