AP Statistics 4.10 Carrying Out a Test for the Difference Between Two Population Means Study Notes - New Syllabus

4.10.A.1 Test Statistic for a Two-Sample t-Test

After stating the hypotheses and verifying the required conditions, the next step in a two-sample t-test is to calculate the test statistic.

The test statistic measures how many standard errors the observed difference between the sample means is from the difference stated in the null hypothesis.

For most AP Statistics problems, the null hypothesis states

\( H_0:\mu_1-\mu_2=0 \)

so the test statistic compares the observed sample difference with 0.

Test Statistic Formula

\( t=\dfrac{(\bar{x}_1-\bar{x}_2)-(\mu_1-\mu_2)}{\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}} \)

Since the null hypothesis usually states

\( \mu_1-\mu_2=0 \),

the formula simplifies to

\( t=\dfrac{\bar{x}_1-\bar{x}_2}{\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}} \)

Where:

• \( \bar{x}_1,\bar{x}_2 \) = Sample means
• \( s_1,s_2 \) = Sample standard deviations
• \( n_1,n_2 \) = Sample sizes
• \( \mu_1-\mu_2 \) = Difference stated in the null hypothesis (usually 0)

Degrees of Freedom

The test statistic follows a t-distribution when the null hypothesis is true.

The degrees of freedom are calculated using technology.

The degrees of freedom always satisfy

\( \min(n_1-1,\;n_2-1)\le df\le n_1+n_2-2 \)

Graphing calculators and statistical software automatically compute the appropriate degrees of freedom.

Interpreting the Test Statistic

• A large positive \(t\)-value provides evidence that Population 1 has a larger mean than Population 2.
• A large negative \(t\)-value provides evidence that Population 1 has a smaller mean than Population 2.
• A \(t\)-value close to 0 indicates that the observed sample difference is close to the difference expected under the null hypothesis.

Important AP Exam Notes

• The test statistic measures how unusual the observed difference is if the null hypothesis is true.
• Use the simplified formula only when the null hypothesis states \( \mu_1-\mu_2=0 \).
• The test statistic follows a t-distribution.
• The degrees of freedom are determined using technology.
• A larger absolute value of \(t\) generally leads to a smaller p-value.

4.10.A.2 Calculating the p-value for a Two-Sample t-Test

After calculating the test statistic, the next step in a two-sample t-test is to determine the p-value.

The p-value measures how likely it is to obtain a test statistic as extreme as the one observed, assuming that the null hypothesis is true.

For a two-sample t-test, the p-value is found using the appropriate t-distribution.

On the AP Statistics Exam, the p-value is usually determined using a graphing calculator or other statistical technology.

Finding the p-value Using Technology

On a TI-84 Graphing Calculator:

STAT → TESTS → 2-SampTTest

Enter either:

• The raw data (Lists L1 and L2), or
• The summary statistics (\(\bar{x}_1,\;s_1,\;n_1,\;\bar{x}_2,\;s_2,\;n_2\)).

Select the appropriate alternative hypothesis:

• \( \mu_1>\mu_2 \)
• \( \mu_1<\mu_2 \)
• \( \mu_1\ne\mu_2 \)

Then select Calculate.

The calculator displays:

• The test statistic (\(t\))
• The p-value
• The degrees of freedom (\(df\))

Relationship Between the Test Statistic and the p-value

Important AP Exam Notes

• The p-value is calculated assuming the null hypothesis is true.
• Use the t-distribution, not the normal distribution.
• The p-value can be found using a t-table, but the AP Exam typically expects the use of technology.
• A smaller p-value indicates stronger evidence against the null hypothesis.
• The p-value itself is not the probability that the null hypothesis is true.