AP Statistics 3.13.A Two-Proportion Test Statistics and p-Values

Practise calculating two-proportion z-statistics and p-values by pooling under the null, respecting the alternative direction, and using normal or simulated tails.

Exam points

Pool successes across both groups under H0 to compute the two-proportion z-test standard error.
Use the alternative direction to choose one tail or double the corresponding tail probability.
Estimate a randomization p-value by counting simulated differences at least as extreme as observed.

[Maximum number: 5]

A new anti-spam software program is field-tested on 250 randomly selected spam emails and on 750 randomly selected legitimate emails. The results are summarized in the following table (positive test = program labels email as spam):
\begin{tabular}[t]{|l|l|l|l|}
\hline & Spam & Legitimate & Total \\
\hline Positive
test & 205 & 90 & 295 \\
\hline Negative
test & 45 & 660 & 705 \\
\hline Total & 250 & 750 & 1,000 \\
\hline
\end{tabular}
The software developer claims that the proportion of all spam emails for which the tests are positive equals the proportion of all legitimate emails for which the tests are negative.

Figure for Question 3.13.A—Calculate an appropriate test statistic and p-value for testing a hypothesis for the difference between two… question 1 — AP Statistics

Complete the inference procedure, including calculations.

Let $p_1$ be the proportion of spam emails testing positive and $p_2$ the proportion of legitimate emails testing negative. Test $H_0:p_1=p_2$ against $H_a:p_1\ne p_2$ . The samples are random and independent and are less than 10 percent of their populations. With pooled $\hat p=0.865$ , expected successes and failures are 216.25, 33.75, 648.75, and 101.25, all at least 10. The statistic is z=-2.404 and the two-sided p-value is 0.0162.

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3.13.A—Calculate an appropriate test statistic and p-value for testing a hypothesis for the difference between two… question 1