Unit 6 test: Inference for Proportions

In a random sample of 600 adults in a large state, 408 say they have a library card. Assume the conditions are met. Find a 90% confidence interval for the proportion of all adults in the state who have a library card. Give both endpoints, rounded to three decimal places. (For 90% confidence, $z^* = 1.645$.)

A poll reports that 52% of likely voters support a candidate, with a margin of error of 3 percentage points at 95% confidence. Which statement is correct?

A researcher wants to estimate a population proportion within 0.025 with 99% confidence and has no prior estimate of $p$. What is the smallest sample size that guarantees this? (For 99% confidence, $z^* = 2.576$.)

Which change would not make a confidence interval for a proportion narrower?

A manufacturer claims that 15% of its customers return for a second purchase within a month. A marketing analyst believes the true proportion is higher. She selects a random sample of 300 customers and finds that 54 returned. Which set of hypotheses should she test?

For the analyst's study (54 of 300 customers returned; $H_0: p = 0.15$), compute the test statistic $z$. Round to two decimal places.

Using $z \approx 1.46$ and $H_a: p > 0.15$, find the P-value. Round to three decimal places.

With a P-value of about 0.073, which is the correct conclusion for the analyst's test at $\alpha = 0.05$?

Based on the decision in the previous problem, which error could the analyst have made, and what would it mean?

A researcher is planning a test of $H_0: p = 0.5$ against $H_a: p > 0.5$. Which combination of choices gives the greatest power against the alternative $p = 0.6$?

A factory's defect rate has been 10%. After a process change, engineers will inspect a random sample of 300 items and test $H_0: p = 0.10$ versus $H_a: p < 0.10$ at $\alpha = 0.05$, rejecting $H_0$ if $\hat{p} \le 0.0715$. If the true defect rate after the change is $p = 0.05$, what is the power of the test? Round to two decimal places.

A school district randomly assigns 900 families to receive either a printed newsletter or an email newsletter about a volunteer program. Of 450 families who got the printed version, 198 signed up; of 450 who got the email, 162 signed up. For testing $H_0: p_P - p_E = 0$ versus $H_a: p_P - p_E > 0$, compute the test statistic $z$. Round to two decimal places.

The newsletter test gives a P-value of about 0.007. Which conclusion is best at $\alpha = 0.05$?

In an independent random sample of 240 residents of Town A, 132 favor a new park; in a random sample of 260 residents of Town B, 117 favor it. Assume the conditions are met. Find a 90% confidence interval for $p_A - p_B$. Give both endpoints, rounded to three decimal places.