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AP Statistics Unit 6: Inference for Categorical Data: Proportions

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Unit 6: Inference for Categorical Data: Proportions

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Question 1

A 95% confidence interval for the proportion of adults who read daily is $(0.31, 0.47)$ . A school administrator says, "95% of adults read daily between 31% and 47% of the time." What is wrong with this interpretation?

A.The interval should be centered at the mean, not the proportion.B.The interval estimates the population proportion, not the frequency of individual behavior; 'between 31% and 47%' refers to confidence about a fixed parameter.C.The interval is too wide; a 95% interval should be narrower.D.The confidence level should be stated as a range, not a single value.

Question 2

The $p$ -value for a one-sided test is 0.03. Which of the following is a correct interpretation of this $p$ -value?

A.There is a 3% probability that

H_{0}

is true.B.If

H_{0}

were true, there is a 3% probability of obtaining a test statistic at least as extreme as the one observed.C.There is a 97% probability that the alternative hypothesis is correct.D.The data support

H_{0}

with 97% confidence.

Question 3

A researcher wants to estimate the proportion of adults who exercise regularly. She samples 400 adults and finds 160 exercise regularly. What is the 95% confidence interval for the true proportion?

(0.35, 0.45)

(0.352, 0.448)

(0.36, 0.44)

(0.40, 0.40)

Question 4

A 90% confidence interval for the proportion of defective items is $(0.04, 0.12)$ . A quality engineer claims the defect rate is 15%. Is the claim consistent with the interval at the 10% significance level?

A.Yes; since 15% is close to 12%, it is consistent.B.No; since 15% falls outside the 90% CI, a two-sided test at

α = 0.10

would reject

H_{0} : p = 0.15

.C.Cannot determine without computing the

p

-value.D.Yes; a 90% CI at

α = 0.10

can only reject proportions below 0.04.

Question 5

Researchers conduct a two-sample $z$ -test for proportions comparing recovery rates between two treatments. They obtain $p$ -value = 0.08 with $α = 0.05$ . A skeptic says the treatment difference is statistically significant. Is the skeptic correct?

A.Yes;

p

-value = 0.08 is close to

α = 0.05

, which is nearly significant.B.No; since

p

-value = 0.08 >

α = 0.05

, we fail to reject

H_{0}

and cannot conclude a significant difference.C.Yes; any

p

-value below 0.10 indicates significance.D.No; a two-sided test should be used, which would double the

p

-value to 0.16.

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Question 6

A large school district conducts a two-sample $z$ -test and finds a statistically significant difference in graduation rates between two schools ( $p$ -value = 0.002). The school with the lower rate has $\overset{p}{^}_{1} = 0.91$ vs. $\overset{p}{^}_{2} = 0.93$ . What is the most important caveat when reporting these results?

A.The test should not have been performed because both rates are above 0.90.B.Statistical significance does not imply practical significance; a 2-percentage-point difference may not be educationally meaningful.C.A

p

-value of 0.002 proves the schools are different in graduation policy.D.With a large enough sample, any small difference will become statistically significant, so no conclusion is possible.

Question 7

For the test $H_{0} : p = 0.80$ vs. $H_{a} : p < 0.80$ using the data from the previous question ( $n = 100$ , $\overset{p}{^} = 0.72$ ), the $z$ -test statistic is approximately $- 2.0$ and the $p$ -value is 0.023. At $α = 0.05$ , what is the correct conclusion?

A.Fail to reject

H_{0}

; there is insufficient evidence that fewer than 80% of customers are satisfied.B.Reject

H_{0}

; there is convincing evidence that fewer than 80% of customers are satisfied.C.Reject

H_{0}

; we have proved the true satisfaction rate is exactly 0.72.D.Accept

H_{0}

; the satisfaction rate is 80%.

Question 8

A researcher increases the confidence level from 90% to 99% while keeping the sample size constant. What happens to the confidence interval?

A.The interval becomes narrower because higher confidence means more precision.B.The interval becomes wider because a higher

z^{*}

is used.C.The interval stays the same because the sample data do not change.D.The interval becomes wider and the point estimate changes.

Question 9

A researcher wants to estimate a population proportion to within ±0.04 with 95% confidence, but has no prior estimate of $p$ . What minimum sample size is needed?

A.385B.601C.25D.2401

Question 10

A poll finds that 54 of 200 surveyed voters support a ballot measure. A reporter states: "We are 95% confident that between 20.5% and 33.5% of all voters support the measure." How should this interval be correctly interpreted?

A.There is a 95% probability that the true population proportion is between 20.5% and 33.5%.B.We are 95% confident that the interval (20.5%, 33.5%) captures the true proportion of all voters who support the measure.C.95% of voters support the measure at a level between 20.5% and 33.5%.D.The interval was constructed by a method that is correct 95% of the time in any given poll.

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Question 11

A random sample of 150 teenagers found 90 who own a smartphone. A random sample of 120 adults found 60 who own a smartphone. Researchers test $H_{0} : p_{teen} = p_{adult}$ vs. $H_{a} : p_{teen} \neq = p_{adult}$ . What is the pooled proportion $\overset{p}{^}_{c}$ ?

\overset{p}{^}_{c} = 0.575

\overset{p}{^}_{c} = \frac{90 + 60}{150 + 120} = \frac{150}{270} \approx 0.556

\overset{p}{^}_{c} = \frac{90}{150} = 0.60

\overset{p}{^}_{c} = \frac{0.60 + 0.50}{2} = 0.55

Question 12

A company claims that 80% of its customers are satisfied. A consumer group surveys 100 customers and finds 72 are satisfied. Which hypothesis should be used to test whether the satisfaction rate is lower than claimed?

H_{0} : p = 0.72

H_{a} : p < 0.72

H_{0} : p = 0.80

H_{a} : p \neq = 0.80

H_{0} : p = 0.80

H_{a} : p < 0.80

H_{0} : p < 0.80

H_{a} : p = 0.80

AP Statistics Unit 6: Inference for Categorical Data: Proportions — Free with Answer Explanations | Test Practice Hub