AP Statistics Unit 3: Inference for Categorical Data: Proportions

From estimators and the sampling distribution of a sample proportion to confidence intervals and significance tests for one proportion and for a difference of two proportions, Type I and Type II errors and power, and chi-square tests for homogeneity and independence in two-way tables.

Topics: 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, 3.14, 3.15 · free response

Key formulas and facts

• Sampling distribution of $\hat p$: $\mu_{\hat p}=p$, $\sigma_{\hat p}=\sqrt{\dfrac{p(1-p)}{n}}$ (10% condition), approximately normal when $np\ge10$ and $n(1-p)\ge10$
• One-proportion z interval: $\hat p\pm z^*\sqrt{\dfrac{\hat p(1-\hat p)}{n}}$; sample size for margin of error $M$: $n\ge\left(\dfrac{z^*}{M}\right)^2p^*(1-p^*)$
• One-proportion z test: $z=\dfrac{\hat p-p_0}{\sqrt{p_0(1-p_0)/n}}$
• Difference of proportions: $\mu_{\hat p_1-\hat p_2}=p_1-p_2$, $\sigma_{\hat p_1-\hat p_2}=\sqrt{\dfrac{p_1(1-p_1)}{n_1}+\dfrac{p_2(1-p_2)}{n_2}}$
• Two-proportion z interval: $(\hat p_1-\hat p_2)\pm z^*\sqrt{\dfrac{\hat p_1(1-\hat p_1)}{n_1}+\dfrac{\hat p_2(1-\hat p_2)}{n_2}}$
• Two-proportion z test: $\hat p_c=\dfrac{x_1+x_2}{n_1+n_2}$ and $z=\dfrac{\hat p_1-\hat p_2}{\sqrt{\hat p_c(1-\hat p_c)\left(\frac1{n_1}+\frac1{n_2}\right)}}$
• Chi-square: expected count $=\dfrac{(\text{row total})(\text{column total})}{\text{table total}}$, $\chi^2=\sum\dfrac{(O-E)^2}{E}$, df $=(r-1)(c-1)$
• P(Type I error) $=\alpha$; power $=1-P(\text{Type II error})$; power rises with $n$, with $\alpha$ and with the distance between the true value and the null value

Inference procedures and conditions

• Four steps for every interval and test. State: the parameter in context, the confidence level or the hypotheses and α. Plan: name the procedure and check the conditions. Do: the statistic, the interval, or the test statistic and p-value. Conclude: answer the question in context.
• Random: a random sample (or random assignment in an experiment); for two samples, the samples are independent.
• 10%: when sampling without replacement, $n\le0.10N$ for each population.
• Large Counts (intervals): at least 10 observed successes and 10 failures in each sample. Large Counts (one-proportion test): $np_0\ge10$ and $n(1-p_0)\ge10$. Two-proportion test: use $\hat p_c$ (or the observed counts) in each group.
• Chi-square: random sample(s) or random assignment, 10% condition, and every expected count at least 5. Homogeneity = separate samples or treatment groups, one variable; independence = one sample, two variables.
• Conclusion of a test: "Because the p-value of ... is less than (greater than) α = ..., we reject (fail to reject) $H_0$. There is (is not) convincing evidence that [$H_a$ in context]." Never "accept $H_0$".
• Interpreting an interval: "We are C% confident that the interval from ... to ... captures [the parameter in context]." Interpreting the confidence level: in many random samples, about C% of the intervals would capture the parameter.

Worked examples

• One-proportion interval: 228 of 600 randomly chosen adults rent their home. $\hat p=0.38$, $SE=\sqrt{0.38(0.62)/600}\approx0.01982$, and a 95% interval is $0.38\pm1.960(0.01982)=(0.341,\ 0.419)$.
• Sample size: a 90% interval with margin of error 0.02 and no prior estimate needs $n\ge(1.6449/0.02)^2(0.25)\approx1690.96$, so $n=1691$ (with the table value 1.645 you get 1692; either is fine as long as you round up).
• One-proportion test: $H_0: p=0.25$, $H_a: p>0.25$; 66 of 220 successes gives $\hat p=0.30$, $z=\frac{0.30-0.25}{\sqrt{0.25(0.75)/220}}\approx1.713$ and p-value $\approx0.0434$, so at α = 0.05 reject $H_0$.
• Two-proportion interval: 90 of 150 versus 70 of 140 gives $\hat p_1-\hat p_2=0.10$ and $SE\approx0.05819$; a 95% interval is $(-0.014,\ 0.214)$, which includes 0.
• Two-proportion test: 45 of 100 versus 30 of 100 gives $\hat p_c=0.375$ and $z=\frac{0.15}{\sqrt{0.375(0.625)(0.02)}}\approx2.191$; two-sided p-value $\approx0.0285$.
• Chi-square: for the table with rows (20, 30) and (30, 20), every expected count is 25 and $\chi^2=4(25/25)=4$ with df = 1; p-value $\approx0.0455$.
• Power: testing $H_0: p=0.5$ against $H_a: p>0.5$ with $n=100$ and α = 0.05 rejects when $\hat p>0.5+1.645(0.05)\approx0.582$; if $p=0.6$, power $=P(\hat p>0.582)\approx0.641$.

Common traps

• Writing hypotheses or interval interpretations about $\hat p$ instead of p.
• Checking Large Counts for a test with $\hat p$ instead of $p_0$, or using the pooled proportion for a confidence interval.
• Saying "there is a 95% probability that p is in the interval": the method, not the computed interval, has the 95% capture rate.
• Interpreting the p-value as the probability that $H_0$ is true.
• Concluding "we accept $H_0$" or "$H_0$ is true" after failing to reject.
• Adding standard deviations instead of variances for a difference of independent proportions.
• Using observed counts instead of expected counts for the chi-square condition, or calling a one-sample, two-variable design a test for homogeneity.
• Treating statistical significance as practical importance: a tiny difference can be significant with a huge sample.

Practice questions

120 multiple-choice questions (4 options each, as on the new exam) and 3 free-response questions worth 10 points each. A calculator is allowed throughout. Try each one before opening the answer. Each line of a scoring guide is one point, scored 0 or 1 and earned independently of the other lines, the style of the new course's 10-point free-response scoring (no E/P/I part scores). A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Topic 3.1: Estimators: bias and variability

8 multiple-choice questions.

Question 1 · Multiple choice

In a random sample of 420 adults in a county, 147 said they had donated blood in the past year. What is the point estimate of the proportion of all adults in the county who donated blood?

1. 0.147
2. 0.350
3. 0.420
4. 0.538

Question 2 · Multiple choice

A random sample of 250 of the 6,000 trees in a state park found 38 infected with a fungus. Using the sample proportion, estimate the number of infected trees in the park.

1. 380
2. 912
3. 1012
4. 1520

Question 3 · Multiple choice

What does it mean for a statistic to be an unbiased estimator of a parameter?

1. It comes from a sample of at least 30.
2. It always equals the value of the parameter.
3. The mean of its sampling distribution equals the value of the parameter.
4. It has a small standard deviation.

Question 4 · Multiple choice

Four estimators are each simulated many times. Estimator W is centered on the parameter with a large spread; X is centered above the parameter with a small spread; Y is centered on the parameter with a small spread; Z is centered below with a large spread. Which estimator is best?

1. X, because it has the smallest spread.
2. Z, because it is the most variable.
3. W, because unbiased estimators are always best.
4. Y, because it is unbiased and has low variability.

Question 5 · Multiple choice

The sample range is a biased estimator of the population range. In which direction is it biased?

1. It is unbiased for large samples only.
2. It has no predictable direction of bias.
3. It tends to underestimate, because a sample rarely includes the population's smallest and largest values.
4. It tends to overestimate, because samples have outliers.

Question 6 · Multiple choice

A random sample of 80 students at a large school found 52 who had a part-time job. What is the point estimate of the proportion of students at the school WITHOUT a part-time job?

1. 0.280
2. 0.350
3. 0.538
4. 0.650

Question 7 · Multiple choice

How does increasing the sample size affect the sampling distribution of an unbiased estimator such as a sample proportion?

1. The bias decreases but the variability stays the same.
2. The center stays at the parameter and the variability decreases.
3. Both the center and the variability stay the same.
4. The center moves toward 0.5 and the variability decreases.

Question 8 · Multiple choice

A school board polled a random sample of 64 parents: 40 supported a later start time, 18 opposed it and 6 had no opinion. What is the point estimate of the proportion of all parents who oppose the later start time?

1. 0.1800
2. 0.2812
3. 0.3103
4. 0.4500

Topic 3.2: The sampling distribution of a sample proportion

8 multiple-choice questions.

Question 9 · Multiple choice

Suppose 35% of all customers of an online store use a coupon. For random samples of 200 customers, what is the standard deviation of the sampling distribution of the sample proportion $\hat p$?

1. 0.0011
2. 0.0337
3. 0.0418
4. 0.4770

Question 10 · Multiple choice

For random samples of 40 students from a large school where 18% of students walk to school, is the sampling distribution of $\hat p$ approximately normal?

1. No: 40 is less than 10% of the school.
2. No: $np=7.2$, which is less than 10.
3. Yes: $np=7.2$ and $n(1-p)=32.8$ are both at least 5.
4. Yes: $n=40$ is at least 30.

Question 11 · Multiple choice

About 62% of a city's voters support a park bond. For a random sample of 300 voters, what is the probability that more than 65% of the sample support the bond?

1. 0.0300
2. 0.1422
3. 0.4613
4. 0.8578

Question 12 · Multiple choice

About 62% of a city's voters support a park bond. For random samples of 300 voters, what is the mean of the sampling distribution of $\hat p$?

1. 0.03
2. 0.38
3. 0.62
4. 0.65

Question 13 · Multiple choice

A researcher takes a random sample of 150 students without replacement from a college of 1,200 students. Is the 10% condition met for using the standard deviation formula for $\hat p$?

1. Yes: 150 is less than 1,200.
2. No: 150 is more than 10% of 1,200 (that is, 120).
3. Yes: the sample is random.
4. It depends on the value of $\hat p$.

Question 14 · Multiple choice

A manufacturer says 4% of its light bulbs are defective. For a random sample of 500 bulbs, what is the probability that the sample proportion defective is less than 0.03?

1. 0.0300
2. 0.1269
3. 0.4000
4. 0.8731

Question 15 · Multiple choice

For random samples of size n from a population with p = 0.5, the standard deviation of $\hat p$ is 0.025. What is n?

1. 20
2. 100
3. 200
4. 400

Question 16 · Multiple choice

Why does the Large Counts condition matter for the sampling distribution of $\hat p$?

1. It makes sure $\hat p$ is unbiased.
2. When $np$ and $n(1-p)$ are both at least 10, the sampling distribution of $\hat p$ is approximately normal.
3. It makes sure the sample is random.
4. It makes sure the observations are independent.

Topic 3.3: Confidence interval for one proportion

8 multiple-choice questions.

Question 17 · Multiple choice

In a random sample of 400 commuters in a city, 156 said they use public transit at least once a week. Which is a 95% confidence interval for the proportion of all the city's commuters who use transit weekly?

1. $(0.294,\ 0.486)$
2. $(0.342,\ 0.438)$
3. $(0.350,\ 0.430)$
4. $(0.367,\ 0.413)$

Question 18 · Multiple choice

What is the critical value $z^*$ for a 98% confidence interval?

1. 1.960
2. 2.054
3. 2.326
4. 2.576

Question 19 · Multiple choice

A random sample of 1,000 adults found 55% in favor of a policy. What is the margin of error for a 99% confidence interval for the population proportion?

1. 0.0157
2. 0.0308
3. 0.0405
4. 0.0448

Question 20 · Multiple choice

A polling firm wants a 95% confidence interval for a proportion with a margin of error of at most 0.03 and has no prior estimate (use $p^*=0.5$). What is the smallest sample size it needs?

1. 33
2. 534
3. 1067
4. 1068

Question 21 · Multiple choice

A random sample of 120 households in a town found 84 with a smart speaker. Which is a 90% confidence interval for the proportion of all households in the town with a smart speaker?

1. $(0.618,\ 0.782)$
2. $(0.631,\ 0.769)$
3. $(0.650,\ 0.750)$
4. $(0.668,\ 0.732)$

Question 22 · Multiple choice

A student wants a 95% confidence interval for the proportion of left-handed students at her school from a random sample of 50 students, 6 of whom are left-handed. Which condition for a one-sample z interval is NOT met?

1. Large Counts: there are only 6 successes.
2. Random: the sample is random.
3. All conditions are met.
4. 10%: the school has more than 500 students.

Question 23 · Multiple choice

A study wants a 99% confidence interval for a proportion with margin of error at most 0.04. A pilot study suggests the proportion is about 0.20. What is the smallest sample size needed?

1. 385
2. 663
3. 664
4. 1037

Question 24 · Multiple choice

In a random sample of 520 adults in a state, 312 said they read news online every day. Which is a 95% confidence interval for the proportion of all adults in the state who do?

1. $(0.545,\ 0.655)$
2. $(0.550,\ 0.650)$
3. $(0.558,\ 0.642)$
4. $(0.579,\ 0.621)$

Topic 3.4: Using a confidence interval for p to judge a claim

8 multiple-choice questions.

Question 25 · Multiple choice

A 95% confidence interval for the proportion of a city's residents who support a new bike lane is (0.512, 0.584). A council member claims that only half of residents support it. Is the claim plausible?

1. Yes: the interval has 95% confidence, so any value is possible.
2. No: the true proportion must be 0.548.
3. No: 0.5 is not in the interval, so the interval gives convincing evidence that more than half support it.
4. Yes: 0.5 is close to 0.512.

Question 26 · Multiple choice

A 90% confidence interval for a population proportion is (0.284, 0.356). What is the point estimate $\hat p$?

1. 0.036
2. 0.284
3. 0.320
4. 0.356

Question 27 · Multiple choice

Using the same 90% interval (0.284, 0.356), what is the margin of error?

1. 0.018
2. 0.036
3. 0.072
4. 0.320

Question 28 · Multiple choice

A 95% confidence interval for the proportion of teens who own a smartwatch is (0.21, 0.27). Which is the correct interpretation of the interval?

1. 95% of teens own a smartwatch.
2. We are 95% confident that the interval from 0.21 to 0.27 captures the proportion of all teens who own a smartwatch.
3. There is a 95% chance that the sample proportion is between 0.21 and 0.27.
4. 95% of all samples will give a proportion between 0.21 and 0.27.

Question 29 · Multiple choice

What does "95% confident" mean for a confidence interval for p?

1. There is a 95% probability that p is in this particular interval.
2. The sample proportion is correct 95% of the time.
3. 95% of the data values fall in the interval.
4. If we took many random samples and built an interval from each, about 95% of the intervals would capture p.

Question 30 · Multiple choice

With the same sample data, how does a 99% confidence interval for p compare with a 90% interval?

1. It is wider, because the sample proportion changes.
2. It has the same width but a different center.
3. It is narrower, because we are more confident.
4. It is wider, because $z^*$ is larger.

Question 31 · Multiple choice

A survey with n = 400 gave a margin of error of 0.05. If the survey is repeated with the same confidence level and about the same $\hat p$, what sample size would cut the margin of error to 0.025?

1. 200
2. 800
3. 1600
4. 3200

Question 32 · Multiple choice

A company says 20% of its customers return a product. A 99% confidence interval from a random sample of returns is (0.162, 0.231). What does the interval say about the company's claim?

1. The interval proves the return rate is exactly 0.20.
2. 0.20 is not plausible because the interval is wide.
3. 0.20 is in the interval, so it is a plausible value; the data do not give convincing evidence against the claim.
4. The claim is false because the center of the interval is not 0.20.

Topic 3.5: Setting up a test for one proportion

8 multiple-choice questions.

Question 33 · Multiple choice

A school says 70% of its graduates attend college. A counselor suspects the rate is lower. Which hypotheses should she test?

1. $H_0: p<0.70$ and $H_a: p=0.70$
2. $H_0: \hat p=0.70$ and $H_a: \hat p<0.70$
3. $H_0: p=0.70$ and $H_a: p\ne0.70$, where p is the proportion in her sample
4. $H_0: p=0.70$ and $H_a: p<0.70$, where p is the proportion of all the school's graduates who attend college

Question 34 · Multiple choice

To test $H_0: p=0.25$ with a random sample of 60, is the Large Counts condition met?

1. No: $np_0=15$ is less than 30.
2. Yes: $np_0=15$ and $n(1-p_0)=45$ are both at least 10.
3. Yes, because the sample is random.
4. No: $n$ must be at least 100.

Question 35 · Multiple choice

To test $H_0: p=0.04$ with a random sample of 150, is the Large Counts condition met?

1. No: $np_0=6$ is less than 10.
2. Yes: $n(1-p_0)=144$ is at least 10.
3. Yes: $n=150$ is large.
4. It cannot be checked until the data are collected.

Question 36 · Multiple choice

A coin-making machine is supposed to produce coins that land heads 50% of the time. An engineer will test whether the machine's coins are unbalanced in either direction. Which alternative hypothesis is appropriate?

1. $H_a: p\ne0.5$
2. $H_a: \hat p\ne0.5$
3. $H_a: p>0.5$
4. $H_a: p<0.5$

Question 37 · Multiple choice

Which procedure should be used to test a claim that more than 40% of a town's adults have a library card, using a random sample of 300 adults?

1. A chi-square test for independence
2. A one-sample z test for a proportion
3. A two-sample z test for a difference in proportions
4. A one-sample t test for a mean

Question 38 · Multiple choice

For a test of $H_0: p=0.30$ with n = 250, what is the standard deviation of $\hat p$ used in the test statistic (computed from $p_0$)?

1. 0.0008
2. 0.0290
3. 0.0458
4. 0.0917

Question 39 · Multiple choice

Why are the hypotheses for a significance test always about a parameter and not a statistic?

1. Statistics cannot be written with symbols.
2. We already know the statistic from the sample; the test asks about the unknown population value.
3. The null hypothesis must always be false.
4. Parameters change from sample to sample.

Question 40 · Multiple choice

A researcher chooses a significance level of $\alpha=0.01$ instead of $\alpha=0.05$. What does this mean?

1. She will need stronger evidence (a p-value below 0.01) before rejecting $H_0$.
2. Her test will be two-sided.
3. Her sample will be larger.
4. She is more likely to reject a true $H_0$.

Topic 3.6: P-values

8 multiple-choice questions.

Question 41 · Multiple choice

A test of $H_0: p=0.5$ against $H_a: p>0.5$ gives a standardized test statistic z = 1.83. What is the p-value?

1. 0.0336
2. 0.0446
3. 0.0672
4. 0.9664

Question 42 · Multiple choice

A two-sided test about a proportion gives z = −2.14. What is the p-value?

1. 0.0162
2. 0.0324
3. 0.0647
4. 0.9838

Question 43 · Multiple choice

A test of whether more than 60% of a store's customers are satisfied gives a p-value of 0.03. What is the correct interpretation?

1. 3% of customers are satisfied.
2. Assuming 60% of all customers are satisfied, there is about a 0.03 probability of getting a sample proportion as large as or larger than the one observed, by chance alone.
3. There is a 0.03 probability that the alternative hypothesis is false.
4. There is a 0.03 probability that 60% of customers are satisfied.

Question 44 · Multiple choice

To test $H_0: p=0.30$ against $H_a: p>0.30$, a student simulated 25 samples of size 50 assuming p = 0.30 (dotplot). Her actual sample had $\hat p=0.38$. Use the simulation to estimate the p-value.

1. 0.08
2. 0.16
3. 0.38
4. 0.84

Question 45 · Multiple choice

A test at $\alpha=0.05$ gives a p-value of 0.072. What is the correct decision?

1. Accept $H_0$: it has been proven true.
2. Reject $H_a$, because the p-value is small.
3. Fail to reject $H_0$, because 0.072 > 0.05.
4. Reject $H_0$, because 0.072 > 0.05.

Question 46 · Multiple choice

A test at $\alpha=0.01$ gives a p-value of 0.004. What is the correct decision?

1. Fail to reject $H_0$, because 0.004 < 0.01.
2. Accept $H_a$ with certainty.
3. Reject $H_0$, because 0.004 < 0.01.
4. No decision is possible at this α.

Question 47 · Multiple choice

A test of $H_0: p=0.8$ against $H_a: p<0.8$ gives z = −1.12. What is the p-value?

1. 0.1120
2. 0.1314
3. 0.2627
4. 0.8686

Question 48 · Multiple choice

Which statement about p-values is correct?

1. A large p-value means the sample was biased.
2. The smaller the p-value, the stronger the evidence against $H_0$.
3. The p-value is the probability that $H_0$ is true.
4. A p-value of 0.20 shows that $H_0$ is true.

Topic 3.7: Carrying out a test for one proportion

8 multiple-choice questions.

Question 49 · Multiple choice

A random sample of 520 voters found 287 who support a proposal. For a test of $H_0: p=0.5$ against $H_a: p>0.5$, what is the standardized test statistic?

1. 0.55
2. 1.24
3. 2.37
4. 3.79

Question 50 · Multiple choice

Using the same voter sample (287 of 520 support; $H_a: p>0.5$), what is the p-value?

1. 0.0089
2. 0.0179
3. 0.0500
4. 0.9911

Question 51 · Multiple choice

A city says 15% of its households compost. An environmental group thinks the rate is higher. In a random sample of 220 households, 41 compost. At $\alpha=0.05$, what should the group conclude?

1. Reject $H_0$: there is convincing evidence that more than 15% of households compost.
2. Reject $H_0$: the sample proportion 0.186 is greater than 0.15.
3. Fail to reject $H_0$: there is not convincing evidence that more than 15% of households compost.
4. Accept $H_0$: exactly 15% of households compost.

Question 52 · Multiple choice

An airline claims 90% of its flights arrive on time. A traveler group suspects fewer do. A random sample of 200 flights had 172 on time. At $\alpha=0.05$, what is the conclusion?

1. Reject $H_0$: there is convincing evidence that fewer than 90% of the airline's flights arrive on time.
2. Fail to reject $H_0$: 0.86 is close to 0.90.
3. Fail to reject $H_0$: there is not convincing evidence that fewer than 90% arrive on time.
4. Reject $H_0$: this proves exactly 86% of flights are on time.

Question 53 · Multiple choice

A random sample of 150 students found 63 who prefer e-books. For a two-sided test of $H_0: p=0.5$, what is the p-value?

1. 0.0250
2. 0.0500
3. 0.1000
4. 0.4200

Question 54 · Multiple choice

A random sample of 400 phones from a production line found 18 defective. For a test of $H_0: p=0.03$ against $H_a: p>0.03$, what is the z statistic?

1. 0.04
2. 0.88
3. 1.76
4. 3.52

Question 55 · Multiple choice

Using the phone sample (18 defective out of 400; $H_0: p=0.03$, $H_a: p>0.03$), what is the conclusion at $\alpha=0.05$?

1. Accept $H_a$: exactly 4.5% are defective.
2. Fail to reject $H_0$: 18 is a small number of phones.
3. Fail to reject $H_0$: not convincing evidence that more than 3% are defective.
4. Reject $H_0$: convincing evidence that more than 3% of the phones are defective.

Question 56 · Multiple choice

A test gives a p-value of 0.31 for $H_a: p>0.4$. Which conclusion is written correctly in context?

1. Because 0.31 > 0.05, we accept $H_0$: exactly 40% of members prefer the new schedule.
2. There is a 31% chance that 40% of members prefer the new schedule.
3. Because 0.31 > 0.05, we reject $H_0$: more than 40% prefer the new schedule.
4. Because 0.31 > α = 0.05, we fail to reject $H_0$. We do not have convincing evidence that more than 40% of all members prefer the new schedule.

Topic 3.8: Errors and power

8 multiple-choice questions.

Question 57 · Multiple choice

A factory tests $H_0$: the machine is working properly against $H_a$: the machine needs adjustment. What would a Type I error be?

1. Concluding the machine needs adjustment when it is actually working properly
2. Adjusting a machine that needs adjustment
3. Concluding the machine is working properly when it actually needs adjustment
4. Failing to collect a random sample

Question 58 · Multiple choice

A medical test checks $H_0$: the patient does not have the disease. What would a Type II error be, and what is a consequence?

1. Concluding the patient does not have the disease when they do; the patient might not get needed treatment.
2. The test result is lost; the test must be repeated.
3. Concluding the patient has the disease when they do; treatment begins.
4. Concluding the patient has the disease when they do not; the patient might get unneeded treatment.

Question 59 · Multiple choice

A researcher runs a test at $\alpha=0.05$. If $H_0$ is actually true, what is the probability of a Type I error?

1. 0.025
2. 0.050
3. 0.500
4. 0.950

Question 60 · Multiple choice

A company runs 12 independent significance tests at $\alpha=0.05$, and every null hypothesis is actually true. What is the probability of at least one Type I error?

1. 0.050
2. 0.460
3. 0.540
4. 0.600

Question 61 · Multiple choice

A test of $H_0: p=0.5$ against $H_a: p>0.5$ uses n = 200 and $\alpha=0.05$ (it rejects $H_0$ when $\hat p$ is above about 0.558). If the true proportion is 0.58, what is the power of the test?

1. 0.050
2. 0.420
3. 0.734
4. 0.950

Question 62 · Multiple choice

Which change would increase the power of a significance test, with everything else unchanged?

1. Decreasing the significance level from 0.05 to 0.01
2. Testing a true value of p closer to the null value
3. Increasing the sample size
4. Using a two-sided alternative instead of the correct one-sided alternative

Question 63 · Multiple choice

A test has power 0.83 against a particular alternative value. What is the probability of a Type II error for that alternative?

1. 0.05
2. 0.17
3. 0.83
4. 0.95

Question 64 · Multiple choice

A school tests whether a new tutoring program raises the pass rate. Which error would be worse if the program is very expensive, and which α should the school choose?

1. A Type I error (paying for a program that does not help) is worse, so use a small α such as 0.01.
2. A Type II error is worse, so use α = 0.10.
3. Neither error matters, so α is irrelevant.
4. A Type I error is worse, so use α = 0.10.

Topic 3.9: The sampling distribution of a difference in sample proportions

8 multiple-choice questions.

Question 65 · Multiple choice

In State A, 40% of adults have a passport; in State B, 32% do. Independent random samples of 300 adults from A and 250 from B are taken. What is the mean of the sampling distribution of $\hat p_A-\hat p_B$?

1. -0.08
2. 0.08
3. 0.36
4. 0.72

Question 66 · Multiple choice

Using the same two states (p = 0.40, n = 300 and p = 0.32, n = 250), what is the standard deviation of $\hat p_A-\hat p_B$?

1. 0.0016
2. 0.0288
3. 0.0409
4. 0.0578

Question 67 · Multiple choice

In State A, 40% of adults have a passport; in State B, 32% do. For independent random samples of 300 adults from A and 250 from B, what is the probability that $\hat p_A-\hat p_B$ is greater than 0.15?

1. 0.0434
2. 0.1301
3. 0.1500
4. 0.9566

Question 68 · Multiple choice

Two random samples of 25 are taken from two large schools where 12% and 15% of students play an instrument. Is the sampling distribution of $\hat p_1-\hat p_2$ approximately normal?

1. No: $n_1p_1=3$ and $n_2p_2=3.75$ are less than 10.
2. Yes: the combined sample is 50.
3. Yes: the difference in proportions is small.
4. Yes: both samples are random.

Question 69 · Multiple choice

In Town 1, 55% of residents favor a curfew; in Town 2, 50% do. Random samples of 400 are taken in each town. What is the probability that the sample from Town 2 shows a higher proportion in favor than the sample from Town 1 (that is, $\hat p_1-\hat p_2<0$)?

1. 0.0500
2. 0.0781
3. 0.5000
4. 0.9219

Question 70 · Multiple choice

Why must the two samples be independent to use $\sigma_{\hat p_1-\hat p_2}=\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}$?

1. Independence makes the distribution centered at 0.
2. The formula needs both samples to have the same size.
3. The variances of two random variables add only when the variables are independent.
4. Independent samples are always larger.

Question 71 · Multiple choice

Suppose $p_1=p_2=0.3$ and independent random samples of 150 and 100 are taken. What is the standard deviation of $\hat p_1-\hat p_2$?

1. 0.0000
2. 0.0290
3. 0.0592
4. 0.0832

Question 72 · Multiple choice

For the sampling distribution of $\hat p_1-\hat p_2$ with $p_1=0.62$ and $p_2=0.70$, what is the mean?

1. -0.08
2. 0.08
3. 0.66
4. 1.32

Topic 3.10: Confidence interval for a difference in proportions

8 multiple-choice questions.

Question 73 · Multiple choice

In a random sample of 240 city residents, 132 recycle weekly; in a random sample of 230 suburban residents, 98 do. Which is a 95% confidence interval for $p_{city}-p_{suburb}$?

1. $(-0.016,\ 0.164)$
2. $(0.034,\ 0.214)$
3. $(0.049,\ 0.199)$
4. $(0.074,\ 0.174)$

Question 74 · Multiple choice

Using the recycling samples (132 of 240 city; 98 of 230 suburban), what is the standard error of $\hat p_1-\hat p_2$?

1. 0.0021
2. 0.0323
3. 0.0458
4. 0.0647

Question 75 · Multiple choice

In an experiment, 150 volunteers with a cold took a zinc lozenge and 45 of them recovered within 3 days; 150 took a placebo and 63 recovered within 3 days. Which is a 90% confidence interval for $p_{zinc}-p_{placebo}$?

1. $(-0.228,\ -0.012)$
2. $(-0.210,\ -0.030)$
3. $(-0.170,\ 0.010)$
4. $(0.030,\ 0.210)$

Question 76 · Multiple choice

Which condition must be checked for a two-sample z interval for $p_1-p_2$ that is NOT needed for a one-sample interval?

1. Both populations must be normal.
2. The two samples (or groups) must be independent of each other.
3. Each sample size must be at least 30.
4. The two sample proportions must be equal.

Question 77 · Multiple choice

A random sample of 350 teens found 210 who use a certain app; a random sample of 320 adults found 168 who do. Which is a 99% confidence interval for $p_{teens}-p_{adults}$?

1. $(-0.024,\ 0.174)$
2. $(0.000,\ 0.150)$
3. $(0.026,\ 0.224)$
4. $(0.055,\ 0.095)$

Question 78 · Multiple choice

A 95% confidence interval for $p_1-p_2$ is (0.023, 0.141). What is the margin of error?

1. 0.0295
2. 0.0590
3. 0.0820
4. 0.1180

Question 79 · Multiple choice

Using the teen/adult app samples (210 of 350 teens; 168 of 320 adults), what is the margin of error for a 95% confidence interval for $p_{teens}-p_{adults}$?

1. 0.0386
2. 0.0581
3. 0.0750
4. 0.0985

Question 80 · Multiple choice

Using the zinc lozenge experiment (45 of 150 recovered with zinc; 63 of 150 with placebo), what is the point estimate of $p_{zinc}-p_{placebo}$?

1. -0.18
2. -0.12
3. 0.12
4. 0.72

Topic 3.11: Using an interval for p1 - p2 to judge a claim

8 multiple-choice questions.

Question 81 · Multiple choice

A 95% confidence interval for $p_{new}-p_{old}$, the difference in the proportions of customers who renew with a new versus old loyalty program, is (0.018, 0.094). What does the interval suggest?

1. 0 is not in the interval, so there is convincing evidence that the renewal rate is higher with the new program.
2. The new program's renewal rate is exactly 5.6 percentage points higher.
3. 0 is plausible, so the programs may have the same renewal rate.
4. The interval cannot be used to compare the programs.

Question 82 · Multiple choice

A 90% confidence interval for $p_{men}-p_{women}$, the difference in the proportions who prefer a streaming service, is (−0.042, 0.065). What does the interval suggest?

1. 0 is in the interval, so there is not convincing evidence of a difference between men and women.
2. Exactly 1.15% more men prefer the service.
3. There is convincing evidence of a difference because the interval is not centered at 0.
4. Men are more likely to prefer the service because 0.065 > 0.042.

Question 83 · Multiple choice

A 99% confidence interval for $p_1-p_2$ is (0.03, 0.17). Which statement is correct?

1. There is a 99% chance that $\hat p_1-\hat p_2$ is between 0.03 and 0.17.
2. $p_1-p_2=0.10$ with 99% certainty.
3. We are 99% confident that the interval from 0.03 to 0.17 captures the true difference $p_1-p_2$.
4. 99% of individuals in population 1 have a value between 0.03 and 0.17.

Question 84 · Multiple choice

A 95% confidence interval for $p_A-p_B$ is (−0.153, −0.061). Which conclusion is correct?

1. The interval is invalid because it is negative.
2. All plausible values are negative, so $p_A$ is greater than $p_B$.
3. All plausible values are negative, so there is convincing evidence that $p_B$ is greater than $p_A$.
4. 0 is plausible, so there is no evidence of a difference.

Question 85 · Multiple choice

A confidence interval for $p_1-p_2$ in a randomized experiment with volunteer subjects is entirely above 0. What conclusion is justified?

1. Treatment 1 causes a higher success rate than Treatment 2 for subjects like these volunteers.
2. Nothing, because volunteers were used.
3. Treatment 1 causes a higher success rate for everyone in the population.
4. Treatment 1 is associated with success, but no causal claim is possible.

Question 86 · Multiple choice

Two 95% confidence intervals for the same difference $p_1-p_2$ come from studies with the same design: Study 1 gives (0.01, 0.09) and Study 2 gives (0.02, 0.06). Which study most likely used larger samples?

1. Study 1, because its upper limit is larger
2. Study 1, because its interval is wider
3. They must have used the same sample sizes
4. Study 2, because its interval is narrower

Question 87 · Multiple choice

A 95% interval for $p_1-p_2$ is (0.004, 0.036). A reporter says the treatment "greatly increases" the success rate. What is the best response?

1. The reporter is right because 0 is not in the interval.
2. The difference is statistically significant, but the data suggest it is small: between 0.4 and 3.6 percentage points.
3. The interval shows the treatment decreases the success rate.
4. The difference is not significant because the interval is narrow.

Question 88 · Multiple choice

A 95% confidence interval for $p_{after}-p_{before}$ is (−0.012, 0.088). What is the point estimate of the difference?

1. 0.038
2. 0.050
3. 0.076
4. 0.100

Topic 3.12: Setting up a test for a difference in proportions

8 multiple-choice questions.

Question 89 · Multiple choice

A researcher wants to know whether the proportion of high school students who vape differs between urban and rural schools. Which hypotheses are appropriate?

1. $H_0: \hat p_U-\hat p_R=0$ and $H_a: \hat p_U-\hat p_R\ne0$
2. $H_0: p_U-p_R\ne0$ and $H_a: p_U-p_R=0$
3. $H_0: p_U-p_R=0$ and $H_a: p_U-p_R>0$
4. $H_0: p_U-p_R=0$ and $H_a: p_U-p_R\ne0$

Question 90 · Multiple choice

In a test comparing two proportions, sample 1 has 54 successes out of 180 and sample 2 has 39 out of 170. What is the pooled (combined) sample proportion $\hat p_c$?

1. 0.2290
2. 0.2647
3. 0.2657
4. 0.3000

Question 91 · Multiple choice

For the samples above (54 of 180 and 39 of 170), is the Large Counts condition met for the two-sample z test?

1. Yes: the combined sample is larger than 30.
2. Yes: 54, 126, 39 and 131 (or the expected counts using $\hat p_c$) are all at least 10.
3. No: 39 is less than 50.
4. No: the samples have different sizes.

Question 92 · Multiple choice

Why is the pooled proportion $\hat p_c$ used in the standard error for a test of $H_0: p_1=p_2$?

1. The test assumes $H_0$ is true, so both groups share one proportion, best estimated by combining the samples.
2. Pooling is also used for confidence intervals for $p_1-p_2$.
3. Pooling always gives a smaller standard error.
4. Pooling makes the samples independent.

Question 93 · Multiple choice

A study randomly assigns 300 volunteers to a new flu vaccine or the standard one and records who gets the flu. Which procedure is appropriate to test whether the new vaccine lowers the flu rate?

1. A one-sample z test for p
2. A chi-square test of independence with a one-sided alternative
3. A two-sample z test for $p_1-p_2$ with a one-sided alternative
4. A two-sample t test for $\mu_1-\mu_2$

Question 94 · Multiple choice

In an experiment comparing two treatments, which condition replaces "random samples from two populations"?

1. The subjects were volunteers.
2. The treatments were randomly assigned to the subjects.
3. Each group has at least 30 subjects.
4. The groups have equal sizes.

Question 95 · Multiple choice

Two independent random samples give 72 successes out of 200 and 95 successes out of 250. What is $\hat p_c$?

1. 0.3600
2. 0.3700
3. 0.3711
4. 0.3800

Question 96 · Multiple choice

A researcher compares the proportions of defective items from two machines using random samples of 60 from each, with 3 and 7 defective items. Is the Large Counts condition met?

1. Yes: the total number of defective items is 10.
2. Yes: each sample has 60 items.
3. No: the expected numbers of defective items, $60\hat p_c=5$, are less than 10.
4. Yes: the samples are independent.

Topic 3.13: Carrying out a test for a difference in proportions

8 multiple-choice questions.

Question 97 · Multiple choice

For the samples 54 of 180 and 39 of 170, what is the z statistic for testing $H_0: p_1=p_2$?

1. 0.071
2. 0.747
3. 1.494
4. 2.694

Question 98 · Multiple choice

Using the same samples (54 of 180 vs 39 of 170) and $H_a: p_1\ne p_2$, what is the p-value?

1. 0.0676
2. 0.0706
3. 0.1351
4. 0.8649

Question 99 · Multiple choice

In an experiment, 200 patients got a new drug and 26 had side effects; 200 got the current drug and 44 had side effects. At $\alpha=0.05$, is there convincing evidence that the new drug has a lower side-effect rate ($H_a: p_{new}-p_{current}<0$)?

1. Yes. Accept $H_a$: the new drug's rate is exactly 0.13.
2. Yes. Reject $H_0$: there is convincing evidence that the new drug has a lower side-effect rate.
3. No. Fail to reject $H_0$: there is not convincing evidence of a lower rate.
4. No. The samples are the same size, so no test is possible.

Question 100 · Multiple choice

A school compares the proportions of students who pass a placement test after two prep courses: 88 of 160 passed after Course A and 79 of 150 after Course B (independent random samples). At $\alpha=0.05$, is there convincing evidence that Course A has a higher pass rate?

1. Accept $H_0$: the pass rates are equal.
2. Reject $H_0$: 0.55 > 0.527.
3. Reject $H_0$: Course A has a higher pass rate.
4. Fail to reject $H_0$: there is not convincing evidence that Course A's pass rate is higher.

Question 101 · Multiple choice

Using the drug experiment (26 of 200 vs 44 of 200 with side effects), what is the z statistic for $H_0: p_{new}-p_{current}=0$?

1. -2.37
2. -1.18
3. -0.09
4. 2.37

Question 102 · Multiple choice

Using the placement samples (88 of 160 for A; 79 of 150 for B) and $H_a: p_A>p_B$, what is the p-value?

1. 0.0230
2. 0.3402
3. 0.6598
4. 0.6805

Question 103 · Multiple choice

Independent random samples of 400 adults in 2015 and 400 adults in 2025 found 150 and 117 who said they read a print newspaper weekly. At $\alpha=0.05$, is there convincing evidence that the proportion changed ($H_a: p_{2015}\ne p_{2025}$)?

1. Yes: the 2025 proportion is exactly 0.2925.
2. No: fail to reject $H_0$; there is not convincing evidence of a change.
3. No: the difference is only 33 people.
4. Yes: reject $H_0$; there is convincing evidence that the proportion changed.

Question 104 · Multiple choice

A two-sample z test for proportions gives a p-value of 0.0004 in a large observational study of coffee drinking and insomnia. Which conclusion is appropriate?

1. There is convincing evidence of a difference in insomnia rates between coffee drinkers and non-drinkers, but we cannot conclude coffee causes insomnia.
2. Coffee causes insomnia.
3. The difference must be large because the p-value is small.
4. The null hypothesis is true with probability 0.0004.

Topic 3.14: Setting up a chi-square test for homogeneity or independence

8 multiple-choice questions.

Question 105 · Multiple choice

Independent random samples of 100 middle school and 120 high school students in a district were asked their favorite sport to watch (table). Which test is appropriate to compare the distributions?

1. A chi-square test for independence
2. A one-sample z test for a proportion
3. A two-sample z test for proportions
4. A chi-square test for homogeneity

Question 106 · Multiple choice

Using the favorite-sport table, what is the expected count of high school students who prefer volleyball if $H_0$ is true?

1. 29.09
2. 32.00
3. 34.91
4. 42.00

Question 107 · Multiple choice

How many degrees of freedom does the chi-square test for the favorite-sport table have?

1. 2
2. 3
3. 5
4. 6

Question 108 · Multiple choice

A random sample of 500 adults is classified by political party (3 categories) and opinion on a ballot measure (support, oppose, undecided). Which hypotheses are appropriate?

1. $H_0$: party and opinion are associated; $H_a$: they are independent.
2. $H_0$: the mean opinion is the same for each party.
3. $H_0$: party and opinion are independent among all adults; $H_a$: party and opinion are not independent (they are associated).
4. $H_0$: the proportions of support are equal in the sample; $H_a$: they are not.

Question 109 · Multiple choice

For the favorite-sport table, is the expected-count condition met?

1. Yes: every expected count is at least 5 (the smallest is about 29.1).
2. Yes: every observed count is at least 10.
3. No: one observed count is 22, which is less than 30.
4. No: the samples have different sizes.

Question 110 · Multiple choice

A two-way table of a random sample has 4 rows and 3 columns. How many degrees of freedom does the chi-square test of independence have?

1. 6
2. 7
3. 11
4. 12

Question 111 · Multiple choice

In an experiment, 150 volunteers were randomly assigned to Diet A (50) or Diet B (100), and their blood pressure did or did not improve (table). What is the expected count for Diet A, Improved, under $H_0$?

1. 15.0
2. 25.0
3. 30.0
4. 37.5

Question 112 · Multiple choice

Which condition is required for a chi-square test for homogeneity or independence?

1. The table has the same number of rows and columns.
2. All observed counts are at least 10.
3. The data are quantitative.
4. All expected counts are at least 5.

Topic 3.15: Carrying out a chi-square test for homogeneity or independence

8 multiple-choice questions.

Question 113 · Multiple choice

For the favorite-sport table (middle school 48, 30, 22; high school 36, 42, 42), what is the chi-square test statistic?

1. 1.21
2. 4.11
3. 8.21
4. 11.21

Question 114 · Multiple choice

Using the favorite-sport table, what is the p-value of the chi-square test for homogeneity?

1. 0.0165
2. 0.0500
3. 0.1448
4. 0.9835

Question 115 · Multiple choice

At $\alpha=0.05$, what is the conclusion of the chi-square test for the favorite-sport table?

1. Reject $H_0$: there is convincing evidence that the distribution of favorite sport differs between middle and high school students.
2. Accept $H_0$: the distributions are the same.
3. Fail to reject $H_0$: there is not convincing evidence of a difference.
4. Reject $H_0$: high school students prefer basketball.

Question 116 · Multiple choice

Which cell contributes most to the chi-square statistic for the favorite-sport table?

1. Middle school, Volleyball
2. High school, Basketball
3. Middle school, Soccer
4. High school, Soccer

Question 117 · Multiple choice

A random sample of 300 students was classified by whether they listen to music while studying and by their grade band (table). What is the chi-square statistic for a test of independence?

1. 2.04
2. 5.95
3. 9.92
4. 13.92

Question 118 · Multiple choice

Using the music-and-grades table, what is the p-value of the chi-square test of independence?

1. 0.0070
2. 0.0291
3. 0.0500
4. 0.0775

Question 119 · Multiple choice

A chain took random samples of 50 purchases at each of three stores and recorded the payment method (table). At $\alpha=0.05$, what is the conclusion of a chi-square test for homogeneity?

1. Accept $H_0$: the distributions are identical.
2. Fail to reject $H_0$: there is not convincing evidence that the distribution of payment method differs among the three stores.
3. Reject $H_0$: the distribution of payment method differs among the stores.
4. Reject $H_0$: Central has the most card payments.

Question 120 · Multiple choice

A chi-square test of independence gives a very small p-value for a large random sample classified by age group and favorite social media platform. Which statement is correct?

1. Age group causes platform preference.
2. There is convincing evidence of an association between age group and favorite platform in the population.
3. The association must be strong because the p-value is small.
4. Every age group prefers a different platform.

Free response

Write a complete answer, in context, as you would type it on the exam. The scoring guides are ours, written in the new course's style: 10 points per question, each point scored 0 or 1 and earned independently (no E/P/I part scores). They are not College Board scoring guidelines.

Question 121 · Free response (10 points)

A school district says that half of its high school students get at least 8 hours of sleep on school nights. A health teacher suspects that the proportion is greater than 0.5 in her large school. She surveys a random sample of 240 of her school's students; 138 report at least 8 hours of sleep on school nights.

1. (a) State the hypotheses for a test of her suspicion, and define the parameter. [2 points]
2. (b) Name the procedure and check the conditions. [2 points]
3. (c) Calculate the test statistic and the p-value. [3 points]
4. (d) State a conclusion at $\alpha=0.05$ in context. [2 points]
5. (e) Interpret the p-value in context. [1 point]

Question 122 · Free response (10 points)

A city tested two designs for its parking app. A random sample of 380 app users was randomly assigned to see Design A (180 users) or Design B (200 users). Of the Design A users, 117 paid for parking through the app within a week; of the Design B users, 96 did.

1. (a) Construct and interpret a 95% confidence interval for $p_A-p_B$. Show the four steps: state, plan (procedure and conditions), do and conclude. [6 points]
2. (b) Does your interval give convincing evidence that Design A leads to a higher proportion of app payments? Explain. [2 points]
3. (c) Can the city conclude that Design A causes more users to pay through the app? To whom can the result be generalized? Explain. [2 points]

Question 123 · Free response (10 points)

A company took a random sample of 340 of its employees and recorded each employee's work arrangement and job satisfaction (table).

1. (a) Name the appropriate test and state its hypotheses. [2 points]
2. (b) Find the expected count of in-office employees who are dissatisfied, and check the expected-count condition. [2 points]
3. (c) Calculate the test statistic, degrees of freedom and p-value. [3 points]
4. (d) State a conclusion at $\alpha=0.05$, and describe the cell that contributes most to the statistic. [3 points]