AP Statistics Quiz: Setting Up Tests For Population Proportion

What this quiz covers

This quiz focuses on Setting Up Tests For Population Proportion, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $H_0: p=0.70\quad H_a: p\ne0.70$ (correct answer)
2. $H_0: p=0.64\quad H_a: p\ne0.64$
3. $H_0: p=0.70\quad H_a: p<0.70$
4. $H_0: \hat{p}=0.70\quad H_a: \hat{p}\ne0.70$
5. $H_0: p\ne0.70\quad H_a: p=0.70$

Explanation: The skill assessed is setting up hypotheses for testing a reported proportion of 0.70 in AP Statistics, to determine accuracy. The hypotheses H0: p = 0.70 and Ha: p ≠ 0.70 fit a two-sided test for verifying the exact value, as in choice A. A distractor is choice D, substituting ˆp for p. Choice C uses a one-sided alternative, unsuitable for accuracy checks. For a mini-lesson, use ≠ in Ha for assessing reported or exact values to allow rejection for any difference. Null includes equality to the reported figure, about p. This setup evaluates if the proportion matches without directional bias.

Question 2

1. The sample was not random, so the results may be biased.
2. The sample size of 20 is too small to apply the Central Limit Theorem.
3. The population size is too small to yield a valid p-value.
4. The sample size is more than 10% of the population, violating the independence condition. (correct answer)

Explanation: The 10% condition states that when sampling without replacement, the sample size $n$ should be no more than 10% of the population size $N$ to ensure independence between observations. Here, the sample size is $n=20$ and the population is $N=120$. Since $20 > 0.10(120) = 12$, the 10% condition is violated.

Question 3

1. The proportion of the 300 sampled adults who prefer coffee over tea.
2. The mean number of cups of coffee preferred by adults in the city.
3. The proportion of all adults in the city who prefer coffee over tea. (correct answer)
4. The number of adults in the city who prefer coffee over tea.

Explanation: A parameter is a value that describes a characteristic of a population. The researcher is making a claim about 'the majority of adults in a large city,' which refers to the entire population of the city. The characteristic of interest is the proportion who prefer coffee. The sample proportion is a statistic used to estimate this parameter.

Question 4

1. $H_0: p = 0.60$ vs. $H_a: p > 0.60$ (correct answer)
2. $H_0: p > 0.60$ vs. $H_a: p = 0.60$
3. $H_0: p = 0.60$ vs. $H_a: p \ne 0.60$
4. $H_0: \hat{p} = 0.60$ vs. $H_a: \hat{p} > 0.60$

Explanation: The null hypothesis ($H_0$) represents the status quo or the boundary of the claim, stated with equality. The alternative hypothesis ($H_a$) represents what the researcher is trying to find evidence for, which in this case is that the proportion is 'more than 60%.' Therefore, $H_0: p = 0.60$ and $H_a: p > 0.60$ are the correct hypotheses.

Question 5

1. A one-sample z-test for a proportion should be used with $H_0: p = 0.45$. (correct answer)
2. A one-sample t-test for a mean should be used with $H_0: \mu = 0.45$.
3. A one-sample z-test for a proportion should be used with $H_0: p \ne 0.45$.
4. A two-sample z-test for proportions should be used with $H_0: p_1 - p_2 = 0$.

Explanation: The problem involves testing a claim about a single population proportion (the proportion of adults in the town with a favorable opinion). Thus, a one-sample z-test for a proportion is appropriate. The null hypothesis is based on the national claim, representing the status quo, so $H_0: p = 0.45$.

Question 6

1. The population of all possible coin flips must be approximately normal.
2. The sample size of 40 must be less than 10% of all possible coin flips.
3. The number of heads and tails observed in the 40 flips must both be at least 10.
4. Assuming the coin is fair, the expected number of heads and tails must both be at least 10. (correct answer)

Explanation: The Large Counts condition for a significance test requires using the hypothesized proportion $p_0$. To test if a coin is biased, the null hypothesis is that it is fair, so $H_0: p = 0.5$. The expected counts are $40(0.5) = 20$ heads and $40(0.5) = 20$ tails. Since both are at least 10, the condition is met. Using observed counts is incorrect.

Question 7

1. A one-sample z-test for a population proportion (correct answer)
2. A one-sample t-test for a population mean
3. A two-sample z-test for a difference in population proportions
4. A chi-square test for goodness-of-fit

Explanation: The scenario involves a single categorical variable (support for the candidate) from one population (registered voters in the city). The goal is to test a claim about a single population proportion. Therefore, a one-sample z-test for a population proportion is the appropriate procedure.

Question 8

1. $H_0: p=0.50$; $H_a: p\ne 0.50$ (correct answer)
2. $H_0: p=0.54$; $H_a: p\ne 0.54$
3. $H_0: \hat{p}=0.50$; $H_a: \hat{p}\ne 0.50$
4. $H_0: p\ne 0.50$; $H_a: p=0.50$
5. $H_0: p=0.50$; $H_a: p>0.50$

Explanation: This question tests setup for testing if approval differs from exactly 50%. Correct choice A: H0: p = 0.50 and Ha: p ≠ 0.50, for two-sided testing. Distractors use sample p-hat = 0.54 in H0, like option B, or p-hat in C. Option D reverses H0 and Ha. Mini-lesson: 'Differs from' calls for ≠ in Ha, checking both directions. Equality is always in H0, about p. Sample proportion is for the z-test, not hypotheses.

Question 9

1. $H_0: p=0.62$; $H_a: p\ne 0.62$
2. $H_0: p=0.55$; $H_a: p\ne 0.55$ (correct answer)
3. $H_0: p\ne 0.55$; $H_a: p=0.55$
4. $H_0: \hat{p}=0.55$; $H_a: \hat{p}\ne 0.55$
5. $H_0: p=0.55$; $H_a: p>0.55$

Explanation: This question tests hypothesis setup for a principal's belief that the proportion of students eating breakfast differs from 55%, requiring a two-sided test. The correct hypotheses are H0: p = 0.55 and Ha: p ≠ 0.55, as in choice B, matching the 'different from' claim. A distractor is using the sample p-hat = 0.62 in H0, like option A, which confuses sample with population. Option C reverses H0 and Ha, violating the rule that H0 includes equality. Mini-lesson: For two-sided tests, Ha uses ≠ to check for any difference, while one-sided tests use < or >. Hypotheses focus on p, the unknown population proportion. The sample data informs the test statistic but not the hypotheses statements.

Question 10

1. $H_0: p=0.104$; $H_a: p>0.104$
2. $H_0: p=0.08$; $H_a: p>0.08$ (correct answer)
3. $H_0: p\ge 0.08$; $H_a: p<0.08$
4. $H_0: \hat{p}=0.08$; $H_a: \hat{p}>0.08$
5. $H_0: p=0.08$; $H_a: p\ne 0.08$

Explanation: This question involves hypotheses to test if defect rate exceeds 8%, challenging the 'no more than' claim. Choice B is correct: H0: p = 0.08 and Ha: p > 0.08, for a right-tailed test. Distractors include sample p-hat = 0.104 in H0, like option A, or p-hat in D. Option C reverses direction and structure. Mini-lesson: To test exceeding a maximum, use > in Ha, with H0 as equality. Hypotheses are about p; sample informs evidence against H0. This setup places the burden on showing a higher rate.

Question 11

1. $H_0: p=0.65$; $H_a: p>0.65$ (correct answer)
2. $H_0: p=0.65$; $H_a: p<0.65$
3. $H_0: \hat{p}=0.65$; $H_a: \hat{p}>0.65$
4. $H_0: p=0.60$; $H_a: p>0.60$
5. $H_0: p>0.65$; $H_a: p=0.65$

Explanation: This question involves testing a claim about a proportion being greater than a specific value. The nonprofit claims MORE THAN 65% have volunteered, so we test if p > 0.65. The null hypothesis states H₀: p = 0.65 (the boundary value), and the alternative reflects the claim: Hₐ: p > 0.65. Choice B has the wrong direction. Choice C incorrectly uses the sample proportion (p̂). Choice D uses the sample proportion value (0.60) instead of the claimed threshold. Choice E reverses the null and alternative hypotheses. Note that even though the sample proportion (0.60) is less than 65%, we still set up hypotheses to test the original claim, not based on what the sample shows.

Question 12

1. $H_0: p=0.54$; $H_a: p<0.54$
2. $H_0: p=0.60$; $H_a: p<0.60$ (correct answer)
3. $H_0: p<0.60$; $H_a: p=0.60$
4. $H_0: \hat{p}=0.60$; $H_a: \hat{p}<0.60$
5. $H_0: p=0.60$; $H_a: p>0.60$

Explanation: This question tests setting up hypotheses for believing a documentary viewing proportion is less than 60%. Choice B is correct: H0: p = 0.60 and Ha: p < 0.60, matching the 'less than' belief. Distractors include using sample p-hat = 0.54 in H0, like option A, or p-hat in option D. Option C reverses H0 and Ha incorrectly. Mini-lesson: For 'less than' claims, Ha uses <, making it left-tailed, with H0 as equality. Hypotheses concern p, not p-hat. The sample proportion helps compute the z-score but doesn't appear in the hypotheses.

Question 13

1. $H_0: p=0.10$; $H_a: p\ne 0.10$ (correct answer)
2. $H_0: p=0.10$; $H_a: p<0.10$
3. $H_0: \hat{p}=0.10$; $H_a: \hat{p}\ne 0.10$
4. $H_0: p=0.075$; $H_a: p\ne 0.075$
5. $H_0: p\ne 0.10$; $H_a: p=0.10$

Explanation: This question tests setting up hypotheses for a two-sided test about a population proportion. The principal suspects the proportion is DIFFERENT FROM 10%, indicating a two-sided alternative. The null hypothesis is H₀: p = 0.10 (the value being tested against), and the alternative is Hₐ: p ≠ 0.10 (different from 10%). Choice B uses a one-sided alternative when the claim is about any difference. Choice C incorrectly uses the sample proportion (p̂) in the hypotheses. Choice D uses the sample proportion value (0.075) instead of the stated comparison value. Choice E reverses the null and alternative hypotheses. The phrase "different from" always indicates a two-sided test, regardless of what the sample data shows.

Question 14

1. $H_0: p=0.85$; $H_a: p\ne 0.85$ (correct answer)
2. $H_0: p=0.85$; $H_a: p<0.85$
3. $H_0: \hat{p}=0.85$; $H_a: \hat{p}\ne 0.85$
4. $H_0: p\ne 0.85$; $H_a: p=0.85$
5. $H_0: p=0.80$; $H_a: p\ne 0.80$

Explanation: This question involves a two-tailed test for a specific claimed proportion. The restaurant chain claims the proportion is exactly 0.85, and we want to test if the true proportion is "different from" this value. This language indicates a two-tailed test with H₀: p = 0.85 and H_a: p ≠ 0.85. Choice A correctly represents these hypotheses. The sample proportion of 0.800 might suggest the true value is lower, but since we're testing for any difference, we use the two-sided alternative. Never reverse the null and alternative hypotheses (as in choice D) or use sample statistics (p̂) in place of population parameters. The phrase "different from" always signals a two-tailed test with ≠ in the alternative hypothesis.

Question 15

1. $H_0: p=0.30$; $H_a: p<0.30$ (correct answer)
2. $H_0: p=0.30$; $H_a: p>0.30$
3. $H_0: \hat{p}=0.30$; $H_a: \hat{p}<0.30$
4. $H_0: p<0.30$; $H_a: p=0.30$
5. $H_0: p=38/160$; $H_a: p<38/160$

Explanation: This question involves setting up hypotheses to test if a proportion is lower than a reported value. The district reports 30% have no internet access, but the community group suspects it's LOWER. The null hypothesis states the reported value: H₀: p = 0.30, and the alternative hypothesis reflects the suspicion: Hₐ: p < 0.30. Choice B has the wrong direction (greater than). Choice C incorrectly uses the sample proportion (p̂) in the hypotheses. Choice D reverses the null and alternative hypotheses. Choice E uses the sample proportion value instead of the reported population value. The key principle is that we test the claim or reported value in the null hypothesis, and the alternative reflects what we're trying to show evidence for.

Question 16

1. $H_0: p=0.35$; $H_a: p<0.35$
2. $H_0: p=0.306$; $H_a: p>0.306$
3. $H_0: \hat{p}=0.35$; $H_a: \hat{p}>0.35$
4. $H_0: p=0.35$; $H_a: p>0.35$ (correct answer)
5. $H_0: p>0.35$; $H_a: p=0.35$

Explanation: This question assesses hypotheses for believing gym attendance exceeds 35%. Choice D is correct: H0: p = 0.35 and Ha: p > 0.35, aligning with 'more than' for right-tailed. Distractors use sample p-hat ≈ 0.306 in B, or p-hat in C. Option A has wrong direction. Mini-lesson: 'More than' uses > in Ha, H0 with equality on p. Sample data tests H0 but isn't in statements. This tests if evidence supports higher attendance.

Question 17

1. $H_0: p=0.25$; $H_a: p<0.25$ (correct answer)
2. $H_0: p=0.183$; $H_a: p<0.183$
3. $H_0: \hat{p}=0.25$; $H_a: \hat{p}<0.25$
4. $H_0: p\le 0.25$; $H_a: p>0.25$
5. $H_0: p=0.25$; $H_a: p\ne 0.25$

Explanation: This question evaluates the ability to set up hypotheses for a one-proportion test where a consumer group suspects the coupon proportion is lower than 25%. The appropriate hypotheses are H0: p = 0.25 and Ha: p < 0.25, as in choice A, aligning with the suspicion of a lower true proportion. Distractors include using the sample proportion p-hat = 0.183 in H0, like option B, or incorrectly using p-hat symbols, as in option C. Option D reverses the direction, which would test for a higher proportion instead. Mini-lesson: Hypotheses are statements about the population proportion p, not the sample; H0 assumes the advertised value is true with equality, and Ha specifies the direction of deviation based on the claim. This setup leads to a left-tailed test. Always ensure H0 has the equality to maintain the burden of proof on Ha.

Question 18

1. $H_0: p=0.87$; $H_a: p<0.87$
2. $H_0: p\ge 0.90$; $H_a: p<0.90$
3. $H_0: p=0.90$; $H_a: p<0.90$ (correct answer)
4. $H_0: \hat{p}=0.90$; $H_a: \hat{p}<0.90$
5. $H_0: p=0.90$; $H_a: p>0.90$

Explanation: This question involves setting up hypotheses to test if an on-time delivery rate is below the claimed 90%, for a one-proportion z-test. Choice C is correct: H0: p = 0.90 and Ha: p < 0.90, reflecting the advocate's goal to show it's lower. Distractors include using the sample p-hat = 0.87 in H0, as in option A, or using p-hat symbols, like option D. Option B incorrectly places the inequality in H0, which should always have equality. Mini-lesson: When a claim is 'at least' a value, test Ha as less than to challenge it, with H0 as equal to the boundary value. This creates a left-tailed test. Hypotheses are about p, not observed sample proportions.

Question 19

1. $H_0: p=0.55$; $H_a: p\ne 0.55$ (correct answer)
2. $H_0: \hat{p}=0.55$; $H_a: \hat{p}\ne 0.55$
3. $H_0: p=0.50$; $H_a: p\ne 0.50$
4. $H_0: p\ne 0.55$; $H_a: p=0.55$
5. $H_0: p=0.55$; $H_a: p>0.55$

Explanation: This question involves setting up a two-sided hypothesis test for a population proportion. The health center believes the proportion is DIFFERENT FROM 55%, which indicates a two-sided alternative hypothesis. The null hypothesis states H₀: p = 0.55 (the value we're testing against), and the alternative hypothesis is Hₐ: p ≠ 0.55 (different from 55%). Choice B incorrectly uses the sample proportion (p̂) instead of the population parameter. Choice C tests against 50% instead of the stated 55%. Choice D reverses the null and alternative hypotheses. Choice E uses a one-sided alternative when the claim is about a difference in either direction. The key insight is that "different from" always leads to a two-sided test with the ≠ symbol in the alternative hypothesis.

Question 20

1. $H_0: p=0.40$; $H_a: p>0.40$ (correct answer)
2. $H_0: p=0.45$; $H_a: p>0.45$
3. $H_0: \hat{p}=0.40$; $H_a: \hat{p}>0.40$
4. $H_0: p=0.40$; $H_a: p<0.40$
5. $H_0: p\ge 0.40$; $H_a: p<0.40$

Explanation: This question requires setting up hypotheses for a right-tailed test about a population proportion. The health center believes the proportion is "greater than 0.40," which indicates we're testing if p > 0.40. The null hypothesis always contains the equality, so H₀: p = 0.40, and the alternative hypothesis is H_a: p > 0.40. Choice A correctly represents these hypotheses. The sample proportion of 0.450 is given as context but should not appear in the hypotheses themselves. A common error is using the sample statistic (p̂) instead of the population parameter (p) in the hypotheses. When a claim uses "greater than" language, we're conducting a right-tailed test with the inequality pointing to the right in the alternative hypothesis.