AP Statistics Quiz: Setting Up Tests For Population Proportion
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Setting Up Tests For Population ProportionQuestion 1 of 20

A restaurant chain reports that 70% of customers rate their experience as "satisfied" or "very satisfied." A random sample of 150 customers is surveyed, and 96 give a satisfied rating (p^=0.64\hat{p}=0.64). Which hypotheses are appropriate for determining whether the reported proportion is accurate?

H0:p=0.70Ha:p≠0.70H_0: p=0.70\quad H_a: p\ne0.70
H0:p=0.64Ha:p≠0.64H_0: p=0.64\quad H_a: p\ne0.64
H0:p=0.70Ha:p<0.70H_0: p=0.70\quad H_a: p<0.70
H0:p^=0.70Ha:p^≠0.70H_0: \hat{p}=0.70\quad H_a: \hat{p}\ne0.70
H0:p≠0.70Ha:p=0.70H_0: p\ne0.70\quad H_a: p=0.70
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AP Statistics Quiz

AP Statistics Quiz: Setting Up Tests For Population Proportion

Practice Setting Up Tests For Population Proportion in AP Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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.

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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.

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Question 1

A restaurant chain reports that 70% of customers rate their experience as "satisfied" or "very satisfied." A random sample of 150 customers is surveyed, and 96 give a satisfied rating (p^=0.64\hat{p}=0.64). Which hypotheses are appropriate for determining whether the reported proportion is accurate?

  1. H0:p=0.70Ha:p≠0.70H_0: p=0.70\quad H_a: p\ne0.70 (correct answer)
  2. H0:p=0.64Ha:p≠0.64H_0: p=0.64\quad H_a: p\ne0.64
  3. H0:p=0.70Ha:p<0.70H_0: p=0.70\quad H_a: p<0.70
  4. H0:p^=0.70Ha:p^≠0.70H_0: \hat{p}=0.70\quad H_a: \hat{p}\ne0.70
  5. H0:p≠0.70Ha:p=0.70H_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

An environmental scientist is studying a specific fish species in a small, isolated lake containing 120 of these fish. The scientist wants to test a hypothesis about the proportion of fish with a certain genetic marker. A simple random sample of 20 fish is captured. Why is a one-sample z-test for a proportion potentially inappropriate in this situation?

  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 nn should be no more than 10% of the population size NN to ensure independence between observations. Here, the sample size is n=20n=20 and the population is N=120N=120. Since 20>0.10(120)=1220 > 0.10(120) = 12, the 10% condition is violated.

Question 3

A researcher wishes to test whether the majority of adults in a large city prefer coffee over tea. A random sample of 300 adults from the city will be surveyed. What is the parameter of interest for this significance test?

  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

A city's mayor claims that more than 60% of residents are in favor of a new public park initiative. To test this claim, an independent polling agency conducts a survey of a random sample of residents. Let pp represent the true proportion of residents who favor the initiative. Which of the following are the correct null and alternative hypotheses?

  1. H0:p=0.60H_0: p = 0.60 vs. Ha:p>0.60H_a: p > 0.60 (correct answer)
  2. H0:p>0.60H_0: p > 0.60 vs. Ha:p=0.60H_a: p = 0.60
  3. H0:p=0.60H_0: p = 0.60 vs. Ha:p≠0.60H_a: p \ne 0.60
  4. H0:p^=0.60H_0: \hat{p} = 0.60 vs. Ha:p^>0.60H_a: \hat{p} > 0.60
Explanation: The null hypothesis (H0H_0) represents the status quo or the boundary of the claim, stated with equality. The alternative hypothesis (HaH_a) represents what the researcher is trying to find evidence for, which in this case is that the proportion is 'more than 60%.' Therefore, H0:p=0.60H_0: p = 0.60 and Ha:p>0.60H_a: p > 0.60 are the correct hypotheses.

Question 5

A national poll claims that 45% of adults have a favorable opinion of a certain policy. A local community leader believes the proportion in their town is different. A random sample of 200 town residents is surveyed. Which statement correctly identifies the inference procedure and null hypothesis?

  1. A one-sample z-test for a proportion should be used with H0:p=0.45H_0: p = 0.45. (correct answer)
  2. A one-sample t-test for a mean should be used with H0:μ=0.45H_0: \mu = 0.45.
  3. A one-sample z-test for a proportion should be used with H0:p≠0.45H_0: p \ne 0.45.
  4. A two-sample z-test for proportions should be used with H0:p1−p2=0H_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 H0:p=0.45H_0: p = 0.45.

Question 6

A student wants to test if a coin is biased towards landing on heads. The student flips the coin 40 times. Which of the following is a required condition to perform a valid one-sample z-test for the proportion of heads?

  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 p0p_0. To test if a coin is biased, the null hypothesis is that it is fair, so H0:p=0.5H_0: p = 0.5. The expected counts are 40(0.5)=2040(0.5) = 20 heads and 40(0.5)=2040(0.5) = 20 tails. Since both are at least 10, the condition is met. Using observed counts is incorrect.

Question 7

A political campaign wants to test the claim that the proportion of registered voters in a city who support their candidate is 0.55. They survey a simple random sample of 400 voters. Which of the following is the most appropriate statistical test for this investigation?

  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

A political analyst claims that exactly 50% of adults in a state approve of the governors performance. A random sample of 1000 adults finds that 540 approve (so p^=540/1000 \hat{p}=540/1000). Which hypotheses are appropriate to test whether approval differs from the analysts claim?

  1. H0:p=0.50H_0: p=0.50; Ha:p≠0.50H_a: p\ne 0.50 (correct answer)
  2. H0:p=0.54H_0: p=0.54; Ha:p≠0.54H_a: p\ne 0.54
  3. H0:p^=0.50H_0: \hat{p}=0.50; Ha:p^≠0.50H_a: \hat{p}\ne 0.50
  4. H0:p≠0.50H_0: p\ne 0.50; Ha:p=0.50H_a: p=0.50
  5. H0:p=0.50H_0: p=0.50; Ha:p>0.50H_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

A school principal believes that the proportion of students who eat school breakfast is different from 55%. In a random sample of 200 students, 124 report eating school breakfast (so p^=124/200 \hat{p}=124/200). Which hypotheses are appropriate for testing the principals belief using a one-proportion zz test?

  1. H0:p=0.62H_0: p=0.62; Ha:p≠0.62H_a: p\ne 0.62
  2. H0:p=0.55H_0: p=0.55; Ha:p≠0.55H_a: p\ne 0.55 (correct answer)
  3. H0:p≠0.55H_0: p\ne 0.55; Ha:p=0.55H_a: p=0.55
  4. H0:p^=0.55H_0: \hat{p}=0.55; Ha:p^≠0.55H_a: \hat{p}\ne 0.55
  5. H0:p=0.55H_0: p=0.55; Ha:p>0.55H_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

A phone manufacturer claims that no more than 8% of its new phones are defective. A quality inspector wants to test whether the defect rate exceeds this value. In a random sample of 500 phones, 52 are found to be defective (so p^=52/500 \hat{p}=52/500). Which hypotheses are appropriate?

  1. H0:p=0.104H_0: p=0.104; Ha:p>0.104H_a: p>0.104
  2. H0:p=0.08H_0: p=0.08; Ha:p>0.08H_a: p>0.08 (correct answer)
  3. H0:p≥0.08H_0: p\ge 0.08; Ha:p<0.08H_a: p<0.08
  4. H0:p^=0.08H_0: \hat{p}=0.08; Ha:p^>0.08H_a: \hat{p}>0.08
  5. H0:p=0.08H_0: p=0.08; Ha:p≠0.08H_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

A nonprofit organization claims that more than 65%65\% of adults in a county have volunteered in the past year. A random sample of n=220n=220 adults finds 132132 have volunteered (so p^=0.60\hat{p}=0.60). Which hypotheses are appropriate to test the nonprofit's claim?

  1. H0:p=0.65H_0: p=0.65; Ha:p>0.65H_a: p>0.65 (correct answer)
  2. H0:p=0.65H_0: p=0.65; Ha:p<0.65H_a: p<0.65
  3. H0:p^=0.65H_0: \hat{p}=0.65; Ha:p^>0.65H_a: \hat{p}>0.65
  4. H0:p=0.60H_0: p=0.60; Ha:p>0.60H_a: p>0.60
  5. H0:p>0.65H_0: p>0.65; Ha:p=0.65H_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

A streaming service claims that 60% of its subscribers watch at least one documentary each month. A market analyst believes the true proportion is less than 60%. In a random sample of 150 subscribers, 81 report watching at least one documentary (so p^=81/150 \hat{p}=81/150). Which hypotheses are appropriate?

  1. H0:p=0.54H_0: p=0.54; Ha:p<0.54H_a: p<0.54
  2. H0:p=0.60H_0: p=0.60; Ha:p<0.60H_a: p<0.60 (correct answer)
  3. H0:p<0.60H_0: p<0.60; Ha:p=0.60H_a: p=0.60
  4. H0:p^=0.60H_0: \hat{p}=0.60; Ha:p^<0.60H_a: \hat{p}<0.60
  5. H0:p=0.60H_0: p=0.60; Ha:p>0.60H_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

A principal suspects that the proportion of all students at a high school who arrive late at least once per week is different from 10%10\%. A random sample of n=120n=120 students finds 99 who arrive late at least once per week (so p^=0.075\hat{p}=0.075). Which hypotheses are appropriate?

  1. H0:p=0.10H_0: p=0.10; Ha:p≠0.10H_a: p\ne 0.10 (correct answer)
  2. H0:p=0.10H_0: p=0.10; Ha:p<0.10H_a: p<0.10
  3. H0:p^=0.10H_0: \hat{p}=0.10; Ha:p^≠0.10H_a: \hat{p}\ne 0.10
  4. H0:p=0.075H_0: p=0.075; Ha:p≠0.075H_a: p\ne 0.075
  5. H0:p≠0.10H_0: p\ne 0.10; Ha:p=0.10H_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

A restaurant chain claims that the proportion of customers who would recommend the restaurant is 0.850.85. A random sample of n=100n=100 customers finds 8080 would recommend it (sample proportion p^=0.800\hat{p}=0.800). Which hypotheses are appropriate to test whether the true proportion is different from 0.850.85?

  1. H0:p=0.85H_0: p=0.85; Ha:p≠0.85H_a: p\ne 0.85 (correct answer)
  2. H0:p=0.85H_0: p=0.85; Ha:p<0.85H_a: p<0.85
  3. H0:p^=0.85H_0: \hat{p}=0.85; Ha:p^≠0.85H_a: \hat{p}\ne 0.85
  4. H0:p≠0.85H_0: p\ne 0.85; Ha:p=0.85H_a: p=0.85
  5. H0:p=0.80H_0: p=0.80; Ha:p≠0.80H_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

A test of H0:p=0.08H_0:p=0.08 uses n=80n=80. Which condition fails?

  1. n(1−p0)≥10n(1-p_0)\ge10
  2. n≤0.10Nn\le0.10N
  3. Sample n≥30n\ge30
  4. np0≥10np_0\ge10 (correct answer)
Explanation: For a one-proportion z test, both np0 and n(1-p0) must be at least 10. Here 80 times 0.08 = 6.4, so np0 >= 10 fails. The other side is 80 times 0.92 = 73.6, which passes. The tempting mistake is checking n(1-p0) or the sample size, but those conditions are satisfied; the expected successes are too few.

Question 16

For H0:p=0.5H_0:p=0.5, p^=0.6\hat p=0.6, n=100n=100, which is the z-test statistic?

  1. z=−.1.25/100z=\frac{-.1}{\sqrt{.25/100}}
  2. z=.1.24/100z=\frac{.1}{\sqrt{.24/100}}
  3. z=.1.25/100z=\frac{.1}{\sqrt{.25/100}} (correct answer)
  4. z=.1.25/100z=\frac{.1}{\sqrt{.25}/100}
Explanation: Subtract the null proportion from the sample proportion: 0.6 - 0.5 = 0.1. Under H0, the standard error is sqrt(0.5(0.5)/100) = sqrt(0.25/100), so z = 0.1 / sqrt(0.25/100). A tempting mistake is using 0.24 in the denominator, but that uses p-hat instead of the null p = 0.5, and the test statistic is based on H0.

Question 17

A random sample of 120 from 1000 without replacement is used for a z-test. Which condition is violated?

  1. np0≥10np_0\ge10
  2. n(1−p0)≥10n(1-p_0)\ge10
  3. n≤0.10Nn\le0.10N (correct answer)
  4. Random sampling
Explanation: The sample is drawn without replacement from a finite population, so the 10% condition must hold: n must be at most 10% of N. Here 120 is greater than 0.10(1000)=100, so that condition is violated. The most tempting wrong answer is np0 >= 10, but that depends on the unknown p0; with only n=120 it cannot be the condition that is violated.

Question 18

To challenge a claim that support is at least 40%, which setup is correct?

  1. H0:p=0.40H_0:p=0.40, Ha:p>0.40H_a:p>0.40
  2. H0:p=0.40H_0:p=0.40, Ha:p<0.40H_a:p<0.40 (correct answer)
  3. H0:p<0.40H_0:p<0.40, Ha:p=0.40H_a:p=0.40
  4. H0:p=0.40H_0:p=0.40, Ha:p≠0.40H_a:p\ne0.40
Explanation: Since the claim is support is at least 40%, you're trying to find evidence it's actually less. Set the null as equality at 0.40 and the alternative as p < 0.40. The tempting opposite setup p > 0.40 would test whether support exceeds the claim, which strengthens it rather than challenging it.

Question 19

A student writes H0:p^=0.5H_0:\hat p=0.5, Ha:p^<0.5H_a:\hat p<0.5. Which correction is needed?

  1. Use pp in both hypotheses (correct answer)
  2. Use p^\hat p in the hypotheses
  3. Make the alternative two-sided
  4. The setup is already correct
Explanation: Hypotheses are about the population parameter p, not the sample statistic p-hat. The sample proportion p-hat comes from data and is used in the test statistic, so it belongs in calculations, not in H0 or Ha. The tempting mistake is to think the setup is already correct because p-hat is the value you calculate, but hypotheses must state a claim about p.

Question 20

A school district reports that 30%30\% of all households in the district have no home internet access. A community group suspects the true proportion is lower. In a random sample of n=160n=160 households, 3838 report no home internet access (so p^=38/160\hat{p}=38/160). Which hypotheses are appropriate?

  1. H0:p=0.30H_0: p=0.30; Ha:p<0.30H_a: p<0.30 (correct answer)
  2. H0:p=0.30H_0: p=0.30; Ha:p>0.30H_a: p>0.30
  3. H0:p^=0.30H_0: \hat{p}=0.30; Ha:p^<0.30H_a: \hat{p}<0.30
  4. H0:p<0.30H_0: p<0.30; Ha:p=0.30H_a: p=0.30
  5. H0:p=38/160H_0: p=38/160; Ha:p<38/160H_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.