AP®StatisticsCollege BoardExam QuestionsUnit 3: Inference for Categorical Data: ProportionsInference for Proportions

Inference for Proportions (College Board AP® Statistics): Exam Questions

Syllabus Edition

First teaching 2026

First exams 2027

9 mins9 questions
1
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1 mark

The owner of a clothing company suspects that the proportion of customers who buy scarves is greater at their Barn Street store than at their Lake Street store in January. During the month of January, the owner took a random sample of 450 customers at both locations and recorded the number of people who bought scarves at each store.

Assuming the conditions for inference are met, which of the following test procedures should be used by the owner of the clothing company?

  • A matched-pairs t-test for a mean difference

  • A one-sample t-test for a mean

  • A two-sample z-test for a difference between two proportions

  • A one-sample z-test for a proportion

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2
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1 mark

Let p be the population proportion and n be the size of a random sample taken from the population. Which of the following cases has the greatest standard deviation for the sampling distribution of the sample proportion, p with hat on top?

  • When n = 80 and p is close to 0

  • When n = 80 and p is close to \frac{1}{2}

  • When n = 8000 and p is close to 0

  • When n = 8000 and p is close to \frac{1}{2}

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3
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1 mark

A 95% confidence interval for the proportion of a city's residents who do not vote is (0.342, 0.410). What is the point estimate for the proportion of city residents who do not vote on which this interval was constructed?

  • 0.342

  • 0.376

  • 0.410

  • 0.068

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4
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1 mark

A sociologist wants to estimate the average number of hours workers spend commuting to work each day in a particular city. They plan to take a random sample of 500 workers and ask them the question: How many hours do you spend commuting to work each day? Let mu represent the population mean (the mean number of hours spent commuting each day for all workers in the city) and let x with bar on top represent the sample mean (the mean number of hours spent commuting each day for the 500 randomly selected workers). Is the sample mean, x with bar on top, an unbiased estimator of the population mean, mu?

  • No, because the sample mean, \bar{x}, is unlikely to be equal to the population mean, \mu.

  • Yes, because the wording of the question "How many hours do you spend commuting to work each day?" is unbiased.

  • Yes, because the standard deviation of the sample mean, \bar{x}, is small when the sample size is 500.

  • Yes, because the mean of all possible sample means, \bar{x}, is equal to the population mean, \mu.

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5
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1 mark

A random sample of 280 medical students from a university shows that 210 are interested in cardiology. What would be a 99% confidence interval for the proportion of all medical students at the university who are interested in cardiology?

  • 0.75 \pm 2.326 \sqrt{\frac{0.75 \left(0.25\right)}{280}}

  • 210 \pm 2.576 \sqrt{\frac{0.5 \left(0.5\right)}{280}}

  • 210 \pm 2.576 \sqrt{\frac{0.75 \left(0.25\right)}{280}}

  • 0.75 \pm 2.576 \sqrt{\frac{0.75 \left(0.25\right)}{280}}

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6
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1 mark

In a particular country, 65% of people under the age of 20 would vote for a new president, while 60% of those aged 20 or over would do the same. A simulation was performed in which a random sample of 300 people under the age of 20 and 270 people aged 20 or over were asked if they would vote for a new president. The difference in the sample proportions of those who answered yes, p with hat on top subscript under space 20 end subscript minus p with hat on top subscript 20 space or space over end subscript, was found. The simulation was run 5,000 times, and the sampling distribution of the difference in sample proportions is shown below. Which of the following a, b, and c values are the most likely?

Histogram showing the distribution of the difference in sample proportions. The X-axis is labeled with points a, b, and c.

  • a = -0.08, b = 0, and c = 0.08

  • a = 0.49, b = 0.65, and c = 0.81

  • a = -0.031, b = 0.05, and c = 0.131

  • a = 0.045, b = 0.05, and c = 0.055

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7
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1 mark

A gardener is worried about a specific plant disease in their garden. From a random sample of 50 plants, the gardener will create a 90% confidence interval for the proportion of plants in their garden that have the disease. Which of the following statements is true?

  • The probability that the confidence interval will include the population proportion is 0.9.

  • The population proportion will be in the confidence interval.

  • The probability that the confidence interval will include the sample proportion is 0.9.

  • The population proportion will be equal to the sample proportion.

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8
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1 mark

It is known that 12% of new cell phones from a certain company break after six months. A quality control inspector follows a random sample of 250 customers who bought new phones from the company for six months to see if they have broken or not. What is the best approximation to the probability that less than 10% of the 250 phones sampled are broken? Note that z represents the standard normal variable.

  • P\left(z < \frac{0.1 - 0.12}{\sqrt{\frac{0.12(0.88)}{250}}}\right)

  • P\left(z < \frac{0.12 - 0.1}{\sqrt{\frac{(0.12)(0.88)}{250}}}\right)

  • P\left(z < \frac{0.1 - 0.12}{\sqrt{\frac{(0.1)(0.9)}{250}}}\right)

  • P\left(z < \frac{0.12 - 0.1}{\sqrt{\frac{(0.5)(0.5)}{250}}}\right)

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9
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1 mark

A train company's operations manager plans to take a random sample of passengers to create a 95% confidence interval and estimate the proportion of passengers who are satisfied with their recent journey. Which of the following is the smallest sample size that will result in a margin of error of no more than 7.5 percentage points?

  • 125

  • 150

  • 175

  • 200

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