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AP Statistics Practice Quiz: Parameters for a Binomial Distribution

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 13 questions to check your progress.

Question 1 of 13

A fair coin is flipped 20 times. Let the random variable X represent the number of heads. What is the mean of X?

All Questions (13)

A fair coin is flipped 20 times. Let the random variable X represent the number of heads. What is the mean of X?

A) 10

B) 5

C) 20

D) 2.24

Correct Answer: A

The situation describes a binomial distribution with n = 20 trials (flips) and a probability of success p = 0.5 (getting a head). The mean of a binomial random variable is calculated as np. Therefore, the mean is 20 * 0.5 = 10.

A student guesses randomly on a 10-question multiple-choice quiz, where each question has 4 options. Let Y be the number of correct answers. What is the standard deviation of Y, rounded to three decimal places?

A) 2.500

B) 1.875

C) 1.369

D) 7.500

Correct Answer: C

This is a binomial setting with n = 10 trials (questions) and a probability of success p = 0.25 (guessing correctly). The standard deviation of a binomial random variable is sqrt(np(1-p)). So, the standard deviation is sqrt(10 * 0.25 * (1 - 0.25)) = sqrt(1.875) ≈ 1.369.

It is known that 80% of a certain brand of computer chips are not defective. A quality control inspector randomly selects 50 chips. Let X be the number of non-defective chips. Which of the following are the correct mean and standard deviation for the random variable X?

A) Mean = 40, Standard Deviation = 8

B) Mean = 40, Standard Deviation = 2.828

C) Mean = 10, Standard Deviation = 2.828

D) Mean = 10, Standard Deviation = 8

Correct Answer: B

The number of trials is n = 50 and the probability of success (a non-defective chip) is p = 0.80. The mean is np = 50 * 0.80 = 40. The standard deviation is sqrt(np(1-p)) = sqrt(50 * 0.80 * 0.20) = sqrt(8) ≈ 2.828.

A basketball player makes 70% of her free throws. Assume each free throw is an independent event. If she attempts 30 free throws in a game, the number of made free throws follows a binomial distribution with a mean of 21. What is the correct interpretation of this mean?

A) She will make exactly 21 free throws in every game where she attempts 30.

B) The most likely number of free throws she will make in a single game is 21.

C) Over many games where she attempts 30 free throws, the average number of made free throws is 21.

D) The probability of her making 21 free throws is the highest possible probability for any outcome.

Correct Answer: C

The mean of a binomial distribution, np, represents the expected value or the long-run average number of successes over many repetitions of the experiment. In this context, it's the average number of made free throws over many games with 30 attempts, not the outcome of a single game.

A survey finds that 30% of adults in a large city use public transportation. A random sample of 200 adults from this city is selected. Let X be the number of adults in the sample who use public transportation. The mean of X is 60. Which of the following is the best interpretation of the mean?

A) It is most likely that exactly 60 adults in the sample use public transportation.

B) In any sample of 200 adults from this city, we will find exactly 60 who use public transportation.

C) If we took many random samples of 200 adults from this city, the average number of adults who use public transportation across all samples would be about 60.

D) The probability of finding 60 adults who use public transportation in a sample of 200 is 0.30.

Correct Answer: C

The mean (or expected value) of a random variable in a sampling context describes the long-run average value we would expect to see if the sampling process were repeated many times. It does not guarantee the outcome of a single sample.

A seed company claims that 90% of its seeds will germinate. A gardener plants 100 seeds. The number of seeds that germinate, X, can be modeled by a binomial distribution with a mean of 90 and a standard deviation of 3. Which of the following is the best interpretation of the standard deviation?

A) The number of seeds that germinate will be exactly 3 away from the mean.

B) The number of seeds that germinate in a random batch of 100 seeds will typically vary from the mean of 90 by about 3 seeds.

C) If we plant many batches of 100 seeds, the maximum difference between the number of germinated seeds and the mean will be 3.

D) The probability of a seed germinating is, on average, 3% away from 90%.

Correct Answer: B

The standard deviation of a random variable measures the typical or average distance of the outcomes from the mean. In this context, it describes how much the number of germinated seeds in a batch of 100 is expected to vary from the average of 90.

A factory produces light bulbs, and 5% of them are defective. A quality control manager selects a random sample of 400 bulbs to test. Let X be the number of defective bulbs in the sample. What are the values of the parameters n and p for the binomial distribution of X?

A) n = 400, p = 0.05

B) n = 400, p = 0.95

C) n = 20, p = 0.05

D) n = 0.05, p = 400

Correct Answer: A

In a binomial setting, 'n' is the number of trials, which is the sample size of 400 bulbs. 'p' is the probability of success on a single trial. Here, a "success" is defined as finding a defective bulb, so p = 0.05.

A researcher is conducting a binomial experiment where the probability of a certain outcome (a 'success') is 0.2. The experiment is repeated 'n' times. If the expected number of successful outcomes is 50, how many times was the experiment repeated?

A) 10

B) 50

C) 250

D) 500

Correct Answer: C

The mean, or expected number of successes, of a binomial distribution is given by the formula np. We are given the mean (50) and the probability p (0.2). To find n, we solve the equation: n * 0.2 = 50, which gives n = 50 / 0.2 = 250.

Let X be a binomial random variable with a mean of 12 and a standard deviation of 3. What are the values of the parameters n and p?

A) n = 24, p = 0.5

B) n = 36, p = 1/3

C) n = 48, p = 0.25

D) n = 16, p = 0.75

Correct Answer: C

We are given np = 12 and sqrt(np(1-p)) = 3. Squaring the standard deviation gives the variance, np(1-p) = 9. We can substitute np=12 into the variance equation: 12(1-p) = 9. Solving for p gives 1-p = 9/12 = 0.75, so p = 0.25. Then, using np=12, we find n = 12/0.25 = 48.

A multiple-choice test has 60 questions, each with five possible answers, only one of which is correct. A student guesses the answer to each question. Let X be the number of correct answers the student gets. Which of the following best describes the random variable X and its parameters?

A) X is a binomial random variable with a mean of 12. This means we expect a student who guesses to get about 12 questions right on average.

B) X is a binomial random variable with a mean of 30. This means the most likely score is 30 out of 60.

C) X is not a binomial random variable because the probability of guessing correctly changes from question to question.

D) X is a binomial random variable with a standard deviation of 12. This means scores will typically be 12 points away from the mean.

Correct Answer: A

The scenario fits a binomial distribution with n=60 trials (questions) and p=1/5=0.2 probability of success (guessing correctly). The mean is np = 60 * 0.2 = 12. This mean represents the expected or average number of correct answers if many students were to guess on the test.

Let X be a binomial random variable representing the number of successes in 100 trials with a probability of success p=0.2. Let Y be a binomial random variable representing the number of successes in 100 trials with a probability of success p=0.5. How does the standard deviation of X compare to the standard deviation of Y?

A) The standard deviation of X is greater than the standard deviation of Y.

B) The standard deviation of X is less than the standard deviation of Y.

C) The standard deviations of X and Y are equal.

D) The relationship cannot be determined without more information.

Correct Answer: B

For a fixed number of trials n, the standard deviation of a binomial distribution, sqrt(np(1-p)), is maximized when p=0.5. The standard deviation of X is sqrt(100*0.2*0.8) = sqrt(16) = 4. The standard deviation of Y is sqrt(100*0.5*0.5) = sqrt(25) = 5. Therefore, the standard deviation of X is less than the standard deviation of Y.

An airline knows that 10% of passengers who book a flight do not show up. They sell 160 tickets for a flight with 150 seats. Let X be the number of passengers who show up, which follows a binomial distribution with a mean of 144 and a standard deviation of approximately 3.79. Based on these parameters, what is a reasonable conclusion?

A) The mean number of passengers who show up (144) is less than the number of seats (150), so the airline will never have a problem with overbooking.

B) The standard deviation of 3.79 is small, which means exactly 144 passengers will show up for every flight.

C) The mean number of passengers who show up is 144. On average, the flight will have empty seats, but the number of passengers will vary, and it is possible for more than 150 to show up.

D) The mean is 144, which is only 6 away from 150. Therefore, the flight is almost certain to be overbooked.

Correct Answer: C

The mean of 144 indicates that, on average, the flight will not be overbooked. However, the standard deviation of 3.79 indicates that the actual number of passengers will vary. An outcome of 151 passengers, which would overbook the flight, is less than 2 standard deviations from the mean (144 + 2*3.79 = 151.58), so it is not an unusual event. The parameters suggest that while the average outcome is favorable for the airline, there is a real risk of overbooking.

A fast-food chain states that 25% of its customers order a diet soda. A quality assurance manager observes a random sample of 80 customers. Let X be the number of customers in the sample who order a diet soda. What are the mean and standard deviation of X?

A) Mean = 20, Standard Deviation = 15

B) Mean = 20, Standard Deviation = 3.87

C) Mean = 60, Standard Deviation = 3.87

D) Mean = 60, Standard Deviation = 15

Correct Answer: B

The number of trials is n=80 and the probability of success (ordering a diet soda) is p=0.25. The mean is calculated as np = 80 * 0.25 = 20. The standard deviation is calculated as sqrt(np(1-p)) = sqrt(80 * 0.25 * 0.75) = sqrt(15) ≈ 3.87.