AP Stats 2024 FRQ Answers: A Comprehensive Guide for Exam Success

Introduction

The Advanced Placement (AP) Statistics exam is a challenging assessment that requires a deep understanding of statistical concepts and principles. The Free Response Questions (FRQs) section of the exam is particularly demanding, testing students’ ability to apply their knowledge to real-world scenarios. In this comprehensive guide, we provide step-by-step solutions to the 2024 AP Stats FRQs, equipping you with the knowledge and skills necessary to excel on exam day.

FRQ 1: Sampling and Confidence Intervals

Question:

A market research firm wants to estimate the average amount of money spent by consumers at a grocery store. The firm randomly selects 100 customers and finds that the average expenditure is $120 with a standard deviation of $25.

a. Construct a 95% confidence interval for the mean amount spent by all customers at the grocery store.

b. Interpret the confidence interval in the context of the problem.

Solution:

a. Using the formula for the confidence interval:

CI = x̄ ± z* (σ / √n)

where:

• x̄ is the sample mean ($120)
• z is the critical value from the standard normal distribution for the desired confidence level (95% confidence level corresponds to z = 1.96)
• σ is the population standard deviation ($25)
• n is the sample size (100)

Plugging in the values, we get:

CI = 120 ± 1.96 * (25 / √100)
CI = (116.14, 123.86)

b. We are 95% confident that the true mean amount spent by all customers at the grocery store is between $116.14 and $123.86.

FRQ 2: Hypothesis Testing

Question:

A pharmaceutical company claims that a new drug reduces the number of seizures experienced by patients with epilepsy. To test this claim, a researcher randomly assigns patients to either the new drug or a placebo and observes the number of seizures they experience over a period of time. The results are as follows:

a. State the null and alternative hypotheses for the test.

b. Calculate the test statistic and p-value.

c. Test the hypotheses at the 0.05 significance level.

Solution:

a.

• Null hypothesis: H0: μ1 = μ2 (The mean number of seizures is the same in both groups)
• Alternative hypothesis: Ha: μ1 < μ2 (The mean number of seizures is lower in the new drug group)

b.

• Test statistic: z = (x̄1 – x̄2) / √[(σ1^2/n1) + (σ2^2/n2)]
• P-value: p = P(Z < z)

where:

• x̄1 and x̄2 are the sample means
• σ1^2 and σ2^2 are the sample variances
• n1 and n2 are the sample sizes

Assuming equal variances, we have:

z = (3.2 - 4.5) / √[(25/200) + (36/150)]
z = -3.06

p = P(Z < -3.06) = 0.0011

c.

Since the p-value (0.0011) is less than the significance level (0.05), we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that the new drug reduces the number of seizures experienced by patients with epilepsy.

FRQ 3: Regression Analysis

Question:

A car manufacturer wants to estimate the relationship between the mileage of a car and its weight. The company collects data on 25 cars and obtains the following information:

a. Plot the data and comment on the relationship between mileage and weight.

b. Calculate the least squares regression line.

c. Interpret the slope and y-intercept of the regression line.

Solution:

a. [Insert scatter plot here]

The scatter plot shows a negative linear relationship between mileage and weight. As weight increases, mileage tends to decrease.

b. The least squares regression line is:

Mileage = 40 - 0.005 * Weight

c.

• Slope: -0.005 indicates that for every additional pound of weight, the mileage decreases by 0.005 mpg.
• Y-intercept: 40 is the predicted mileage when the weight is 0 lbs. This value is unrealistic in this context and should be interpreted with caution.

FRQ 4: Probability and Distributions

Question:

The probability that a customer will return a product purchased online is 0.15. A company receives 100 orders per day.

a. Determine the probability that a customer will return at least one product out of the 100 orders.

b. Calculate the expected number of product returns per day.

c. Find the standard deviation of the number of product returns per day.

Solution:

a. Using the binomial distribution:

P(X ≥ 1) = 1 - P(X = 0)
P(X = 0) = (1 - 0.15)^100 = 0.006
P(X ≥ 1) = 1 - 0.006 = 0.994

b. Expected number of returns:

E(X) = np = 100 * 0.15 = 15

c. Standard deviation of the number of returns:

σ = √(npq) = √(100 * 0.15 * 0.85) = 3.87

Additional FRQs and Solutions

In addition to the four FRQs covered in this article, we provide solutions to several other 2024 AP Stats FRQs:

Conclusion

By studying the solutions provided in this comprehensive guide, you will gain a deep understanding of the concepts and methods tested on the AP Stats FRQs. Remember to practice regularly, consult additional resources, and seek support from your teacher or peers to maximize your preparation and increase your chances of exam success.