AP Statistics 4.3 Justifying a Claim Based on a Confidence Interval for a Population Mean or Population Mean Difference- Exam Style Questions - MCQs - New Syllabus

Question

A pharmaceutical company is testing a new allergy medication. In previous studies, the average relief time after taking the standard medication was 45 minutes. The company believes the new medication works faster.

A random sample of 36 patients was given the new medication. The sample mean relief time was 41.2 minutes with a standard deviation of 9.6 minutes.

Which of the following is the most appropriate set of hypotheses to test whether the new medication works faster than the standard medication?

(A)

(B)

(C)

(D)

(E)

▶️ Answer/Explanation

The parameter of interest is the population mean relief time, denoted by \( \mu \). Hypotheses must always be written in terms of a population parameter, not a sample statistic.

Since the company believes the new medication works faster, the mean relief time should be less than 45 minutes. Therefore, the alternative hypothesis should be:
\(H_a:\mu < 45\)

The null hypothesis represents the historical value:
\(H_0:\mu = 45\)

This is a left-tailed test comparing the new medication to the established average relief time.

✅ Answer: (B)

Question

Concerned parents claim the speed of drivers in a school zone exceeds the posted speed of 35 miles per hour (mph). To support their claim, the parents will conduct a hypothesis test. Let \( \mu \) represent the true average speed of cars in the school zone. In a random sample of 50 drivers in the school zone, the average speed was 37 mph. Which set of hypotheses below are appropriate to test this claim?

(A)
\( H_0:\mu = 35 \text{ mph} \)
\( H_a:\mu > 35 \text{ mph} \)

(B)
\( H_0:\mu < 35 \text{ mph} \)
\( H_a:\mu > 35 \text{ mph} \)

(C)
\( H_0:\mu = 37 \text{ mph} \)
\( H_a:\mu > 37 \text{ mph} \)

(D)
\( H_0:\mu < 37 \text{ mph} \)
\( H_a:\mu > 37 \text{ mph} \)

(E)
\( H_0:\mu = 35 \text{ mph} \)
\( H_a:\mu = 37 \text{ mph} \)

▶️ Answer/Explanation

The claim is that drivers are traveling faster than the posted speed limit of 35 mph. Therefore, the alternative hypothesis should reflect a population mean speed that is greater than 35 mph.

The null hypothesis uses the benchmark value being tested:
\( H_0:\mu = 35 \text{ mph} \)
\( H_a:\mu > 35 \text{ mph} \)

Notice that hypotheses are always written in terms of the population parameter \( \mu \), not the sample mean of 37 mph. The sample mean is used later when calculating the test statistic.

✅ Answer: (A)

Question

You prepare 8 cups of coffee with cream. For each one, you flip a coin to decide whether to pour the cream into the coffee or the coffee into the cream. After you stir the cups well, you present each of the 8 cups to your friend. Your friend tastes each one and tells you whether they think the cream was poured into the coffee or the coffee into the cream. The variable of interest is \(X\) = the number of cups out of 8 that they guess correctly. How would you calculate the probability that your friend gets at least 7 cups correct by guessing?

(A) Find the area above 7 under the curve in the normal distribution with
\( \mu = 4 \) and
\( \sigma = \sqrt{2} \).

(B) Find \(P(X \geq 7)\) using
\( \left(\frac{1}{2}\right)^7 \).

(C) Find
\(P(X=7)+P(X=8)\)
using the binomial distribution with
\(n=8\) and
\(p=0.5\).

(D) Find
\(P(X=7)+P(X=8)\)
using the geometric distribution with
\(p=0.5\).

(E) Find the proportion of cups correct out of 8:
\( \frac{7}{8}=0.875 \).

▶️ Answer/Explanation

Each cup represents an independent trial with two possible outcomes: a correct guess or an incorrect guess. If your friend is purely guessing, the probability of a correct guess is:
\( p = 0.5 \)

Therefore,
\( X \sim \text{Binomial}(n=8,\;p=0.5) \)

The probability of getting at least 7 cups correct is:
\( P(X \geq 7) \)
\( = P(X=7)+P(X=8) \)

Since this is a binomial setting, the correct method is to use the binomial distribution and add the probabilities of exactly 7 and exactly 8 correct guesses.

✅ Answer: (C)

AP Statistics 4.3 Justifying a Claim Based on a Confidence Interval for a Population Mean or Population Mean Difference- Exam Style Questions - MCQs - New Syllabus

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