AP Statistics 2.8 Introduction to Random Variables and Probability Distributions- Exam Style Questions - MCQs - New Syllabus

Question

Super Express Mart has a “speedy checkout” lane for people who have 6 items or less. The random variable S is the number of items purchased by a person in the “speedy checkout” lane. S follows the distribution shown in the table below.

The mean of S is 3.84. Which of the following statements is the best interpretation of the mean?

(A) If a person is to be randomly selected from the “speedy checkout” lane, the number of items purchased by that person is 3.84 items.
(B) Since a person who goes through the “speedy checkout” lane can’t buy 3.84 items, the mean number of items purchased by a person in the “speedy checkout” should be 4.
(C) Most people who go through the “speedy checkout” get 3 or 4 items.
(D) If a few people are randomly selected from the “speedy checkout” lane, the average number of items they purchased would be 3.84 items.
(E) The average number of items purchased by all people using the “speedy checkout” lane is 3.84 items.

▶️ Answer/Explanation

The mean of a random variable represents its long-run average value. It does not mean that every individual observation will equal the mean, nor does it require the mean to be a whole number.

A value of:

\(E(S)=3.84\)

means that if we consider all customers using the speedy checkout lane (or a very large number of such customers), the average number of items purchased per customer would be approximately 3.84 items.

Choices (A), (B), and (C) incorrectly interpret the mean for individual customers, while (D) refers only to a few people rather than the long-run average. Therefore, the best interpretation is that the overall average number of items purchased is 3.84.

✅ Answer: (E)

Question

A company hires workers to answer phone call questions concerning technical issues with cable television service. Data suggests that the relationship between the number of days on the job, x, and the time (in minutes) it takes an employee to resolve a problem, y, is well modeled by the linear equation

\( \hat{y} = 32.04 – 1.6x \).

A randomly chosen work phone call showed that an employee who had been working for 5 days solved a problem in 20 minutes.

What is the residual for this phone call, and did this employee resolve the problem faster or slower than expected?

(A) 4.04; The employee was slower than expected.
(B) 4.04; The employee was faster than expected.
(C) -4.04; The employee was slower than expected.
(D) -4.04; The employee was faster than expected.
(E) 24.04; The employee was slower than expected.

▶️ Answer/Explanation

First find the predicted resolution time for an employee with 5 days on the job:

\( \hat{y} = 32.04 – 1.6(5) \)
\( \hat{y} = 32.04 – 8 \)
\( \hat{y} = 24.04 \)

The actual time was:
\( y = 20 \)

Residual:
\( \text{Residual} = y – \hat{y} \)
\( = 20 – 24.04 \)
\( = -4.04 \)

A negative residual means the actual time was less than predicted. Since the employee took fewer minutes than expected, the problem was resolved faster than expected.

✅ Answer: (D)

Question

In an online game, the probability that a geode contains a red crystal is \(0.08\). The color of the crystal in each geode is independent.

Conrad will stop opening geodes after finding a red crystal or when \(4\) geodes have been opened, whichever comes first. Let \(Y\) represent the number of geodes Conrad will open.

Probability Distribution for \(Y\)

Number of geodes Conrad will open, \(y\)	\(1\)	\(2\)	\(3\)	\(4\)
Probability, \(P(Y=y)\)	\(0.08\)	\(0.0736\)

What is \(P(Y=3)\)?

(A) \(0.0064\)
(B) \(0.067712\)
(C) \(0.0736\)
(D) \(0.778688\)

▶️ Answer/Explanation

For Conrad to open exactly \(3\) geodes, the first \(2\) geodes must not contain red crystals, and the third geode must contain a red crystal.
The probability of a non-red crystal is
\(1-0.08=0.92\).
Therefore,
\(P(Y=3)=(0.92)^2(0.08)\)
\(P(Y=3)=0.8464(0.08)=0.067712\).

✅ Answer: (B)

AP Statistics 2.8 Introduction to Random Variables and Probability Distributions- Exam Style Questions - MCQs - New Syllabus

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