AP Statistics 2.5 Mutually Exclusive Events- Exam Style Questions - MCQs - New Syllabus

Question

The height of each student at a large high school was measured and recorded. Suppose one student at this school is randomly selected. Let A be the event that the student is taller than 60 inches; let B be the event that the student is taller than 70 inches; and let C be the event that the student is shorter than 65 inches. Which of the following sets of events are mutually exclusive?

(A) Events A and B
(B) Events A and C
(C) Events \((A \cup B)\) and C
(D) Events \((A \cup C)\) and B
(E) Events \((A \cap C)\) and B

▶️ Answer/Explanation

Define the events:

\(A=\{\text{height} > 60\}\)
\(B=\{\text{height} > 70\}\)
\(C=\{\text{height} < 65\}\)

Notice that:

\(A \cap C=\{60 < \text{height} < 65\}\)

Any student in \(A \cap C\) must have a height less than 65 inches, while any student in \(B\) must have a height greater than 70 inches. Therefore, no student can belong to both events simultaneously.

\((A \cap C)\cap B=\varnothing\)

Since the intersection is empty, the events are mutually exclusive.
✅ Answer: (E)

Question

An electricity company summarized data from October to December regarding the electricity usage (kilowatt hours) of its customers at different times of the day and the corresponding outside air temperature (degrees Fahrenheit). A scatterplot of the data indicated a linear pattern and the correlation between electricity usage and outside air temperature for this data was found to be \( r = -0.65 \).

Which of the following statements must be true?

(A) If the temperatures were converted to degrees Celsius, the correlation between electricity usage and outside air temperature would change.
(B) Electricity usage tends to be higher on warmer days.
(C) A decrease in outside air temperature causes an increase in electricity usage.
(D) Because \( r < 0 \), the relationship between outside air temperature and electricity usage is weak.
(E) The linear relationship between outside air temperature and electricity usage is moderately strong.

▶️ Answer/Explanation

The correlation coefficient is:
\( r = -0.65 \)

The negative sign indicates a negative association, meaning electricity usage tends to increase as temperature decreases. The magnitude,
\( |r| = 0.65 \),
suggests a moderately strong linear relationship.

Correlation does not imply causation, so choice (C) is incorrect. Converting Fahrenheit to Celsius is a linear transformation and does not change the correlation coefficient, so choice (A) is incorrect. A negative correlation is not necessarily weak, eliminating choice (D).

Therefore, the statement that must be true is that the linear relationship is moderately strong.

✅ Answer: (E)

Question

An experiment has three mutually exclusive outcomes, A, B, and C. If \(P(A)=0.12\), \(P(B)=0.61\), and \(P(C)=0.27\), which of the following must be true?

I. A and C are independent.
II. \(P(A \text{ and } B)=0\)
III. \(P(B \text{ or } C)=P(B)+P(C)\)

(A) I only
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II, and III

▶️ Answer/Explanation

Since A, B, and C are mutually exclusive outcomes, no two of them can occur at the same time.

Therefore,
\( P(A \cap B)=0 \)
so statement II is true.

Also, because B and C are mutually exclusive,
\( P(B \cup C)=P(B)+P(C) \)
Thus, statement III is true.

Statement I is false. If A and C were independent, then:
\( P(A \cap C)=P(A)P(C) \)
\( = (0.12)(0.27)=0.0324 \)
But because A and C are mutually exclusive,
\( P(A \cap C)=0 \)
Since \(0 \ne 0.0324\), A and C are not independent.

Therefore, only statements II and III must be true.

✅ Answer: (D)

AP Statistics 2.5 Mutually Exclusive Events- Exam Style Questions - MCQs - New Syllabus

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