Answered

The two-way table shows the distribution of gender to favorite film genre for the senior class at Mt. Roce High School.




ComedyHorrorDramaActionTotal

M		3616162896

F		54222444144

Total		90384072240






Which statement is true?


AThe probability of randomly selecting a student who has a favorite genre of drama and is also female is about 17 percent.
BEvent F for female and event D for drama are independent events.
CThe probability of randomly selecting a male student, given that his favorite genre is horror, is $\frac{16}{40}$.
DEvent M for male and event A for action are independent events.

Verified Answer :

  • The probability \(P(\text{Drama and Female}) = \frac{24}{240} = 10\%\).
  • The events \(F\) and \(D\) are independent because \(P(F \cap D) = P(F) \cdot P(D) = 0.1\).
  • The conditional probability \(P(M|H) = \frac{16}{38} eq \frac{16}{40}\).
  • The events \(M\) and \(A\) are not independent because \(P(M \cap A) = \frac{28}{240} \approx 0.117 eq 0.12\).
  • The true statement is: \(\boxed{\text{Event F for female and event D for drama are independent events.}}\)

Explanation

  1. Analyze the two-way table data

To determine which statement is true, we first analyze the provided two-way table for Mt. Roce High School, which lists the distribution of gender and preferred film genre for 240 students.

Row and column totals:

  • Male (M): 96
  • Female (F): 144
  • Comedy: 90
  • Horror: 38
  • Drama: 40
  • Action: 72
  • Total students: 240
  1. Evaluate the statements

We evaluate each statement one by one:

  1. Probability of selecting a student who prefers drama and is female:

The number of female students who chose drama is 24. The total number of students is 240.
The probability \(P(\text{Drama and Female}) = \frac{24}{240} = 0.10\), which is \(10\%\). The statement claims it is about \(17\%\), so this statement is false.

  1. Are events \(F\) and \(D\) independent?

Two events \(A\) and \(B\) are independent if \(P(A \cap B) = P(A) \cdot P(B)\).

  • \(P(F) = \frac{144}{240} = 0.6\)
  • \(P(D) = \frac{40}{240} = \frac{1}{6}\)
  • \(P(F) \cdot P(D) = 0.6 \cdot \frac{1}{6} = 0.1\)
  • \(P(F \cap D) = \frac{24}{240} = 0.1\)

Since \(P(F \cap D) = P(F) \cdot P(D)\), the events \(F\) and \(D\) are independent. This statement is true.

  1. Probability of selecting a male student, given the favorite genre is horror:

The conditional probability \(P(M|H) = \frac{P(M \cap H)}{P(H)} = \frac{16}{38}\).
The statement claims it is \(\frac{16}{40}\), so this statement is false.

  1. Are events \(M\) and \(A\) independent?
  • \(P(M) = \frac{96}{240} = 0.4\)
  • \(P(A) = \frac{72}{240} = 0.3\)
  • \(P(M) \cdot P(A) = 0.4 \cdot 0.3 = 0.12\)
  • \(P(M \cap A) = \frac{28}{240} = \frac{7}{60} \approx 0.1167\)

Since \(P(M \cap A) eq P(M) \cdot P(A)\), the events \(M\) and \(A\) are not independent. This statement is false.

Examples

Understanding two-way tables is essential for data analysis in many fields, such as marketing research and social sciences. For example, a business might use such a table to determine if customer demographics (like age or gender) are independent of product preferences. This helps in tailoring targeted advertising campaigns by identifying which segments of the population are more likely to prefer certain products.

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