Question

according to survey results from the pew research center, 30% of people made new years resolutions. of the 5,140 people who were surveyed, 16% were in the 18 - 29 age bracket. of the people who were surveyed, 7.9% were in the 18 - 29 age bracket and made new years resolutions. suppose that a person from the survey is randomly selected.
a. are the events \made a new years resolution\ and \18 - 29 years old\ independent? justify your answer.
b. is a person between the ages of 18 - 29 more likely, less likely, or equally likely to make a new years resolution than all survey responders? explain.

Explanation:

Step1: Recall the formula for independent events

For two events \(A\) and \(B\), \(P(A\cap B)=P(A)\times P(B)\) if they are independent. Let \(A\) be the event “Made a New Year’s resolution” and \(B\) be the event “18 - 29 years old”.
We know \(P(A) = 0.3\), \(P(B)=0.16\), and \(P(A\cap B)=0.079\)

Step2: Calculate \(P(A)\times P(B)\)

\(P(A)\times P(B)=0.3\times0.16 = 0.048\)

Step3: Compare \(P(A\cap B)\) and \(P(A)\times P(B)\)

Since \(P(A\cap B)=0.079
eq0.048 = P(A)\times P(B)\), the events are not independent.

Step4: Calculate the conditional probability for part b

The conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\)
Substitute \(P(A\cap B) = 0.079\) and \(P(B)=0.16\)
\(P(A|B)=\frac{0.079}{0.16}=0.49375\)
Since \(P(A) = 0.3\) and \(P(A|B)=0.49375>0.3\)

Answer:

a. The events “Made a New Year’s resolution” and “18 - 29 years old” are not independent. Because \(P(A\cap B)=0.079
eq P(A)\times P(B)=0.048\)
b. A person between the ages of 18 - 29 is more likely to make a New Year’s resolution than all survey responders. Since \(P(A|B)=\frac{0.079}{0.16}=0.49375>0.3 = P(A)\)