Question

the following data represent the number of different communication activities used by a random sample of teenagers in a given week. complete parts (a) through (d).
activities 0 1 - 2 3 - 4 5+ total
male 22 81 60 37 200
female 22 51 57 70 200
total 44 132 117 107 400
(a) are the events \male\ and \0 activities\ independent?

Explanation:

Step1: Calculate the probability of being male, $P(M)$

The total number of males is 200 and the total number of teenagers is 400. So $P(M)=\frac{200}{400}=\frac{1}{2}$.

Step2: Calculate the probability of having 0 activities, $P(0)$

The total number of teenagers with 0 activities is 44 and the total number of teenagers is 400. So $P(0)=\frac{44}{400}=\frac{11}{100}$.

Step3: Calculate the probability of being male and having 0 activities, $P(M\cap0)$

The number of males with 0 activities is 22 and the total number of teenagers is 400. So $P(M\cap0)=\frac{22}{400}=\frac{11}{200}$.

Step4: Check the independence condition

Two events $A$ and $B$ are independent if $P(A\cap B)=P(A)\times P(B)$. Here, $P(M)\times P(0)=\frac{1}{2}\times\frac{11}{100}=\frac{11}{200}$, and $P(M\cap0)=\frac{11}{200}$. Since $P(M\cap0) = P(M)\times P(0)$, the events are independent.

Answer:

Yes, because $P(\text{male}\cap0\text{ activities})=P(\text{male})\times P(0\text{ activities})$ are equal.