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).
$\begin{array}{lcccccc}\text { Activities } & \mathbf{0} & \mathbf{1 - 2} & \mathbf{3 - 4} & \mathbf{5 +} & \text { Total } \square \\ \text { Male } & 22 & 81 & 60 & 37 & 200 \\ \text { Female } & 22 & 51 & 56 & 71 & 200 \\ \text { Total } & 44 & 132 & 116 & 108 & 400\end{array}$
(a) Are the events "male" and "0 activities" independent?
because are
$P($ male $)$ and $P(0$ activities)
$P$ (male) and $P($ male $\mid 0$ activities)
$P(0$ activities) and $P($ male $\mid 0$ activities)

Step 1 ：Given data represents the number of different communication activities used by a random sample of teenagers in a given week. We are asked to determine if the events 'male' and '0 activities' are independent.

Step 2 ：To determine if two events are independent, we need to check if the probability of one event occurring does not affect the probability of the other event occurring. In other words, if the events A and B are independent, then \(P(A \text{ and } B) = P(A) \times P(B)\).

Step 3 ：First, we calculate the probability of the event 'male'. This is given by the formula \(P(\text{male}) = \frac{\text{number of males}}{\text{total number of people}}\). From the given data, we have 200 males out of a total of 400 people, so \(P(\text{male}) = \frac{200}{400} = 0.5\).

Step 4 ：Next, we calculate the probability of the event '0 activities'. This is given by the formula \(P(0 \text{ activities}) = \frac{\text{number of people with 0 activities}}{\text{total number of people}}\). From the given data, we have 44 people with 0 activities out of a total of 400 people, so \(P(0 \text{ activities}) = \frac{44}{400} = 0.11\).

Step 5 ：Then, we calculate the probability of the event 'male and 0 activities'. This is given by the formula \(P(\text{male and 0 activities}) = \frac{\text{number of males with 0 activities}}{\text{total number of people}}\). From the given data, we have 22 males with 0 activities out of a total of 400 people, so \(P(\text{male and 0 activities}) = \frac{22}{400} = 0.055\).

Step 6 ：Finally, we check if \(P(\text{male and 0 activities}) = P(\text{male}) \times P(0 \text{ activities})\). Substituting the calculated probabilities, we get \(0.055 = 0.5 \times 0.11\). Since the two sides of the equation are equal, we can conclude that the events 'male' and '0 activities' are independent.

Step 7 ：\(\boxed{\text{Final Answer: The events 'male' and '0 activities' are independent.}}\)