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). $\begin{array}{lcccccc}\text { Activities } & 0 & 1-2 & 3-4 & 5+ & \text { Total } \\ \text { Male } & 22 & 81 & 60 & 37 & 200 \\ \text { Female } & 22 & 51 & 56 & 71 & 200 \\ \text { Total } & 44 & 132 & 116 & 108 & 400\end{array}$ (a) Are the events "male" and "0 activities" independent? Yes, because $P($ male) and $P($ male $\mid 0$ activities) are equal. (b) Are the events "female" and "5 + activities" independent? No, because $P($ female) and $P($ female $5+$ activities) are not equal. (c) Are the events "1 - 2 activities" and "5 + activities" mutually exclusive? because is

Step 1 ：Given the total number of teenagers is 400, the number of males is 200, the number of females is 200, the number of 0 activities is 44, the number of 1-2 activities is 132, the number of 3-4 activities is 116, the number of 5+ activities is 108, the number of males with 0 activities is 22, and the number of females with 5+ activities is 71.

Step 2 ：Calculate the probability of each event. The probability of being male is \(\frac{200}{400} = 0.5\), the probability of 0 activities is \(\frac{44}{400} = 0.11\), the probability of being female is \(\frac{200}{400} = 0.5\), and the probability of 5+ activities is \(\frac{108}{400} = 0.27\).

Step 3 ：Calculate the probability of both events occurring. The probability of being male and having 0 activities is \(\frac{22}{400} = 0.055\), and the probability of being female and having 5+ activities is \(\frac{71}{400} = 0.1775\).

Step 4 ：Check if the events are independent. The events 'male' and '0 activities' are independent because \(0.5 \times 0.11 = 0.055\), which is equal to the probability of both events occurring. However, the events 'female' and '5+ activities' are not independent because \(0.5 \times 0.27 \neq 0.1775\).

Step 5 ：Check if the events are mutually exclusive. The events '1-2 activities' and '5+ activities' are not mutually exclusive because they can both occur at the same time.

Step 6 ：\(\boxed{\text{(a) The events 'male' and '0 activities' are independent.}}\)

Step 7 ：\(\boxed{\text{(b) The events 'female' and '5+ activities' are not independent.}}\)

Step 8 ：\(\boxed{\text{(c) The events '1-2 activities' and '5+ activities' are not mutually exclusive.}}\)