Teen Communication The following data represent the number of different communication activities (e.g., cell phone, text messaging, e-mail, Internet, and so on) used by a random sample of teenagers over the past 24 hours.
(a) Are the events “male” and “0 activities” independent? Justify your answer.
(b) Are the events “female” and “5+ activities” independent? Justify your answer.
(c) Are the events “1-2 activities” and “3-4 activities” mutually exclusive? Justify your answer.
(d) Are the events “male” and “1-2 activities” mutually exclusive? Justify your answer.
(a) No, the events “male” and “0 activities” are not independent. This is because the number of activities used by a male teenager could influence the probability of the event “0 activities” occurring.
(b) No, the events “female” and “5+ activities” are not independent. This is because the gender of the teenager could influence the probability of the event “5+ activities” occurring.
(c) Yes, the events “1-2 activities” and “3-4 activities” are mutually exclusive. This is because it is impossible for a teenager to use both 1-2 activities and 3-4 activities in the same 24 hour period.
(d) No, the events “male” and “1-2 activities” are not mutually exclusive. This is because it is possible for a male teenager to use 1-2 activities in the same 24 hour period.
To answer these questions, we need additional information about the data. Specifically, we need the data in the form of a contingency table that shows the frequencies or proportions of each combination of variables. With this information, we can calculate the probabilities and determine if the events are independent or mutually exclusive.
Once we have the contingency table, we can use the formulas for independence and mutual exclusivity to determine the relationships between the events. Let's go through each question:
(a) To determine if the events "male" and "0 activities" are independent, we need to calculate the probabilities of these events occurring. Suppose we have a contingency table that shows the frequencies of each combination of gender and number of activities. We would look at the frequency of "male and 0 activities" and compare it to the product of the frequencies of "male" and "0 activities" separately. If the observed frequency matches the expected frequency calculated using the product rule, then the events are considered independent. However, if the observed frequency is significantly different from the expected frequency, then the events are dependent.
(b) Similarly, to determine if the events "female" and "5+ activities" are independent, we would calculate the probabilities using the same approach as in (a). We would compare the observed frequency of "female and 5+ activities" to the expected frequency calculated from the product of the frequencies of "female" and "5+ activities".
(c) Mutual exclusivity means that the events cannot both occur at the same time. If the events "1-2 activities" and "3-4 activities" are mutually exclusive, it means that no individuals in the sample can be classified in both categories. To determine mutual exclusivity, we would examine the frequencies of "1-2 activities" and "3-4 activities" in the contingency table. If there are no individuals who fall into both categories, then the events are mutually exclusive.
(d) To determine if the events "male" and "1-2 activities" are mutually exclusive, we would again look at the frequencies in the contingency table. If there are no individuals who are both male and fall into the category of "1-2 activities", then the events are mutually exclusive.
In summary, to answer these questions, we need the contingency table representing the given data. From there, we can use the formulas for independence and mutual exclusivity to determine the relationships between the events.
(a) To determine if the events "male" and "0 activities" are independent, we need to compare the probabilities of each event occurring individually to the probability of both events occurring together. Let's assume that the events are independent.
Let P(male) represent the probability of selecting a male teenager, and P(0 activities) represent the probability of a teenager having 0 activities in the past 24 hours.
If the events are independent, then P(male and 0 activities) should be equal to the product of P(male) and P(0 activities).
However, if we find that P(male and 0 activities) is not equal to the product of P(male) and P(0 activities), then the events are not independent.
So, let's say P(male and 0 activities) = x.
To check for independence, we would need data or information about the probability of males having 0 activities and their intersections.
Without this data, we cannot determine the independence between "male" and "0 activities" based on the given information. Therefore, we cannot justify our answer.
(b) Similarly, to determine if the events "female" and "5+ activities" are independent, we need more information.
Let P(female) represent the probability of selecting a female teenager, and P(5+ activities) represent the probability of a teenager having 5 or more activities in the past 24 hours.
We need data or information about the probability of females having 5+ activities and their intersections, which are not provided. Therefore, we cannot determine the independence between "female" and "5+ activities" based on the given information. We cannot justify our answer.
(c) Two events are mutually exclusive if they cannot occur at the same time. To determine if the events "1-2 activities" and "3-4 activities" are mutually exclusive, we need to consider if it is possible for a teenager to have both 1-2 and 3-4 activities.
Since the events "1-2 activities" and "3-4 activities" represent different ranges of possible activities, it is possible for a teenager to have both 1-2 activities and 3-4 activities. For example, a teenager could have used their cell phone (1 activity) and sent text messages (1 activity), totaling 2 activities, while also using email (1 activity) and browsing the internet (1 activity), totaling 4 activities.
Therefore, the events "1-2 activities" and "3-4 activities" are not mutually exclusive.