5.3: Associations in Categorical Variables

Transcription

1 5.3: Associations in Categorical Variables Now we will consider how to use probability to determine if two categorical variables are associated. Conditional Probabilities Consider the next example, where we investigate probabilities that involve a condition of some kind. 1

2 Ex 1: Use the two way table to find the following probabilities for a randomly selected student from the class with the given characteristics: Party Affiliation Marijuana Use Democrat Republican Other Total Yes No Total P(student who approves marijuana use and is a Democrat) P(student Democrat who approves marijuana use) Are these the same thing? If not, how are they different? 2

3 In the previous example, notice how the phrase student Democrat imposes a condition on the student selected. We must be careful to choose a student already known to be a Democrat. For this reason, we call the second probability in Example 1 a conditional probability. If A is the event the student approves of recreational marijuana use, and B is the event the student is a democrat, then we use the notation P(student approves marijuana use given they are a Democrat) = P(A B) which we read as the probability of A given B. It is the probability of event A occurring, given that event B has already occurred. 3

4 Ex 2: Suppose I select a card at random from a deck of 52 cards. Let A be the event the selected card is a spade, and B the event the card is a black card. Find P(A) Find P(A B) 4

5 Ex 3: In a study, 600 adult males were observed as to whether they developed cancer or not. They were also classified into groups based on whether they smoked or not. The results are given in the table. Suppose a person is randomly selected from this group. Developed Cancer Smoked Yes No Yes No a. Find the probability that the person smoked and developed cancer. 5

6 b. Find the probability that a smoker developed cancer. c. Find the probability that a nonsmoker developed cancer. d. Based upon these results, do you think there is an association between smoking and cancer? Why? 6

7 e. Find the probability that a cancer victim smoked. f. Look at your answers for parts (b) and (e). Is P(B A) = P(A B)? Is P(B A) = 1/P(A B)? 7

8 When computing the probability of A given that B has occurred, we are effectively reducing our sample space to only the outcomes in event B. So we find the probability of A occurring within B, and divide by the probability of B. That is: 8

9 Ex 4: A check of dorm rooms on a college campus revealed that 38% had refrigerators, 52% had TVs, and 21% had both a TV and a refrigerator. Suppose a dorm room is randomly selected. Find the probability that: a. the room has a TV and a refrigerator b. a room with a TV has a refrigerator c. a room with a refrigerator has a TV 9

10 We can also use the conditional probability rule to find the probability of A and B occurring: 10

11 Ex 5: Suppose a person is randomly selected from the group discussed in Example 3. a. Find the probability that the person smoked. b. Use the results from Example 3b and part (a) above to find the probability that the selected person smoked and developed cancer. c. Does this agree with the result we obtained in Example 3a? 11

12 Independent and Dependent Events Let s return to Example 3 one more time. The table is shown again for convenience. Developed Cancer Smoked Yes No Yes No We found: P(cancer smoker) = P(cancer nonsmoker) = Now find: P(cancer) = 12

13 If cancer is NOT associated with smoking, we would expect that the probability of developing cancer would be the same for smokers and nonsmokers. That is, we would expect P(cancer smoker) = P(cancer nonsmoker) = P(cancer) It should not matter if the person smoked or not. Smoking should not have any influence on the chances of developing cancer. But it DOES, which indicates that there IS an association between the two. 13

14 We say that two events are independent if they are not associated, meaning the knowledge that one event has occurred has no influence on the probability that the other event will occur. Here is the formal definition: A and B are independent if: P(A B) = P(A) or P(B A) = P(B) Otherwise, we say the two events are dependent. 14

15 Ex 6: Consider the experiment of tossing a coin two times and recording the results. Are the tosses independent events? That is, if I know that my first flip was heads, does that affect the probability of getting heads on the second flip? 15

16 Ex 7: Suppose I draw two random cards from a full deck of cards and check to see if the card is an ace each time. Are the two draws independent or dependent if: a. the first card is not replaced? b. the first card is replaced and the deck reshuffled before drawing a second card? 16

17 Ex 8: Suppose we roll a fair die. Define events: A = {the number is odd} B = {the number is greater than 2} Are A and B independent? 17

18 Ex 9: The following table shows data from our class survey (after omitting students who responded as other ). Students were asked if they believe in an afterlife, and if they had ever falsely called in sick for work. Determine if falsely calling in sick is independent of believing in an afterlife. Believe in Afterlife Called in Sick Yes No Yes 29 8 No

19 Independence is a very important concept in statistics, and is used frequently. As we will see in the next formula, many things are easier to compute if we have independence. Most of what we focus on in this course assumes independence, but you need to be able to decide for yourself if that is a valid assumption for each case. Notes about independence: 1. The property of independence doesn t always match one s intuition. The only way to check for independence is by performing the necessary calculations; it cannot be seen in a Venn diagram. 2. If two events A and B are mutually exclusive, then they must be dependent. This does not work in reverse; that is, two dependent events are not necessarily mutually exclusive. 19

20 Sequences of Independent Events If events A and B are independent, then the probability of the intersection of events A and B is equal to the product of the probabilities of events A and B. 20

21 Ex 10: Consider flipping a fair coin 10 times. a. Suppose the first 9 flips are all heads. What is the probability of flipping heads on the tenth flip? b. What is the probability of flipping 10 heads in a row? 21

22 Ex 11: I toss a fair coin and roll a fair die. Find the probability of tossing a tails and rolling a number greater than 3. 22

23 Ex 12: On any given day, a web site is expected to fail with probability In one standard work week (Monday Friday) what is the probability that the web site fails at least once? 23

24 Ex 13: According to the National Cancer Institute, 0.44% of women in their 30s have breast cancer. Also, a screening mammogram will correctly detect the presence of breast cancer 80% of the time. What is the probability that a randomly chosen woman in her thirties who goes in for a screening mammogram will have breast cancer and test positive for it? 24