Introduction to the Practice of Statistics

Introduction to the Practice of Statistics Sixth Edition Moore, McCabe Section 4.5 Homework Answers

4.101 Binge drinking and gender. In a college population, students are classified by gender and whether or not they are frequent binge drinkers. Here are the probabilities.

Men Women Binge drinker 0.11 0.12 Not binge drinker 0.32 0.45

(b) What is the probability that a randomly selected student is not a binge drinker?

P(Not binge drinker) = P(Not binge drinker And Men OR Not binge drinker And Women)

= 0.32 + 0.45 = 0.77

Lets break down the sentence to see if we can detect what if anything is suppose to be a “given” (that is we know that this event has occurred).

What is the probability that a randomly selected male… Are we asking the chance of selecting a male or are we saying the person is a male (no speculating)? My interpretation is that we know the student is male, but are asking if the student is a not binge drinker (not known yet).

Here is the notation for the question: P(not binge drinker | male)

What would be the question if being a male was also a speculation?

What is the probability that a randomly selected student is a male that does not binge drinker? A student is chosen at random; what ist he probability that the student is male and not a binge drinker?

But the question is What is the probability that a randomly selected male student is not a binge drinker?

0.32 P(not binge drinker | male) =  0.7442 0.43

(d) Explain why your answers to (b)and (c) are different. Use languagethat would be understood by somone who has not studied the material in this chapter.

Question (a) asks the chance a student is not a binge drinker which can mean male or female, but in part (b) the question is specifically directed a males. 4.102 Find some probabilities. Refer to the previous exercise.

Men Women Binge drinker 0.11 0.12 Not binge drinker 0.32 0.45

(a) Find the probability that a randomly selected student is a male binge drinker, and find the probability that a randomly selected student is a female binge drinker.

Find the probability that a randomly selected student is a male binge drinker: I want a male binge drinker. Compare this statement to question (b) in 101.

(b) 101: What is the probability that a randomly selected male student is not a binge drinker? (a) 102: Find the probability that a randomly selected student is a male binge drinker.

Look at the statement after the word “is” for each sentence; 101 wants not a binge drinker, and before the “is” word you get the description of the student: male.

Now look at the statement after the word “is” for the sentence in 102: a male binge drinker. This shows I want both male and binge drinker.

So the question translates to P(male AND binge drinker) = 0.11

“the probability that a randomly selected student is a female binge drinker.”

P(female and binge drinker) = 0.12.

(b) Find the probability that a student is a binge drinker, given that the student is male, and find the probability that a student is a binge drinker, given that the student is female.

“Find the probability that a student is a binge drinker, given that the student is male.”

0.11 P(binge drinker | male) =  0.2558 0.43 “Find the probability that a student is a binge drinker, given that the student is female.”

0.12 P(binge drinker | female) =  0.2105 0.57 (c) Your answer for part (a) gives a higher probability for females, while your answer for part (b) gives a higher probability for males. Interpret your answers in terms of the question of whether there are gender differences in binge drinking behavior. Decide which comparison you prefer and explain the reasons for your preference.

Men Women Total Binge drinker 0.11 0.12 0.23 Not binge drinker 0.32 0.45 0.77 Total 0.43 0.57 1

Question (a) asks the question to the entire sample sample space (sample space is a word used in the subject of probability, while the same meaning in statistics is population). Now look at the totals: women make up 57% of the sample space while men make up 43% of the sample space. Thus, just by having a larger number of women in this populaiton, the result from (a) which is shown in the table above, say that there are more women binge drinkers in this population, because there are more women period.

But question (b) breaks apart that sample space according to gender. Now, while the male population is smaller than the women, it has a larger proportion of binge drinkers, which is what the probabilities

0.11 P(binge drinker | male) =  0.2558 0.43

0.12 P(binge drinker | female) =  0.2105 show. A little more than a fourth of males are binge 0.57 drinkers, while females are a little more than a fifth of the population.

4.103 Attendance at 2 – year and 4 – year colleges. In a large national population of college students, 61% attend 4 – year institutions and the rest attend 2-year institutions. Males make up 44% of the students in the 4- year institutions and 41% of the students in the 2-year institutions.

(a) Find the four probabilities for each combination of gender and type of institutions in the following table. Be sure your probabilities sum up to 1.

Men Women Total 4-year institution 0.2684 0.3416 0.61 2- year institution 0.1599 0.2301 0.39 Total 0.4283 0.5717 1

Males make up 44% of the students in the 4-year institutions: P(male and 4 - year) x P(males | 4 year) = 0.44 = = : x = 0.44(0.61) = 0.2684. P(4 - year) 0.61 and 41% of the students in the 2-year institutions: P(male and 2 - year) y P(males | 2 year) = 0.41 = = : y = 0.41(0.39) = 0.1599. P(2 - year) 0.39 (b) Consider randomly selecting a female student from this population. What is the probability that she attends a 4-year institution?

“Consider randomly selecting a female student from this population.” It is a given that the student is female.

0.3416 P(4-year | woman) =  0.5977 0.5715