Problem 3 [13 marks]. In this problem, you will be examining data to estimate pr

Question

Problem 3 [13 marks]. In this problem, you will be examining data to estimate proba bility tables for the joint random variables of marijuana use and dance/party participation Dataset obtained from (http://www.stat.ufl.edu/ winner/datasets.html). You can read more about the format of the data and variable descriptions here. The study was conducted to determine if behaviour correlates with marijuana use among middle class youths and was published in 1976 (a bit outdated but interesting nonetheless!). The data is summarized below: Party/Dance Participation Not At All Somewhat A Great Deaow Totals 118 40 54 32 Never 40 1/month 17 Totals a) (2 marks) Using the count data in this problem, how many middle class youths were surveyed? Briefly explain how you might use the data presented to estimate each joint probability? For example, how could you use these counts to estimate the propor- tion of middle class youths or the probability that a middle class youth somewhat uses marijuana less than once a month and participates in dance/parties a great deal? b) (2 marks) Create an estimated joint probability table using the data provided. Verify c) (3 marks) Find the marginal probability mass table for marijuana use, and briefly d) (3 marks) What is the probability that a middle class youth that consumes marijuana e) (3 marks) Find the conditional probability mass table for dance/party participation that your probability sum to describe what information this table con veys daily will never participate at dance/parties? for middle class youths who rarely (

Explanation / Answer

We can enter the data into an excel table, and the result would be something like the below:

a. So the total middle-class youth surveyed is the summation of total no. of students provided since all the different sets are mutually exclusive (a student who parties a great deal, cannot belong the group of students, who do not dance at all)

To find the joint probability of an event, we just need to find the intersection of the particular row and column in the table and divide it by the total no. of students,

For example, the proportion of students who use marijuana greater than once a day and who parties a great deal is 32/617

b. We can easily construct the joint probability table by dividing each value in the table obtained above by 617, to arrive at each joint probability.

We can clearly observe that the summation of probabilities is 1

c. The required information is already reflected in the above table, the marginal probabilities are nothing but the cumulative probabilities for a particular event. for example, the proportion of students who party a great deal is given by the column total under the Great deal Header.

d. The probability that a middle-class youth that consumes marijuana daily will never participate in dance/parties is 0, as the number of such people is ZERO. (Italicized and underlined in the above table)

e. Given that the youth belong to the rare consumption category:

We are left with the above subset. Now to obtain the conditional probabilities, ie.e the probability that a youth dances/parties a great deal, given that he rarely uses marijuana is given by 0.065/0.159 = 0.41

similarly, for others,

Party/Dance participation Not at all Somewhat A great deal Row totals Marijuana use Never 40 213 118 371 less than 1/month 3 55 40 98 grt. than 1/month 1 44 54 99 grt than 1/day 0 17 32 49 Column totals 44 329 244 617

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