Answer:
Explanation:
First, order the information provided:
Table: "Who is better at getting deals?"
Who Is Better?
Respondent I Am My Spouse We Are Equal
Husband 278 127 102
Wife 290 111 102
<u>a. Develop a joint probability table and use it to answer the following questions. </u>
The<em> joint probability table</em> shows the same information but as proportions. Hence, you must divide each number of the table by the total number of people in the set of responses.
1. Number of responses: 278 + 127 + 102 + 290 + 111 + 102 = 1,010.
2. Calculate each proportion:
- 278/1,010 = 0.275
- 127/1,010 = 0.126
- 102/1,010 = 0.101
- 290/1,010 = 0.287
- 111/1,010 = 0.110
- 102/1,010 = 0.101
3. Construct the table with those numbers:
<em>Joint probability table</em>:
Respondent I Am My Spouse We Are Equal
Husband 0.275 0.126 0.101
Wife 0.287 0.110 0.101
Look what that table means: it tells that the joint probability of being a husband and responding "I am" is 0.275. And so for every cell: every cell shows the joint probability of a particular gender with a particular response.
Hence, that is why that is the joint probability table.
<u>b. Construct the marginal probabilities for Who Is Better (I Am, My Spouse, We Are Equal). Comment.</u>
The marginal probabilities are calculated for each for each row and each column of the table. They are shown at the margins, that is why they are called marginal probabilities.
For the colum "I am" it is: 0.275 + 0.287 = 0.562
Do the same for the other two colums.
For the row "Husband" it is 0.275 + 0.126 + 0.101 = 0.502. Do the same for the row "Wife".
Table<em> Marginal probabilities</em>:
Respondent I Am My Spouse We Are Equal Total
Husband 0.275 0.126 0.101 0.502
Wife 0.287 0.110 0.101 0.498
Total 0.562 0.236 0.202 1.000
Note that when you add the marginal probabilities of the each total, either for the colums or for the rows, you get 1. Which is always true for the marginal probabilities.
<u>c. Given that the respondent is a husband, what is the probability that he feels he is better at getting deals than his wife? </u>
For this you use conditional probability.
You want to determine the probability of the response be " I am" given that the respondent is a "Husband".
Using conditional probability:
- P ( "I am" / "Husband") = P ("I am" ∩ "Husband) / P("Husband")
- P ("I am" ∩ "Husband) = 0.275 (from the intersection of the column "I am" and the row "Husband)
- P("Husband") = 0.502 (from the total of the row "Husband")
- P ("I am" ∩ "Husband) / P("Husband") = 0.275 / 0.502 = 0.548
<u>d. Given that the respondent is a wife, what is the probability that she feels she is better at getting deals than her husband?</u>
You want to determine the probability of the response being "I am" given that the respondent is a "Wife", for which you use again the formula for conditional probability:
- P ("I am" / "Wife") = P ("I am" ∩ "Wife") / P ("Wife")
- P ("I am" / "Wife") = 0.287 / 0.498
- P ("I am" / "Wife") = 0.576
<u>e. Given a response "My spouse," is better at getting deals, what is the probability that the response came from a husband?</u>
You want to determine: P ("Husband" / "My spouse")
Using the formula of conditional probability:
- P("Husband" / "My spouse") = P("Husband" ∩ "My spouse")/P("My spouse")
- P("Husband" / "My spouse") = 0.126/0.236
- P("Husband" / "My spouse") = 0.534
<u>f. Given a response "We are equal" what is the probability that the response came from a husband? What is the probability that the response came from a wife?</u>
<u>What is the probability that the response came from a husband?</u>
- P("Husband" / "We are equal") = P("Husband" ∩ "We are equal" / P ("We are equal")
- P("Husband" / "We are equal") = 0.101 / 0.502 = 0.201
<u>What is the probability that the response came from a wife:</u>
- P("Wife") / "We are equal") = P("Wife" ∩ "We are equal") / P("We are equal")
- P("Wife") / "We are equal") = 0.101 / 0.498 = 0.208
, so that we rewrite the above as
of the line chosen, which means the limit is path-dependent and thus does not exist.