Answer:
- a. Refer to the table below
- b. Refer to the table below
Explanation:
To begin with, organize the data provided:
Table: "Who excels at obtaining deals?"
Who Excels?
Respondent I Am My Spouse We are Equal
Husband 278 127 102
Wife 290 111 102
a. Create a joint probability table and utilize it to respond to the ensuing inquiries.
The joint probability table presents identical details expressed as proportions. The values from the table need to be divided by the total number of responses involved.
1. Total responses: 278 + 127 + 102 + 290 + 111 + 102 = 1,010.
2. Determine 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 containing these values:
Joint probability table:
Respondent I Am My Spouse We Are Equal
Husband 0.275 0.126 0.101
Wife 0.287 0.110 0.101
This table illustrates that the joint probability of identifying as a husband while choosing 'I am' equals 0.275. Each cell conveys the joint probability associated with each gender's response.
Consequently, this delineates the purpose of a joint probability table.
b. Generate marginal probabilities for Who Excels (I Am, My Spouse, We Are Equal). Provide commentary.
Marginal probabilities are computed for each row and column of the table, indicated in the margins, which is their namesake.
For the column titled "I am," it amounts to: 0.275 + 0.287 = 0.562
Similarly, perform calculations for the other two columns.
For the row designated 'Husband,' it would thus be 0.275 + 0.126 + 0.101 = 0.502. Apply the same for the row labeled 'Wife.'
Table Marginal probabilities:
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
Notably, when summing the marginal probabilities for both rows and columns, the results will always equate to 1. This is a consistent truth for marginal probabilities.
c. Given the respondent is a husband, what is the likelihood that he believes he is better at securing deals than his wife?
This requires the utilization of conditional probability.
The goal here is to ascertain the probability of the response being "I am" when the respondent identifies as a "Husband."
Using conditional probability:
- P ( "I am" / "Husband") = P ("I am" ∩ "Husband) / P("Husband")
- P ("I am" ∩ "Husband) = 0.275 (obtained from the intersection of columns "I am" and rows "Husband")
- P("Husband") = 0.502 (derived from total of row "Husband")
- P ("I am" ∩ "Husband) / P("Husband") = 0.275 / 0.502 = 0.548
d. In the instance that the respondent is a wife, what probability exists that she believes she is superior to her husband in acquiring deals?
We seek to identify the probability wherein the response claims "I am" while the respondent is labeled a "Wife," applying the conditional probability formula again:
- P ("I am" / "Wife") = P ("I am" ∩ "Wife") / P ("Wife")
- P ("I am" / "Wife") = 0.287 / 0.498
- P ("I am" / "Wife") = 0.576
e. When responding that "My spouse" is better at scoring deals, what is the likelihood that the claim originated from a husband?
We aim to compute: P ("Husband" / "My spouse")
Applying the conditional probability formula:
- 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
f. When the response indicates "We are equal," what likelihood exists that this response is from a husband? What is the chance that it hails from a wife?
What is the likelihood that this response came from a husband?
- 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
What is the chance the response originated from a wife:
- 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
