Concept explainers
To find: The value of joint distribution, marginal distribution and the conditional distribution for the first and second scenario.
Answer to Problem 21E
Solution: Joint Distribution for first scenario is:
Time Period | |||
Design | More than a minute | Less than a minute | Total |
Design1 | 24.00% | 26.00% | 50.00% |
Design2 | 10.00% | 40.00% | 50.00% |
Total | 34.00% | 66.00% | 100.00% |
The conditional distribution is:
Design | More than a minute | Less than a minute |
Design1 | 70.58% | 39.4% |
Design2 | 29.42% | 60.6% |
Total | 100.00% | 100.00% |
And,
Time Period | |||
Design | More than a minute | Less than a minute | Total |
Design1 | 48.00% | 52.00% | 100.00% |
Design2 | 20.00% | 80.00% | 100.00% |
Now, joint distribution for second scenario is:
Response | |||
Student Type | Yes | No | Total |
1st year | 14.60% | 47.40% | 62.00% |
4th year | 20.10% | 17.90% | 38.00% |
Total | 34.70% | 65.30% | 100.00% |
The conditional distribution is,
Student Type | More than a minute | Less than a minute |
1st year | 42.00% | 72.60% |
4th year | 58.00% | 27.40% |
Total | 100.00% | 100.00% |
And,
Response | |||
Student Type | Yes | No | Total |
1st year | 23.50% | 76.45% | 100.00% |
4th year | 53.00% | 47.00% | 100.00% |
Explanation of Solution
Given: In the study, for the first scenario, there are 12 students who look at the 1st design for more than a minute and 5 students who look at the 2nd design for the same time period. But, there are 13 students, who look at 1st design for less than a minute and 20 students who look at 2nd design for less than a minute.
The 2 x 2 table for first scenario is as follows:
Time Period | |||
Design | More than a minute | Less than a minute | Total |
Design1 | 12 | 13 | 25 |
Design2 | 5 | 20 | 25 |
Total | 17 | 33 | 50 |
For the second scenario, there are two types of student, one in the 1st year and other in the 4th year and their responses to the newly proposed core curriculum. Among the 361, 1st year students, 85 respond yes and 276 respond no. While, among the 221, 4th year students, 117 responded yes and 104 responded no.
The 2 x 2 table for the second scenario is as follows:
Response | |||
Student Type | Yes | No | Total |
1st year | 85 | 276 | 361 |
2nd year | 117 | 104 | 221 |
Total | 202 | 380 | 582 |
Calculations:
In the study, for the first scenario, the Joint distribution is computed by dividing the cell element by the total observation. The obtained joint distribution is shown below:
Time Period | |||
Design | More than a minute | Less than a minute | Total |
Design1 | |||
Design2 | |||
Total |
Now, the marginal distribution is computed by dividing the row or column totals by the overall total. Marginal distributions provide information about the individual variables but not about the relationship between two variables.
Thus, the marginal distribution of designs is shown below:
Design | Marginal Distribution |
Design1 | |
Design2 |
And the marginal distribution of time duration is shown below:
Time Period | ||
More than a minute | Less than a minute | |
Marginal probability |
Conditional distribution is obtained by dividing the row or column elements by the sum of the observations in the corresponding row or column. The conditional distribution of Time Period by Design is shown below:
Time Period | |||
Design | More than a minute | Less than a minute | Total |
Design1 | |||
Design2 |
The conditional distribution of Design by Time Period is shown below:
Design | More than a minute | Less than a minute |
Design1 | ||
Design2 | ||
Total |
For the second scenario, the calculations are as follows:
In the study, Joint distribution is computed by dividing the cell element by the total observation. The obtained joint distribution is shown below:
Response | |||
Student Type | Yes | No | Total |
1st year | |||
2nd year | |||
Total |
Now, the marginal distribution is computed by dividing the row or column totals by the overall total. Marginal distributions provide information about the individual variables but not about the relationship between two variables. Thus, the marginal distribution for the types of students is shown below:
Student Type | Marginal Distribution |
1st year | |
2nd year |
Whereas, the marginal distribution of responses is shown below:
Response | ||
Yes | No | |
Marginal Probability |
Conditional distribution is obtained by dividing the row or column elements by the sum of the observations in the corresponding row or column.
The conditional distribution of Responses by Student Type is shown below:
Response | |||
Student Type | Yes | No | Total |
1st year | |||
4th year |
The conditional distribution of Student Type by Responses is shown below:
Student Type | More than a minute | Less than a minute |
1st year | ||
4th year | ||
Total |
Interpretation: The above 2x2 tables for the first scenario show different percentage values for joint, marginal and conditional distributions. The percentage of looking at the 2nd design for more than a minute is 10% and the percentage of looking at the 2nd design for less than a minute is 40%. The value of the marginal distribution for more than a minute is 34% and for less than a minute is 66%. The conditional distributional for the percentage of Time Period by Design for looking at the 1st design for more than a minute is 48%, the percentage of looking at the 2nd design for the same is 20%.
Also, for the second scenario, the percentage of 4th year students responding yes and no for the new curriculum are 20.1% and 17.9% respectively. The marginal distribution for the responding yes and no are 34.7% and 65.3% respectively. The conditional probability of responding no by the 1st year student is 76.45%.
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Chapter 9 Solutions
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