Chapter 9, Problem 21E

To determine

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 1^st design for more than a minute and 5 students who look at the 2^nd design for the same time period. But, there are 13 students, who look at 1^st design for less than a minute and 20 students who look at 2^nd 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 1^st year and other in the 4^th year and their responses to the newly proposed core curriculum. Among the 361, 1^st year students, 85 respond yes and 276 respond no. While, among the 221, 4^th 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	$\left(\frac{\text{12}}{\text{50}}\text{×100=24\%}\right)$	$\left(\frac{\text{13}}{\text{50}}\text{×100=26\%}\right)$	$\left(\frac{\text{25}}{\text{50}}\text{×100=50\%}\right)$
Design2	$\left(\frac{\text{5}}{\text{50}}\text{×100=10\%}\right)$	$\left(\frac{\text{20}}{\text{50}}\text{×100=40\%}\right)$	$\left(\frac{\text{25}}{\text{50}}\text{×100=50\%}\right)$
Total	$\left(\frac{\text{17}}{\text{50}}\text{×100=34\%}\right)$	$\left(\frac{\text{33}}{\text{50}}\text{×100=66\%}\right)$	$\left(\frac{\text{50}}{\text{50}}\text{×100=100\%}\right)$

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	$\frac{\text{25}}{\text{50}}\text{×100=50\%}$
Design2	$\frac{\text{25}}{\text{50}}\text{×100=50\%}$

And the marginal distribution of time duration is shown below:

Time Period
	More than a minute	Less than a minute
Marginal probability	$\frac{\text{17}}{\text{50}}\text{×100=34\%}$	$\frac{\text{33}}{\text{50}}\text{×100=66\%}$

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	$\left(\frac{\text{12}}{\text{25}}\text{×100=48\%}\right)$	$\left(\frac{\text{13}}{\text{25}}\text{×100=52\%}\right)$	$\left(\frac{\text{25}}{\text{25}}\text{×100=100\%}\right)$
Design2	$\left(\frac{\text{5}}{\text{25}}\text{×100=20\%}\right)$	$\left(\frac{\text{20}}{\text{25}}\text{×100=80\%}\right)$	$\left(\frac{\text{25}}{\text{25}}\text{×100=100\%}\right)$

The conditional distribution of Design by Time Period is shown below:

Design	More than a minute	Less than a minute
Design1	$\left(\frac{\text{12}}{\text{17}}\text{×100}\right)\text{=70}\text{.58\%}$	$\left(\frac{\text{13}}{\text{33}}\text{×100}\right)\text{=39}\text{.4\%}$
Design2	$\left(\frac{\text{5}}{\text{17}}\text{×100}\right)\text{=29}\text{.42\%}$	$\left(\frac{\text{20}}{\text{33}}\text{×100}\right)\text{=60}\text{.6\%}$
Total	$\left(\frac{\text{17}}{\text{17}}\text{×100}\right)\text{=100\%}$	$\left(\frac{\text{33}}{\text{33}}\text{×100}\right)\text{=100\%}$

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	$\left(\frac{\text{85}}{\text{582}}\text{×100}\right)\text{=14}\text{.6\%}$	$\left(\frac{\text{276}}{\text{582}}\text{×100}\right)\text{=47}\text{.4\%}$	$\left(\frac{\text{361}}{\text{582}}\text{×100}\right)\text{=62\%}$
2nd year	$\left(\frac{\text{117}}{\text{582}}\text{×100}\right)\text{=20}\text{.1\%}$	$\left(\frac{\text{104}}{\text{582}}\text{×100}\right)\text{=17}\text{.9\%}$	$\left(\frac{\text{221}}{\text{582}}\text{×100}\right)\text{=38\%}$
Total	$\left(\frac{\text{202}}{\text{582}}\text{×100}\right)\text{=34}\text{.7\%}$	$\left(\frac{\text{380}}{\text{582}}\text{×100}\right)\text{=65}\text{.3\%}$	$\left(\frac{\text{582}}{\text{582}}\text{×100}\right)\text{=100\%}$

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	$\left(\frac{\text{361}}{\text{582}}\text{×100}\right)\text{=62\%}$
2nd year	$\left(\frac{\text{221}}{\text{582}}\text{×100}\right)\text{=38\%}$

Whereas, the marginal distribution of responses is shown below:

Response
	Yes	No
Marginal Probability	$\left(\frac{\text{202}}{\text{582}}\text{×100}\right)\text{=34}\text{.7\%}$	$\left(\frac{\text{380}}{\text{582}}\text{×100}\right)\text{=65}\text{.3\%}$

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	$\left(\frac{\text{85}}{\text{361}}\text{×100=23}\text{.5\%}\right)$	$\left(\frac{\text{276}}{\text{361}}\text{×100=76}\text{.45\%}\right)$	$\left(\frac{\text{361}}{\text{361}}\text{×100=100\%}\right)$
4th year	$\left(\frac{\text{117}}{\text{221}}\text{×100=53\%}\right)$	$\left(\frac{\text{104}}{\text{221}}\text{×100=47\%}\right)$	$\left(\frac{\text{221}}{\text{221}}\text{×100=100\%}\right)$

The conditional distribution of Student Type by Responses is shown below:

Student Type	More than a minute	Less than a minute
1st year	$\left(\frac{\text{85}}{\text{202}}\text{×100}\right)\text{=42\%}$	$\left(\frac{\text{276}}{\text{380}}\text{×100}\right)\text{=72}\text{.6\%}$
4th year	$\left(\frac{\text{117}}{\text{202}}\text{×100}\right)\text{=58\%}$	$\left(\frac{\text{104}}{\text{380}}\text{×100}\right)\text{=27}\text{.4\%}$
Total	$\left(\frac{\text{202}}{\text{202}}\text{×100}\right)\text{=100\%}$	$\left(\frac{\text{380}}{\text{380}}\text{×100}\right)\text{=100\%}$

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 2^nd design for more than a minute is 10% and the percentage of looking at the 2^nd 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 1^st design for more than a minute is 48%, the percentage of looking at the 2^nd design for the same is 20%.

Also, for the second scenario, the percentage of 4^th 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 1^st year student is 76.45%.