Chapter 9.1, Problem 18E

To determine

To find: The joint distribution, marginal distribution, and conditional distribution.

Answer to Problem 18E

Solution: In the study, 16.95% private institutions required physical education course and 28.53% of public institutions required the same. 46.61% of private institutions do not require physical education course and 7.90% of public institutions do not require the same. Among those who required the physical education course, 37.27% are private and 62.73% are public institutions, and among those who do not require physical education course, 85.49% are private and 14.51% are public.

Explanation of Solution

Calculation:

In the study, there are 354 higher institutions in which 225 are private institutions and 129 are public. Out of 225 private institutions, 60 require a physical education course. Out of 129 public institutions, 101 require a physical education course. In the study, Joint distribution is computed by dividing the cell element by the total observation. The obtained joint distribution is shown below:

$\begin{array}{cccc}& & \text{Educational institutions}& \\ \begin{array}{c}\text{Physical education}\\ \text{ course}\end{array}& \text{Private}& \text{Public}& \text{Total}\\ \text{Yes}& \left(\frac{\text{60}}{\text{354}}\text{×100=16}\text{.95\%}\right)& \left(\frac{\text{101}}{\text{354}}\text{×100=28}\text{.53\%}\right)& \left(\frac{\text{161}}{\text{354}}\text{×100=45}\text{.48\%}\right)\\ \text{No}& \left(\frac{\text{165}}{\text{354}}\text{×100=46}\text{.61\%}\right)& \left(\frac{\text{28}}{\text{354}}\text{×100=7}\text{.90\%}\right)& \left(\frac{\text{193}}{\text{354}}\text{×100=54}\text{.51\%}\right)\\ \text{Total}& \left(\frac{\text{225}}{\text{354}}\text{×100=63}\text{.66\%}\right)& \left(\frac{\text{129}}{\text{354}}\text{×100=36}\text{.44\%}\right)& \left(\frac{\text{354}}{\text{354}}\text{×100=100\%}\right)\end{array}$

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 Educational institutions is shown below:

	Educational institutions
	Private	Public
Marginal distribution	$\left(\frac{225}{354}\times 100\%=63.56\right)$	$\left(\frac{129}{354}\times 100\%=36.44\%\right)$

The marginal distribution of Physical education course is shown below:

Physical education course	Marginal distribution
Yes	$\left(\frac{161}{354}\times 100\%=45.48\%\right)$
No	$\left(\frac{193}{354}\times 100\%=54.52\%\right)$

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 Educational institutions by Physical education course is shown below:

$\begin{array}{cccc}& & \text{Educational institutions}& \\ \begin{array}{c}\text{Physical education }\\ \text{course}\end{array}& \text{Private}& \text{Public}& \text{Total}\\ \text{Yes}& \left(\frac{\text{60}}{\text{161}}\text{×100=37}\text{.27\%}\right)& \left(\frac{\text{101}}{\text{161}}\text{×100=62}\text{.73\%}\right)& \left(\frac{\text{161}}{\text{161}}\text{×100=100\%}\right)\\ \text{No}& \left(\frac{\text{165}}{\text{193}}\text{×100=85}\text{.49\%}\right)& \left(\frac{\text{28}}{\text{193}}\text{×100=14}\text{.51\%}\right)& \left(\frac{\text{193}}{\text{193}}\text{×100=100\%}\right)\end{array}$

The conditional distribution for Physical education course by each Educational institution is shown below:

Physical education course	Private	Public
Yes	$\left(\frac{60}{225}\times 100=26.66\%\right)$	$\left(\frac{101}{129}\times 100=\text{78}\text{.29}\%\right)$
No	$\left(\frac{165}{225}\times 100=73.33\%\right)$	$\left(\frac{28}{129}\times 100=\text{21}\text{.71}\%\right)$
Total	$\left(\frac{225}{225}\times 100=100\%\right)$	$\left(\frac{129}{129}\times 100=\text{100}\%\right)$

To determine

To explain: The conditional distribution to explain the result of the analysis.

Answer to Problem 18E

Solution: The Conditional distribution of ‘Physical education course by educational institutions’ is more preferable.

Explanation of Solution

The use of physical education course is more informative. A lot many public institutions require physical education course than the private institutions. From the above conditional distribution, a higher percentage of public institutions requires physical education course than the private institutions.