Chapter 2.6, Problem 123E

(a)

To determine

The explanatory variable and response variable.

Answer to Problem 123E

Solution: “Age (years)” of the candidates is the explanatory variable and “Rejected (Yes/No)” is the response variable.

Explanation of Solution

The provided study is conducted to see the healthy teeth. If a person wants a soldier in the Spanish American War, he should have at least 8 teeth. The statement of the requirement is that “As long as an applicant has at least four sound teeth, one above and one below each, from each side of the mouth, and thus aimed at chewing it for the purpose of opposing it, it should be rejected.”

In the study, there are two variables. One is an explanatory variable and the other is a response variable. An explanatory variable is a variable that affects the value of response variable and a response variable measures the result of the study. In the study, “Age (years)” is considered as the explanatory variable and the “Rejected (Yes/No)” is the response variable.

(b)

To determine

To find: The joint distribution.

Answer to Problem 123E

Solution: The joint distribution is provided below:

$\begin{array}{ccccccc}& & & \text{Age}& & & \\ \text{Rejected}& \text{Under 20}& \text{20}–\text{25}& \text{25}–\text{30}& \text{30}–\text{35}& \text{35}–\text{40}& \text{Over 40}\\ \text{Yes}& 0.0002& 0.0019& 0.0033& 0.0053& 0.0086& 0.0114\\ \text{No}& 0.1761& 0.2333& 0.1663& 0.1316& 0.1423& 0.1196\end{array}$

Explanation of Solution

Calculation: The joint distribution is computed by dividing the cell element by the total observation. The obtained joint distribution for under 20 can be calculated as follows:

$\begin{array}{c}\text{Candidate who were rejected }\left(\text{under 20}\right)=\frac{68}{334,321}\\ =0.0002\end{array}$

$\begin{array}{c}\text{Candidate who were not rejected }\left(\text{Under 20}\right)=\frac{58,884}{334,321}\\ =0.1761\end{array}$

Similarly, for other age group, the joint distribution can be calculated by using the above formula.

The joint distribution of the rejection data for the candidates classified by age is provided below:

$\begin{array}{ccccccc}& & & \text{Age}& & & \\ \text{Rejected}& \text{Under 20}& \text{20}–\text{25}& \text{25}–\text{30}& \text{30}–\text{35}& \text{35}–\text{40}& \text{Over 40}\\ \text{Yes}& 0.0002& 0.0019& 0.0033& 0.0053& 0.0086& 0.0114\\ \text{No}& 0.1761& 0.2333& 0.1663& 0.1316& 0.1423& 0.1196\end{array}$

From the table above, each cell shows a probability. For each cell, proportion is computed by dividing the cell entry by the total sample size. The collection of these proportions is the joint distribution of the two categorical variables. Because this is a distribution, the sum of the proportion should be 1. For this, the sum is 0.9999 that is approximately equal to 1.

(c)

To determine

To find: The two marginal distributions.

Answer to Problem 123E

Solution: The marginal distribution of “Rejected” is given below:

Marginal distribution of “Rejected”
Yes	No
$\frac{10,300}{334,321}=0.03081$	$\frac{324,021}{334,321}=0.96919$

The marginal distribution of “Age” is given below:

Marginal distribution of “Age”
Under 20	20 $–$ 25	25 $–$ 30	30 $–$ 35	35 $–$ 40	Over 40
$\begin{array}{l}=\frac{58,952}{334,321}\\ =0.1763\end{array}$	$\begin{array}{l}=\frac{78,639}{334,321}\\ =0.2352\end{array}$	$\begin{array}{l}=\frac{56,711}{334,321}\\ =0.1696\end{array}$	$\begin{array}{l}=\frac{45,777}{334,321}\\ =0.1369\end{array}$	$\begin{array}{l}=\frac{50,456}{334,321}\\ =0.1509\end{array}$	$\begin{array}{l}\frac{=43,786}{334,321}\\ =0.1330\end{array}$

Explanation of Solution

Calculation: Now, the marginal distribution is computed by dividing the row or column totals by the overall total. The marginal distribution of “Rejected” is given below:

Marginal distribution of “Rejected”
Yes	No
$\frac{10,300}{334,321}=0.03081$	$\frac{324,021}{334,321}=0.96919$

The marginal distribution of “Age” is given below:

Marginal distribution of “Age”
Under 20	20–25	25–30	30–35	35–40	Over 40
$\begin{array}{l}=\frac{58,952}{334,321}\\ =0.1763\end{array}$	$\begin{array}{l}=\frac{78,639}{334,321}\\ =0.2352\end{array}$	$\begin{array}{l}=\frac{56,711}{334,321}\\ =0.1696\end{array}$	$\begin{array}{l}=\frac{45,777}{334,321}\\ =0.1369\end{array}$	$\begin{array}{l}=\frac{50,456}{334,321}\\ =0.1509\end{array}$	$\begin{array}{l}\frac{=43,786}{334,321}\\ =0.1330\end{array}$

It is clearer to present these distributions as percent of the table total. The marginal distribution of the “Rejection” and “Age” is provided below:

Marginal distribution of “Rejected”
Yes	No
3%	97%

Marginal distribution of “Age”
Under 20	20–25	25–30	30–35	35–40	Over 40
17.63%	23.52%	16.96%	13.69%	15.09%	13.30%

Interpretation: This distribution does not provide any information about the relationship between the variables. The sum of the proportion should be 1.

(d)

To determine

The conditional distribution that will give the better explanation between the variables.

Answer to Problem 123E

Solution: The conditional distribution of “Rejection given Age” will give the better explanation between the variables.

Explanation of Solution

In the study, there are two variables, that is, explanatory and response variables. An explanatory variable is “Age in years” and response variable is “Rejection (Yes/No).” The conditional distribution of “Rejection” for provided age is appropriate because “Age” is chosen as explanatory variable.

(e)

To determine

To find: The conditional distribution.

Answer to Problem 123E

Solution: The conditional distribution is provided below:

$\begin{array}{ccccccc}& & & \text{Age}& & & \\ \text{Rejected}& \text{Under 20}& \text{20}–\text{25}& \text{25}–\text{30}& \text{30}–\text{35}& \text{35}–\text{40}& \text{Over 40}\\ \text{Yes}& 0.0012& 0.0082& 0.0196& 0.0389& 0.0572& 0.0868\\ \text{No}& 0.9988& 0.9918& 0.9804& 0.9611& 0.9428& 0.9132\end{array}$

Explanation of Solution

Calculation: The conditional distribution is obtained by dividing the row or column elements by the sum of that row or column observation. The conditional distribution of “Rejection” for given age can be calculated as follows:

$\begin{array}{c}\text{Candidate who were rejected }\left(\text{Under 20}\right)=\frac{68}{58,952}\\ =0.0012\end{array}$

$\begin{array}{c}\text{Candidate who were not rejected }\left(\text{Under 20}\right)=\frac{58,884}{58,952}\\ =0.9988\end{array}$

Similarly, the conditional distribution for the others can be calculated by using the above formula.

The conditional distribution of “Rejection for given age” is provided below:

$\begin{array}{ccccccc}& & & \text{Age}& & & \\ \text{Rejected}& \text{Under 20}& \text{20}–\text{25}& \text{25}–\text{30}& \text{30}–\text{35}& \text{35}–\text{40}& \text{Over 40}\\ \text{Yes}& 0.0012& 0.0082& 0.0196& 0.0389& 0.0572& 0.0868\\ \text{No}& 0.9988& 0.9918& 0.9804& 0.9611& 0.9428& 0.9132\end{array}$

Interpretation: When the value of one variable is conditioned in calculating the distribution of the other variable, we obtain a conditional distribution. From the above table, as the age increases, the number of rejected candidates also increases. This happens due to weak teeth in the older age.