Chapter 9.1, Problem 25E

To determine

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

Answer to Problem 25E

Solution: The joint distribution of times by sex is shown below:

Joint Distribution
More and boys	37.23%
More and girls	34.13%
Never and boys	5.39%
Never and girls	7.12%
Once and boys	6.36%
Once and girls	9.77%

The marginal distribution of ‘gender’ is shown below:

	Gender
	Boys	Girls
Marginal distribution	$\left(\frac{963}{1966}\times 100\%=48.98\%\right)$	$\left(\frac{1003}{1966}\times 100\%=51.02\%\right)$

The marginal distribution of ‘times’:

Times	Marginal distribution
More	$\left(\frac{1403}{1966}\times 100\%=71.36\%\right)$
Never	$\left(\frac{246}{1966}\times 100\%=12.51\%\right)$
Once	$\left(\frac{317}{1966}\times 100\%=16.12\%\right)$

The conditional distribution of ‘sex’ by time is shown below:

	Gender
Times	Boys	Girls	Total
More	52.17%	47.83%	100%
Never	43.09%	56.91%	100%
Once	39.43%	60.57	100%

The conditional distribution of ‘time’ by sex is shown below:

Times	Boys	Girls
More	76.01%	66.90%
Never	11.01%	13.96%
Once	12.98%	19.14%
Total	100%	100%

Explanation of Solution

Calculation: Joint distributions are computed by dividing the cell element by the total observation. Hence, the table which shows the joint distribution is given below:

	Gender
Times	Boys	Girls	Total
More	37.23%	34.13%	71.36%
Never	5.39%	7.12%	12.51%
Once	6.36%	9.77%	16.12%
Total	48.98%	51.02%	100%

The joint distribution of Boys who had more sexually harassed can be calculated as:

$\begin{array}{c}\text{Joint distribution}=\frac{\text{732}}{1966}\times 100\\ =37.23\%\end{array}$

The joint distribution of Boys who had never sexually harassed times can be calculated as:

$\begin{array}{c}\text{Joint distribution}=\frac{\text{106}}{1966}\times 100\\ =5.39\%\end{array}$

Thus, the remaining joint distribution can be calculated by using the above formula.

Marginal distributions are computed by dividing the row or column totals by the overall total. Marginal distribution provides information about the individual variables but it does not provide any information about the relationship between two variables. Marginal distribution of Gender:

	Gender
	Boys	Girls
Marginal distribution	$\left(\frac{963}{1966}\times 100\%=48.98\%\right)$	$\left(\frac{1003}{1966}\times 100\%=51.02\%\right)$

Marginal distribution of Times:

Times	Marginal distribution
More	$\left(\frac{1403}{1966}\times 100\%=71.36\%\right)$
Never	$\left(\frac{246}{1966}\times 100\%=12.51\%\right)$
Once	$\left(\frac{317}{1966}\times 100\%=16.12\%\right)$

Conditional distributions of one variable restricted to a single outcome of another variable. Conditional distributions are computed by the row or column elements by the total of that row or column observation. Conditional distribution of gender who had sexually harassment is shown below:

	Gender
Times	Boys	Girls	Total
More	52.17%	47.83%	100%
Never	43.09%	56.91%	100%
Once	39.43%	60.57	100%

The conditional distribution of Boys who had more sexually harassed can be calculated as:

$\begin{array}{ccc}\text{Conditional distribution}& =& \frac{\text{732}}{1403}\times 100\\ & =& 52.17\%\end{array}$

Thus, the remaining conditional distribution of gender by the category time can be calculated by using the above formula.

Now, the conditional distribution of time for each gender category is shown below:

Times	Boys	Girls
More	76.01%	66.90%
Never	11.01%	13.96%
Once	12.98%	19.14%
Total	100%	100%

The conditional distribution of time for each gender can be calculated as:

$\begin{array}{ccc}\text{Conditional distribution}& =& \frac{\text{732}}{963}\times 100\\ & =& 76.01\%\end{array}$

Thus, the remaining conditional distribution of time for each gender can be calculated by using the above formula.

To determine

To explain: Conditional distribution to explain the result of analysis.

Answer to Problem 25E

Solution: The distribution of times by sex is preferable for the analysis.

Explanation of Solution

The distribution of times by sex is preferable and more informative. It shows that boys had witnessed sexual harassment more than girls. In category ‘More’ boys had 10% higher than girls.