Chapter 10, Problem 7CA

(a)

To determine

To calculate:

The values from the incomplete two-way table:

	Gym	Choir	Total
Male			$50$
Female	$23$
Total		$49$	$106$

Answer to Problem 7CA

The completed two-way table is as follows:

	Gym	Choir	Total
Male	$34$	$16$	$50$
Female	$23$	$33$	$56$
Total	$57$	$49$	$106$

Explanation of Solution

Given information:

A partial survey report is given about the preference of female students to take a gym class or choir.

	Gym	Choir	Total
Male			$50$
Female	$23$
Total		$49$	$106$

Calculation:

For completion of the given table, first number the columns and rows as follows:

	Gym	Choir	Total
Male	$\left(\text{1},\text{1}\right)$	$\left(\text{1},\text{2}\right)$	$(1,3)$
Female	$\left(\text{2},1\right)$	$\left(\text{2},\text{2}\right)$	$\left(\text{2},\text{3}\right)$
Total	$\left(\text{3},\text{1}\right)$	$(3,2)$	$(3,3)$

Now, let’s find the missing numbers i.e. $\left(\text{1},\text{1}\right)$ , $\left(\text{1},\text{2}\right)$ , $\left(\text{2},\text{2}\right)$ , $\left(\text{2},\text{3}\right)$ , $\left(\text{3},\text{1}\right)$ .

For, $\left(\text{3},\text{1}\right)$ :

  $\left(\text{3},\text{1}\right)=(3,3)-(3,2)$

Substitute the values,

  $\begin{array}{l}\left(\text{3},\text{1}\right)=106-49\\ \left(\text{3},\text{1}\right)=57\end{array}$

For, $\left(\text{2},\text{3}\right)$ :

  $\left(\text{2},\text{3}\right)=(3,3)-(1,3)$

Substitute the values,

  $\begin{array}{l}\left(\text{2},\text{3}\right)=106-50\\ \left(\text{2},\text{3}\right)=56\end{array}$

For, $\left(\text{2},\text{2}\right)$ :

  $\left(\text{2},\text{2}\right)=(2,3)-(2,1)$

Substitute the values,

  $\begin{array}{l}\left(\text{2},\text{2}\right)=56-23\\ \left(\text{2},\text{2}\right)=33\end{array}$

For, $\left(\text{1},\text{2}\right)$ :

  $\left(\text{1},\text{2}\right)=(3,2)-(2,2)$

Substitute the values,

  $\begin{array}{l}\left(\text{1},\text{2}\right)=49-33\\ \left(\text{1},\text{2}\right)=16\end{array}$

For, $\left(\text{1},\text{1}\right)$ :

  $\left(\text{1},\text{1}\right)=(1,3)-(1,2)$

Substitute the values,

  $\begin{array}{l}\left(\text{1},\text{1}\right)=50-16\\ \left(\text{1},\text{1}\right)=34\end{array}$

Thus, putting the respected values in the table, the completed table is as follow:

	Gym	Choir	Total
Male	$34$	$16$	$50$
Female	$23$	$33$	$56$
Total	$57$	$49$	$106$

(b)

To determine

To calculate:

The probability of a random selected student prefers and is female.

Answer to Problem 7CA

The probability of a random selected student prefers and is female is $0.31$ .

Explanation of Solution

Given information:

From subpart (a)

	Gym	Choir	Total
Male	$34$	$16$	$50$
Female	$23$	$33$	$56$
Total	$57$	$49$	$106$

Calculation:

Find the respective row-female and column-choir number in the cell. This gives the number of women who love choir. Divide the number by the total number of students surveyed and find the probability that the female and love choir is a randomly selected student.

Thus,

The required probability can be found as follows:

  $P(femaleandchoir)=\frac{(2,2)}{(3,3)}$

Substituting the values from the table,

  $\begin{array}{l}P(femaleandchoir)=\frac{33}{106}\\ P(femaleandchoir)=0.31\end{array}$

Thus, the probability of a random selected student prefers and is female is $0.31$ .

(c)

To determine

To calculate:

The probability of a randomly selected male student prefers gym class.

Answer to Problem 7CA

The probability of a randomly selected male student prefers gym class is $0.68$ .

Explanation of Solution

Given information:

From subpart (a)

	Gym	Choir	Total
Male	$34$	$16$	$50$
Female	$23$	$33$	$56$
Total	$57$	$49$	$106$

Calculation:

Consider,

P(Male students choosing Gym) = $P(MG)$

The required probability can be found as follows:

  $P(MG)=\frac{(1,1)}{(1,3)}$

Substituting the values from the table,

  $\begin{array}{l}P(MG)=\frac{34}{50}\\ P(MG)=0.68\end{array}$

Thus, the probability of a randomly selected male student prefers gym class is $0.68$ .