Chapter 5.3, Problem 63E

(a)

To determine

Probability for the female student perceived her body image as about right.

Answer to Problem 63E

Probability for the female student perceived her body image as about right is approx. 0.6550.

Explanation of Solution

Given information:

Data on perceived body image by gender summarized in the two − way table:

  

Calculate the total values in the given two − way table:

  

According to conditional probability,

  $P\left(\text{B}|\text{A}\right)=\frac{P\left(\text{A}\cap \text{B}\right)}{P\left(\text{A}\right)}=\frac{P\left(\text{A}\text{and}\text{B}\right)}{P\left(\text{A}\right)}$

Let

A: About right

O: Overweight

U: Underweight

F: Female

Note that

The information about 1200 U.S college students is provided in the table.

Thus,

The number of possible outcomes is 1200.

Also note that

In the table, 855 of the 1200 college students think their body image is about right.

Thus,

The number of favorable outcomes is 855.

When the number of favorable outcomes is divided by the number of possible outcomes, we get the probability.

  $P\left(\text{A}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{855}{1200}$

Now,

Note that

In the table, 560 female students of the 1200 total students think that their body image is about right.

In this case, the number of favorable outcomes is 560 and number of possible outcomes is 1200.

  $P\left(\text{A}\text{and}\text{F}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{560}{1200}$

Apply conditional probability:

  $P\left(\text{A}|\text{F}\right)=\frac{P\left(\text{A}\text{and}\text{F}\right)}{P\left(\text{A}\right)}=\frac{\frac{560}{1200}}{\frac{855}{1200}}=\frac{560}{855}=\frac{112}{171}\approx 0.6550=65.50\%$

Therefore,

About 65.50% of the students think theirbody image as about right are female students and the probability is approx. 0.6550.

(b)

To determine

Probability for the female student did not perceive her body image as overweight.

Answer to Problem 63E

Probability for the female student did not perceive her body image as overweight is approx. 0.7855.

Explanation of Solution

Given information:

Data on perceived body image by gender summarized in the two − way table:

  

Calculate the total values in the given two − way table:

  

According to complement rule,

  $P\left({\text{A}}^{\text{c}}\right)=P\left(\text{not}\text{A}\right)=1-P\left(\text{A}\right)$

According to conditional probability,

  $P\left(\text{B}|\text{A}\right)=\frac{P\left(\text{A}\cap \text{B}\right)}{P\left(\text{A}\right)}=\frac{P\left(\text{A}\text{and}\text{B}\right)}{P\left(\text{A}\right)}$

Let

A: About right

O: Overweight

U: Underweight

F: Female

Note that

The information about 1200 U.S college students is provided in the table.

Thus,

The number of possible outcomes is 1200.

Also note that

In the table, 760 of the 1200 college students are female students.

Thus,

The number of favorable outcomes is 760.

When the number of favorable outcomes is divided by the number of possible outcomes, we get the probability.

  $P\left(\text{F}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{760}{1200}$

Now,

Note that

In the table, 163 female students of the 1200 total students think that their body image is about overweight.

In this case, the number of favorable outcomes is 163 and number of possible outcomes is 1200.

  $P\left(\text{O}\text{and}\text{F}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{163}{1200}$

Apply conditional probability:

  $P\left(\text{O}|\text{F}\right)=\frac{P\left(\text{O}\text{and}\text{F}\right)}{P\left(\text{F}\right)}=\frac{\frac{163}{1200}}{\frac{760}{1200}}=\frac{163}{760}$

Apply the complement rule:

  $\begin{array}{l}P\left({\left(\text{O}|\text{F}\right)}^{\text{c}}\right)=1-P\left(\text{O}|\text{F}\right)\\ =1-\frac{163}{760}\\ =\frac{597}{760}\\ \approx 0.7855\\ =78.55\%\end{array}$

Therefore,

About 78.55% of the students did not perceive their body image as overweight are female students and the probability is approx. 0.7855.