Chapter 5.3, Problem 98E

(a)

To determine

Event “Diet B” and “Lose weight” are independent when one of the subjects is randomly selected from the event.

Answer to Problem 98E

The events “diet B” and “Lose weight” are independent.

Explanation of Solution

Given information:

Two − way table comparing the effectiveness of three different diets (A, B, and C) on weight loss:

  

The two events are independent, if the probability of occurrence of one event does not affect the probability of occurrence of other event.

For the events “Diet B” and “Lose weight” to be independent,

The product of row total and column total, divided by the table total should be equal to the count in the table.

  $\frac{\text{Total}\text{count}\text{"Yes"×Total}\text{count"B"}}{\text{Table}\text{total}\text{count}}=\frac{180\times 100}{300}=\frac{180}{3}=60$

In the table, we can see that the count in row “Yes” and column “B” is already 60.

Thus,

The events “Diet B” and “Lose weight” are independent.

(b)

To determine

Fill the table such that no association exists between type of diet and whether a subject lost weight.

Answer to Problem 98E

Two − way table:

  

Explanation of Solution

Given information:

Two − way table comparing the effectiveness of three different diets (A, B, and C) on weight loss:

  

Two events are independent, when the probability of occurrence of one event does not affect the probability of occurrence of other event.

Then

The counts will be the product of the row total and the column total, divided by the table total provided in the bottom left corner of the table.

Calculate the counts in the two − way table:

  

Thus,

The two – way table becomes:

  

(c)

To determine

Fill the table such that an association exists between type of diet and whether a subject lost weight.

Answer to Problem 98E

Two − way table:

  

Explanation of Solution

Given information:

Two − way table comparing the effectiveness of three different diets (A, B, and C) on weight loss:

  

For two − way table, where an association exists between type of diet and whether a subject lost weight.

In this part, the count for column “A” and row “Yes” should be different from Part (b).

Suppose, if we choose the count 60 for column “A” and row “Yes” instead of 54 (in Part (b)).

Then

Put 60 in the column “A” and the row “Yes”.

And

Put the remaining counts according to the total counts of the rows and columns.

Thus,

The two − way table becomes: