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Two 鈥� 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 鈥淒iet B鈥� and 鈥淟ose weight鈥� 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.

$$\begin{gathered} \frac{Totalcount"Yes"脳Totalcount"B"}{Tabletotalcount} \\ =\frac{180脳100}{300} \\ =\frac{180}{3} \\ =60 \end{gathered}$$

In the table, we can see that the count in row 鈥淵es鈥� and column 鈥淏鈥� is already $60$.

Thus,

The events 鈥淒iet B鈥� and 鈥淟ose weight鈥� are independent.

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 鈥� way table:

Diet

Thus,

The two 鈥� way table becomes:

Diet

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

In this part, the count for column 鈥淎鈥� and row 鈥淵es鈥� should be different from Part (b).

Suppose, if we choose the count 60 for column 鈥淎鈥� and row 鈥淵es鈥� instead of 54 (in Part (b)).

Then

Put 60 in the column 鈥淎鈥� and the row 鈥淵es鈥�.

And

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

Thus,

The two 鈥� way table becomes:

Diet