Question

# The two-way table summarizes data from an experiment comparing the effectiveness of three different diets (A, B, and C) on weight loss. Researchers ra

Bivariate numerical data

The two-way table summarizes data from an experiment comparing the effectiveness of three different diets (A, B, and C) on weight loss. Researchers randomly assigned 300 volunteer subjects to the three diets. The response variable was whether each subject lost weight over a 1-year period. $$\text{Diet}\ \text{Lost weight?}$$

$$\begin{array}{l|c|c|c|c} & \mathrm{A} & \mathrm{B} & \mathrm{C} & \text { Total } \\ \hline \text { Yes } & & 60 & & 180 \\ \hline \text { No } & & 40 & & 120 \\ \hline \text { Total } & 90 & 100 & 110 & 300 \end{array}$$

Suppose we randomly select one of the subjects from the experiment. Show that the events "Diet B" and "Lost weight" are independent.

2021-06-03

Given:
$$\begin{array}{l|c|c|c|c} & \mathrm{A} & \mathrm{B} & \mathrm{C} & \text { Total } \\ \hline \text { Yes } & & 60 & & 180 \\ \hline \text { No } & & 40 & & 120 \\ \hline \text { Total } & 90 & 100 & 110 & 300 \end{array}$$
Two events are independent, if the probability that one event occurs in no way affects the probability of the other event occurring.
We then show that events "Diet B? and "Lost weight” are independent if the product of the row total (given in last column of the same row) and column total (given in last row of the same column), divided by the table total (bottom left corner table) is equal to the count in the table.
$$\displaystyle\frac{{{180}\cdot{100}}}{{300}}=\frac{{18000}}{{300}}=\frac{{180}}{{3}}={60}$$
Since the count in the row ”Yes” and in the column ”B” is indeed 60, the ene "Thiet, Band "Lost weight” are indenendent.