Q. 42 - The Practice of Statistics for AP

Question

Perform a follow-up analysis of the test in Exercise 40 by finding the individual components of the chi-square statistic. Which cell(s) contributed most to the final result? 

Step-by-Step Solution

Answer

It is noted that we obtained the largest chi-square subtotals for the Placebo and Both. This means that Both medicines together lead to better results and thus this treatment should be used (if the side effects are not too significant).

Step 1: Given information

The given data is

Step 2: Explanation

Observed counts

Expected counts

The expected counts are the row total multiplied by the column total, divided by the sample size$n=6602$

The chi-square statistic is the sum of squared deviations (between observed and expected counts) divided by the expected count:

$$\begin{gathered} {\chi }^2=\frac{(250-205.8128{)}^2}{205.8128}+\frac{(206-205.8128{)}^2}{205.8128}+\frac{(211-206.4368{)}^2}{206.4368}+\frac{(157-205.9376{)}^2}{205.9376} \\ +\frac{(1399-1443.1872{)}^2}{1443.1872}+\frac{(1443-1443.1872{)}^2}{1443.1872}+\frac{(1443-1447.5632{)}^2}{1447.5632}+\frac{(1493-1444.0624{)}^2}{1444.0624} \\ =9.487+0+0.101+11.629+1.353+0+0.14+1.658 \\ =24.243 \end{gathered}$$

It is noted that we obtained the largest chi-square subtotals for the Placebo and Both. This means that Both medicines together lead to better results and thus this treatment should be used (if the side effects are not too significant).