Question

Of the 93 participants in a drug trial who were given a new experimental treatment for​ arthritis, 53 showed improvement. Of the 92 participants given a​ placebo, 48 showed improvement. Construct a​ two-way table for these​ data, and then use a 0.05 significance level to test the claim that improvement is independent of whether the participant was given the drug or a placebo. Complete the following​ two-way table.

Answer 1

Solution:

Here, we have to use chi square test for independence of two categorical variables.

Null hypothesis: H₀: The improvement is independent of whether the participant was given the drug or a placebo.

Alternative hypothesis: H_a: The improvement is not independent of whether the participant was given the drug or a placebo.

We are given level of significance = α = 0.05

Test statistic formula is given as below:

Chi square = ∑[(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

E = row total * column total / Grand total

We are given

Number of rows = r = 2

Number of columns = c = 2

Degrees of freedom = df = (r – 1)*(c – 1) = 1*1 = 1

α = 0.05

Critical value = 3.841459

(by using Chi square table or excel)

A two way table is given as below:

Observed Frequencies (O)
	Improvement
Row variable	Yes	No	Total
Drug	53	40	93
Placebo	48	44	92
Total	101	84	185

Calculation tables for test statistic are given as below:

Expected Frequencies (E)
	Improvement
Row variable	Yes	No	Total
Drug	50.77297	42.22703	93
Placebo	50.22703	41.77297	92
Total	101	84	185

(O - E)
2.227027	-2.22703
-2.22703	2.227027

(O - E)^2/E
0.097683	0.117452
0.098745	0.118729

Chi square = ∑[(O – E)^2/E] = 0.432608

P-value = 0.510712

(By using Chi square table or excel)

P-value > α = 0.05

So, we do not reject the null hypothesis

There is sufficient evidence to conclude that improvement is independent of whether the participant was given the drug or a placebo.