Question

Using the data below, suppose we focus on the proportions of patients who show improvement. Is there a statistically significant difference in the proportions of patients who show improvement between treatments 1 and 2. Run the test at a 5% level of significance.

	Symptoms Worsened	No Effect	Symptoms Improved	Total
Treatment 1	22	14	14	50
Treatment 2	14	15	21	50
Treatment 3	9	12	29	50

Answer 1

Let p₁ = The proportion of patients who show improvement after treatment 1 = 14/50 = 0.28

Let p₂ = The proportion of patients who show improvement after treatment 2 = 21/50 = 0.42

Let = Overall proportion = (14+21)/(50+50) = 35/100 = 0.35

1 - = 0.65

= 0.05

(a) The Hypothesis:

H0: p₁ = p₂ : The proportion of patients who show improvement after treatment 1 is equal to the the proportion of patients who show improvement after treatment 2.

Ha: p₁ p₂ : The proportion of patients who show improvement after treatment 1 is different from the the proportion of patients who show improvement after treatment 2

This is a 2 Tailed Test.

The Test Statistic:

The p Value:    The p value (2 Tail) for Z = -1.47, is; p value = 0.1416

The Critical Value:   The critical value (2 tail) at = , Zcritical = +1.96 and -1.96

The Decision Rule:    If Zobserved is > Zcritical or if Zobserved is < -Zcritical, Then Reject H0.

Also If the P value is < , Then Reject H0

The Decision: Since Z lies in between +1.96 and -1.96, We Fail To Reject H0

Also since P value (0.1416) is > (0.05), We Fail to Reject H0.

The Conclusion: There isn't-insufficient evidence at the 95% significance level to conclude that the proportion of patients who show improvement after treatment 1 is significantly different from the the proportion of patients who show improvement after treatment 2.