Question

The cure rate for a standard treatment of a disease is 45%. Dr. Snyder has perfected a primitive treatment which he claims is much better. As evidence, he says that he has used his new treatment on 50 patients with the disease and cured 25 of them.

a) Construct a 95% confidence interval for the proportion of patients that Dr. Snyder has cured.

b) Can we be confident, at a 95% level of confidence, that this new treatment is better? (Support your answers with evidence from the confidence interval)

Answer 1

a)

p̂ = X / n = 25/50 = 0.5
p̂ ± Z(α/2) √( (p * q) / n)
0.5 ± Z(0.05/2) √( (0.5 * 0.5) / 50)
Z(α/2) = Z(0.05/2) = 1.96
Lower Limit = 0.5 - Z(0.05) √( (0.5 * 0.5) / 50) = 0.361
upper Limit = 0.5 + Z(0.05) √( (0.5 * 0.5) / 50) = 0.639
95% Confidence interval is ( 0.361 , 0.639 )

b)

Since 0.45 contained in confidence interval and some values in confidence interval are less

than 0.45, we do not have sufficient evidence to conclude that this new treatment is better.