Question

Suppose you work for a pharmaceutical company that is testing a new drug to treat ulcerative colitis. You conduct a random sample of 31 patients on the new drug and 10 of them indicated a decrease in symptoms since starting the drug regimen. You construct a 95% confidence interval for this proportion to be ( 0.158 , 0.4871 ). What is the correct interpretation of this confidence interval?

Question 9 options:

	1) We are 95% confident that the proportion of all patients on the drug regimen that have experienced a reduction in symptoms is between 0.158 and 0.4871.
	2) We are 95% confident that the proportion of patients surveyed that have experienced a reduction in symptoms is between 0.158 and 0.4871.
	3) We are certain that 95% of patients will be between 0.158 and 0.4871.
	4) We cannot determine the proper interpretation of this interval.
	5) We are 95% confident that the proportion of patients who should be on this drug is between 0.158 and 0.4871.

Answer 1

TRADITIONAL METHOD
given that,
possible chances (x)=10
sample size(n)=31
success rate ( p )= x/n = 0.3226
I.
sample proportion = 0.3226
standard error = Sqrt ( (0.3226*0.6774) /31) )
= 0.084
II.
margin of error = Z a/2 * (standard error)
where,
Za/2 = Z-table value
level of significance, α = 0.05
from standard normal table, two tailed z α/2 =1.96
margin of error = 1.96 * 0.084
= 0.1646
III.
CI = [ p ± margin of error ]
confidence interval = [0.3226 ± 0.1646]
= [ 0.158 , 0.4871]
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DIRECT METHOD
given that,
possible chances (x)=10
sample size(n)=31
success rate ( p )= x/n = 0.3226
CI = confidence interval
confidence interval = [ 0.3226 ± 1.96 * Sqrt ( (0.3226*0.6774) /31) ) ]
= [0.3226 - 1.96 * Sqrt ( (0.3226*0.6774) /31) , 0.3226 + 1.96 * Sqrt ( (0.3226*0.6774) /31) ]
= [0.158 , 0.4871]
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interpretations:
1. We are 95% sure that the interval [ 0.158 , 0.4871] contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the true population proportion
Answer:
option:1
   We are 95% confident that the proportion of all patients on the drug regimen that have experienced a reduction in symptoms is between 0.158 and 0.4871.