Lecture 10/7

Confidence Intervals (p)

Definition

An a% confidence interval for the population parameter p is an interval constructed from a sample proportion (p) within which you expect p to be with a% confidence (remember that we will only consider the case where n is large):

L': p - z_a(s_p)
U': p + z_a(s_p)
where s_p=sqrt(p(1-p)/n) and z_ais the z value from the standard normal table so that the area between -z and z is a% .

Terminology

s_p=sqrt(p(1-p)/n) is often referred to as the "error" in your estimate of p.
z₉₅(s_p) is often referred to as the "margin of error".
The term "point estimate" refers to the value of the statistic used to estimate the corresponding parameter.

Problem:

Consider the y2k problem above. You are interested in estimating the proportion of non-compliant modules in the portfolio. You select a sample of n=100 modules for examination and discover that 90% of them are non-compliant Construct and interpret a 95% confidence interval for the population proportion p.

Solution:

Since the sample proportion p=0.9 then s_p=sqrt(p(1-p)/n)=sqrt(0.9(1-0.9)/100)=0.03. Also, z_a=z₉₅=1.95, hence the 95% confidence interval for p is [0.9-1.95(0.03), 0.9+1.95(0.03)]=[0.843, 0.957]. Hence, you are 95% confident that the proportion of non-compliant modules in the population is between 84.3% and 95.7%.