Confidence Interval for a Proportion - Summary

STA1003: Fundamental Statistics


Confidence Interval for a Proportion
Overview:
 What is meant by a confidence interval;
 How to calculate a confidence interval for a proportion;
 What is meant by a 90%, 95%, 98% or 99% confidence interval;
 How to find a sample size;
 Margin of error.

Introduction:
 So far, we have taken samples from a known population
 Usually the details about the population are unknown
 So (for example) µ and p are unknown
 In general, we take a sample to find out about the unknown population to
draw conclusions about parameters (inference)




Samples and populations:
 Statistics are numbers describing samples
 Parameters are numbers describing populations
 Usually, Roman letters denote statistics (eg. y, s)
 Usually, Greek letters denote parameters (eg. µ, σ)
 Each statistic has an associated parameter
 Parameters are usually unknown, and estimated by statistics




For Example…
 To estimate the population mean, use the sample mean
 But different samples will produce different sample means y, and hence
different estimates of µ
 How confident are we in our estimate of µ?
 (We’d be more confident in results from larger samples)
 We can construct a confidence interval for µ

 To estimate the population proportion, use the sample proportion
 But different samples will produce different sample proportions pb, and
hence different estimates of p
 How confident are we in our estimate of p?
 (We’d be more confident in results from larger samples)
 We can construct a confidence interval for p

, STA1003: Fundamental Statistics




Recall:




 Because the sampling distribution of pb has a Normal model. . .
 . . . we can use the 68–95–99.7 rule:
o 68% of the sample proportions are within one SE of p
o 95% of the sample proportions are within two SE of p
o 99.7% of the sample proportions are within three SE of p


Sample proportions:




Think from the p point-of-view




Confidence intervals:
 A typical confidence interval looks like this:
The 95% confidence interval for p is from 0.25 to 0.35
or simply (0.25, 0.35).
 Or we write it like this:
 The 95% confidence interval for p is 0.30 ± 0.05
 The ‘±’ means ‘plus and minus’, so it means
 Lower limit = 0.30 − 0.05 = 0.25
 Upper limit = 0.30 + 0.05 = 0.35