Statistics 3011 (Geyer and Jones, Spring 2006) Examples:
Tests and Confidence Intervals for Proportions

One-Sample

R has several functions that do confidence intervals and tests of significance for proportions. They are not done exactly as our textbook recommends, in particular they do not do the plus four intervals. But they do something reasonable.

Confidence Intervals

We redo Example 18.6 in the textbook.

We use the R function prop.test (on-line help)

The interval

95 percent confidence interval: 
 0.4969264 0.8040596 

does not agree with either interval in the textbook,

Rweb:> 0.667 + c(-1,1) * 0.148 
[1] 0.519 0.815 
Rweb:> 0.651 + c(-1,1) * 0.142 
[1] 0.509 0.793 

but isn't that much different either.

The performance of the interval that prop.test calculates is presented on our performance page.

Tests of Significance

We redo Example 18.8 in the textbook.

We use the R function binom.test (on-line help)

The P-value

Rweb:> binom.test(2048, 4040, p = 0.5) 
 
	Exact binomial test 
 
data:  2048 and 4040  
number of successes = 2048, number of trials = 4040, p-value = 0.3869
alternative hypothesis: true probability of success is not equal to 0.5  

is not exactly the same as given in the textbook because R does an exact calculation using the binomial distribution rather than an approximate calculation using the normal distribution. In this case R is right and the book is wrong. Exact is better than approximate.

Two-Sample

Confidence Intervals

We redo Examples 19.2 and 19.3 in the textbook.

We use the R function prop.test (on-line help)

The interval

95 percent confidence interval: 
 0.01710674 0.36364418  

does not agree with either interval in the textbook,

(0.033, 0.347)
(0.029, 0.341)

but isn't that much different either.

Tests of Significance

We redo Examples 19.4 and 19.5 in the textbook.

We use the R function prop.test (on-line help)

The P-value

Rweb:> prop.test(c(91, 117), c(149, 236)) 
 
	2-sample test for equality of proportions with continuity correction 
 
data:  c(91, 117) out of c(149, 236)  
X-squared = 4.4092, df = 1, p-value = 0.03575
alternative hypothesis: two.sided  

is not exactly the same as given in the textbook because R does a continuity correction. One could get the answer in the book by using the optional argument correct = FALSE, but there is no point to that.