Introductory Biostatistics

leading to a 95% confidence interval of

0 : 51 Gð 1 : 96 Þð 0 : 05 Þ¼ð 0 : 41 ; 0 : 61 Þ

(b) In 1980, the estimated rate was

p 2 ¼

43

100

¼ 0 : 43

with its standard error

SEðp 2 Þ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð 0 : 43 Þð 1 
 0 : 43 Þ

100

r

¼ 0 : 05

leading to a 95% confidence interval of

0 : 43 Gð 1 : 96 Þð 0 : 05 Þ¼ð 0 : 33 ; 0 : 53 Þ

It can be seen that the two confidence intervals, one for 1965 and one for
1980, are both quite long and overlapsed, even though the estimated rates show
a decrease of 8% in smoking rate, because the sample sizes are rather small.

Example 4.10 A study was conducted to look at the e¤ects of oral contra-
ceptives (OC) on heart disease in women 40–44 years of age. It is found that
amongn 1 ¼5000 current OC users, 13 develop a myocardial infarction (MI)
over a three-year period, while among n 2 ¼ 10 ;000 non-OC users, seven
develop an MI over a three-year period. Application of the method described
above yields the following 95% confidence intervals for the MI rates:

(a) For OC users, the estimated rate was

p 1 ¼

13

5000

¼ 0 : 0026

with its standard error

SEðp 2 Þ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð 0 : 0026 Þð 1 
 0 : 0026 Þ

5000

r

¼ 0 : 0007

ESTIMATION OF PROPORTIONS 163