CONFIDENCE INTERVALS FOR THE DIFFERENCE IN TWO PROPORTIONS 131and in order to make a 95% confidence interval just as for the population mean, we
would use p f 1.960,. Unfortunately, the standard deviation of p is unknown, so we
must estimate it in some way. The best estimate that we have for 7r is p = .70, so as
an estimate for o,, we use
d53F
or
J.70(1 - .70)/50 = dw = = ,065
The 95% confidence interval is then .70 f (1.96)(.065), or .70 i .13, or .57 to 23.
We are 95% confident that the true recovery rate is somewhere between 57 and 83%.
The formula for the 95% confidence interval is
P f 1.96daTm
If we desire to include the continuity correction of 1/2n = .01, we would add .01 to
.13 and the 95% confidence interval is .70 f .14, or .56 to .84. The formula for the
95% confidence interval with the correction factor included is
P (1.96dEG + %) 1
At this point, a question is often raised concerning the use of p in the formula for
the standard deviation of p. It is true that we do not know 7r and therefore do not
know the exact value of the standard deviation of p. Two remarks may be made in
defending the confidence interval just constructed, in which ~'m has been
substituted for dw. First, it has been established mathematically that if n
is large and if confidence intervals are constructed in this way repeatedly, then 95% of
them actually cover the true value of 7r. Second, even though the p that is used for 7r
in estimating the standard deviation of 7r happens to be rather far from the true value
of 7r, the standard deviation computed differs little from the true standard deviation.
For example, if 7r = .5,
0, = dm = J.25/50 = d@% = ,071
This is not very different from the approximation of .065 that we obtained using
p = .70. Note also that 0, is larger for p = .50 than it is for any other value of p and
c, gets progressively smaller as p gets closer to 0 or 1.
10.5 CONFIDENCE INTERVALS FOR THE DIFFERENCE IN TWO
PROPORTIONSOften, two populations are being studied, each with its own population proportion,
and we wish to estimate the difference between the two population proportions. For
example, it may be of interest to compare the recovery rate of patients with a cer-
tain illness who are treated surgically with the recovery rate of patients with this