Basic Statistics

ESTIMATING THE DIFFERENCE BETWEEN TWO MEANS: PAIRED COMPARISON 89

has a standard normal distribution. Substituting sp for 0 gives a quantity whose

distribution is a t distribution. The d.f.’s for this t distribution are n1 + nz - 2,

which is the number used in the denominator of sg. [Recall that for a single sample,

t = (x - p)/(s/&) had n - 1 d.f., and that n - 1 is the number used in the

denominator of s2 .]

The 95% confidence interval for p1 - p2 may then be computed by substituting

sp for 0 and by changing 1.96, the 97.5% point of the z distribution, to the 97.5%

point of the t distribution with n1 + 122 - 2 d.f.’s. That is, the interval becomes

(ff, - X2) i t[.975]spJl/nl + l/nz

with d.f. = n1 + 122 - 2. The calculation of the 95% confidence interval with o2

unknowncanbeillustratedforpl-pz. Here,nl = 16,nz = 9.x1 = 311.9g,xz =

206.4g. s: = 20.392, si = 7060, and t[.975] = 2.069 for n1 + 122 - 2 = 23 d.f.

First, we compute the pooled variance as

2 (16 - 1)(20,392) + (9 - 1)(7060) 362,360

P 16+9-2

• 
  
  
  
  • -- = 15.755
    23
  
  
  
  

s=

Taking the square root of the pooled variance yields sp = 125.5. The confidence
interval for p1 - p2 for o unknown is

(311.9 - 206.4) & (2.069)(125.5)d& + $

or

105.5 i 2.069(125.5)m

or

105.5 & 108.2

or the interval is from -2.7 to 213.7g. Because the lower limit is negative and the

upper limit is positive, we cannot conclude that the population mean gain in weight

under the supplemented diet is larger than under the standard diet; the difference

p1 - p2 may be either positive or negative, but we might be inclined to think that it

is positive.

In the past example, we have assumed that we have simple random samples from

two normal populations with equal population variances. Methods for determining

if the data are approximately normally distributed were discussed in Section 6.4. A

method for testing if the two variances are equal is given in Section 9.2.

7.6 ESTIMATING THE DIFFERENCE BETWEEN TWO MEANS:

PAIRED COMPARISON

Sometimes in studying the difference between two means it is possible to use pairs

or matched samples advantageously. This device is often quite effective in partially

eliminating the effects of extraneous factors. As mentioned in Chapter 2, matching