Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

5.3. Central Limit Theorem 345

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0.25

0.20

0.15

0.10

0.05

Figure 5.3.1:Theb

(

10 ,^12

)

pmf overlaid by theN

(

5 ,^52

)

pdf.

Since (Y−50)/5 has an approximate normal distribution with mean zero and vari-

ance 1, the probability is approximatelypnorm(.5)-pnorm(-.5)= 0.3829.

The convention of selecting the event 47. 5 <Y < 52 .5, instead of another event,

say, 47. 8 <Y < 52 .3, as the event equivalent to the eventY=48, 49 , 50 , 51 ,52 is

due to the following observation. The probabilityP(Y=48, 49 , 50 , 51 ,52) can be

interpreted as the sum of five rectangular areas where the rectangles have widths

1 and the heights are respectivelyP(Y= 48),...,P(Y = 52). If these rectangles

are so located that the midpoints of their bases are, respectively, at the points

48 , 49 ,...,52 on a horizontal axis, then in approximating the sum of these areas

by an area bounded by the horizontal axis, the graph of a normal pdf, and two

ordinates, it seems reasonable to take the two ordinates at the points 47.5 and 52.5.

This is called thecontinuity correction.

We next present two examples concerning large sample inference for proportions.

Example 5.3.5(Large Sample Inference for Proportions).LetX 1 ,X 2 ,...,Xnbe
a random sample from a Bernoulli distribution withpas the probability of success.
Letp̂be the sample proportion of successes. Then̂p=X. Hence,

̂p−p

√

̂p(1−̂p)/n

D

→N(0,1). (5.3.4)