Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

5.3. Central Limit Theorem 347

variance is constant; in particular, it is free ofp. Hence, we seek a transformation

g(p) such that

g′(p)=

c

√

p(1−p)

,

for some constantc. Integrating both sides and making the change-of-variables

z=p,dz=1/(2

√

p)dp,wehave

g(p)=c

∫

1

√

p(1−p)

dp

=2c

∫

1

√

1 −z^2

dz=2carcsin(z)=2carcsin(

√

p).

Takingc=1/2, for the statisticg

(

X

)

=arcsin

(√

X

)

,weobtain

√

n

[

arcsin

(√

X

)

−arcsin (

√

p)

]D

→N

(

0 ,

1

4

)

.

Several other such examples are given in the exercises.

EXERCISES

5.3.1.LetXdenote the mean of a random sample of size 100 from a distribution

that isχ^2 (50). Compute an approximate value ofP(49<X<51).

5.3.2. LetX denote the mean of a random sample of size 128 from a gamma
distribution withα=2andβ= 4. ApproximateP(7<X<9).

5.3.3.LetYbeb(72,^13 ). ApproximateP(22≤Y≤28).

5.3.4.Compute an approximate probability that the mean of a random sample of
size 15 from a distribution having pdff(x)=3x^2 , 0 <x<1, zero elsewhere, is
between^35 and^45.

5.3.5.LetYdenote the sum of the observations of a random sample of size 12 from
a distribution having pmfp(x)=^16 ,x=1, 2 , 3 , 4 , 5 ,6, zero elsewhere. Compute an
approximate value ofP(36≤Y≤48).
Hint:Since the event of interest isY =36, 37 ,...,48, rewrite the probability as
P(35. 5 <Y < 48 .5).

5.3.6.LetYbeb(400,^15 ). Compute an approximate value ofP(0. 25 <Y/400).

5.3.7.IfYisb(100,^12 ), approximate the value ofP(Y= 50).

5.3.8.LetYbeb(n, 0 .55). Find the smallest value ofnsuch that (approximately)

P(Y/n >^12 )≥ 0 .95.