Question

Grus schemes have a/an 9%chance of succeeding. An agent of the Anti-Villain league obtains access to a simple random sample of 1200 of Grus upcoming schemes. Find the probability? ..less than 101 schemes will succeed. B) more that 95 schemes will succeed c) between 95 and 101 schemes Will succeed d) less than 8.5% schemes will succeed . e) more than 9.5% sheens will succeed . f) between 8.5% and 9.5% schemes will succeed .

Answer 1

Let X denote the number of schemes that succeed. Then

X Bin(n 1200, p 0.09)

Using Normal approximation to Binomial distribution,

X~N(3D пр— 108, о2 — пр(1-р) - 98.28)

a)

100.5 108 P(X <101) P(X 100.5) P(Z -) P(Z -0.76) /98.28

{Due to continuity correction as a result of using Normal approximation for Binomial distribution}

P(Z> 0.76) = 1- P(Z < 0.76) 1-0.77637 0.22363

b)

95.5 108 P(X 95) P(X 95.5) P(Z ) P(Z-1.26) 98.28

{Due to continuity correction as a result of using Normal approximation for Binomial distribution}

P(Z<1.26)=0.89617

c)

P(95X<101) P(95.5 < X 100.5)

{Due to continuity correction as a result of using Normal approximation for Binomial distribution}

100.5 108 95.5 108 ) P(-1.26 Z-0.76) P( 98.28 98.28

P(Z 0.76) - P(Z-1.26) P(Z 0.76) P(Z> 1.26)

= [1 - P(Z < 0.76)]- 1- P(Z < 1.26) [1-0.77637]- 1-0.89617] 0.11980

d)

P(X<0.085(1200)) P(X< 102) P(X 101.5)

{Due to continuity correction as a result of using Normal approximation for Binomial distribution}

101.5 108 ) = P(Z-0.66) P(Z> 0.66) P(Z 98.28

$=1-P(Z<0.66)=1-0.74537=0.25463$