Suppose that x has a binomial distribution with n50 and p6 s

Suppose that x has a binomial distribution with n=50 and p=.6, so that the mean=30 and the standard deviation is 3.4641. Calculate the following probabilities using the normal approximation with the continuity correction (Hint:21<x<40 is the same as 22 ?x?39. Round answers to four decimals.

a) p(x=30)

b)p(x=21)

c)p(x ? 21)

d) p(21?x?40)

e) p(21<x<40)

Solution

X is binomial approximated to normal

with N(30, 3.4641)

For calculation of prob we convert x score to z score first

Thus the converted z values are used to find the prob from std normal distribution table.

Let us find prob

IN continuous function as normal approx. <21 or <=21 makes no difference as in discret.

a) P(x=30) =0

b) P(X=21) = 0 (at a point normal prob =0)

c) P(X<=21) = P(Z<=-2.47) = 0.5-0.4932 = 0.0068

d) P(21<=x<=40) = P(-2.47<=z<=2.75) = 0.4932+0.4970

= 0.9902

e) P(21<x<40) = 0.9902 same as d as point inclusion does not make any difference.