Question

A binomial distribution has p? = 0.26 and n? = 76. Use the normal approximation to the binomial distribution to answer parts ?(a) through ?(d) below.
?a) What are the mean and standard deviation for this? distribution?
?b) What is the probability of exactly 15 ?successes?
?c) What is the probability of 14 to 23 ?successes?
?d) What is the probability of 11 to 18 ?successes

Answer 1

We have given , p=0.26 , n=76

a) Mean =

standard deviation

?b) the probability of exactly 15 ?successes

P[X=15]

=P[15-0.5<X<15+0.5]

=P[14.5<X<15.5]

=P[-1.38<Z<-1.11]

=0.1335-0.0838..........................by using normal probability table.

=0.0497
?c) the probability of 14 to 23 ?successes

P[14<X<23]

=P[14.5<X<22.5]...................using continuity correction

=P[-1.38<Z<0.72]

=0.7642-0.0838.........................................by using normal probability table

=0.6804
?d) the probability of 11 to 18 ?successes

P[11<X<18]

=P[11.5<X<17.5]...........................by using continuity correction

=P[-2.16<Z<-0.59]

=0.2776-0.0154.......................by using normal probability table.

=0.2622