Consider a binomial experiment with 16 trials and probabilit

Consider a binomial experiment with 16 trials and probability 0.45 of success on a single trial. (a) Use the binomial distribution to find the probability of exactly 10 successes. (Round your answer to three decimal places.)
(a) Use the binomial distribution to find the probability of exactly 10 successes. (Round your answer to three decimal places.)
(b) Use the normal distribution to approximate the probability of exactly 10 successes. (Round your answer to three decimal places.)

Solution

Consider a binomial experiment with 16 trials and probability 0.45 of success on a single trial.

P(X=x) = (_nC_x) p^x (1-p)^n-x

P(X=10) = (₁₆C₁₀) 0.45¹⁰ (1-0.45)^16-10

P(X = 10) = 0.075478

Answer =P=0.075   ( 3 decimals)

(b) Use the normal distribution to approximate the probability of exactly 10 successes. (Round your answer to three decimal places.)

Expectation = np = 7.2

Variance = np(1 - p) = 3.96

Standard deviation = 1.99

P( X=10) = P( 9.05<x<10.5)

z value for 9.5, z=( 9.5-7.2)/1.99 =1.16

z value for 10.5, z=( 10.5-7.2)/1.99 =1.66

P( 9.05<x<10.5) = P( 1.16<z<1.66)

=P( z <1.66) –P( z<1.16)

= 0.9515 - 0.877

= 0.0745

Answer =P=0.075   ( 3 decimals)