Question

Suppose 80% of all farms in a certain state produce corn. A simple random sample of 60 the state's farms is drawn. (Use technology. Round your answers to three decimal places.)

Find the probability that the number of farms producing corn is between 40 and 50, inclusive. That is, find

P(40 ≤ x ≤ 50).

Answer 1

P( Farm Producing corn) = 0.8

A sample random of 60 state's farm is drawn, n= 60

Let X be the number of farms producing corn

X~ Binomial ( 60, 0.8)

Since here the sample size is large enough , we will use normal approximation

X~ Normal ( np , np(1-p))

X~ Normal ( 48, 9.6)

P(40 ≤ x ≤ 50) = P( 39.5 < X < 50.5 ) ; using the continuity correction

= P( < < )

= P( -2.74 < z < 0.81)

= P ( z < 0.81 ) - P( z < -2.74)

= P( z < 0.81 ) - 1 + P(z < 2.74)

= 0.79103 - 1 +0.99693

= 0.788