Question

According to a national survey, Americans consume an average of 36 pounds of cookie dough a year with a standard deviation of 12 pounds. A random sample of 100 people is taken. What is the probability the sampling error will be 2 pounds or less?

Answer 1

Solution:

We are given

µ = 36

σ = 12

n = 100

We have to find P(36 - 2 < x̄ < 36 + 2) = P(34< x̄ <38)

P(34< x̄ <38) = P(x̄<38) - P(x̄<34)

Find P(x̄<38)

Z = (x̄ - µ)/[σ/sqrt(n)]

Z = (38 - 36)/(12/sqrt(100))

Z = 1.666667

P(Z<1.666667) = P(x̄<38) = 0.95221

(by using z-table)

Now find P(x̄<34)

Z = (x̄ - µ)/[σ/sqrt(n)]

Z = (34 - 36)/(12/sqrt(100))

Z = -1.66667

P(Z<-1.66667) = P(x̄<34) = 0.04779

(by using z-table)

P(34< x̄ <38) = P(x̄<38) - P(x̄<34)

P(34< x̄ <38) = 0.95221 - 0.04779

P(34< x̄ <38) = 0.90442

Required probability = 0.90442