Question

According to the Pew Research Center, 20% of Americans have not read a book in the past year. Let ˆp be the proportion of people that have not read a book in the past year in a random sample of 662 Americans. Determine the following probabilities. Round probability solutions to four decimal places and ˆp values to two decimal places, if necessary.

Find the probability that in a random sample of 662 Americans, at most 113 people have not read a book in the past year.

P(ˆp≤)=

Find the probability that in a random sample of 662 Americans, between 126 and 166 people have not read a book in the past year.

P(<ˆp<)=

Answer 1

Solution

Given that,

p = 0.20

1 - p = 1 - 0.20 = 0.80

n = 662

= p = 0.20

=  [p ( 1 - p ) / n] =   [(0.20 * 0.80) / 662 ] = 0.0155

a) P( 0.17)

= P[( - ) /   (0.17 - 0.20) / 0.0155]

= P(z -1.94 )

Using z table,

= 0.0262

b) P( 0.19 < < 0.25 )

= P[(0.19 - 0.20) / 0.0155 < ( - ) / < (0.25 - 0.20) / 0.0155]

= P(-0.65 < z < 3.23)

= P(z < 3.23 ) - P(z < -0.65)

Using z table,   

= 0.9994 - 0.2578

= 0.7416