Question

In a local university, 40% of the students live in the dormitories. A random sample of 80 students is selected for a particular study. The probability that the sample proportion (the proportion living in the dormitories) is between 0.30 and 0.50 is

Select one:

a. 0.9328

b. 0.0336

c. 0.0672

d. 0.4664

Answer 1

Given,

p = 0.40 , n = 80

Mean = p = 0.40

Standard deviation (SD) = Sqrt ( p ( 1 - p) / n)

= Sqrt ( 0.4 * 0.6 / 80)

= 0.05477

Using central limit theorem,

P( < p) = P( Z < - mean / SD)

So,

P( 0.30 < < 0.50) = P( < 0.50) - P( < 0.30)

= P( Z < 0.50 - 0.40 / 0.05477) - P( Z < 0.30 - 0.40 / 0.05477)

= P( Z < 1.83) - P( Z < -1.83)

= P( Z < 1.83) - ( 1 - P( Z < 1.83) )

= 0.9664 - ( 1 - 0.9664)

= 0.9328