Question

Suppose that the heights of students are normally distributed with mean 67 inches and standard deviation 3 inches. (a) What is the probability that a randomly chosen student is at least 69 inches tall? (b) What is the probability that the mean height of a random sample of 5 students is at least 69 inches? (c) What is the probability that the mean height of a random sample of 20 students is at least 69 inches?

Answer 1

Given,

= 67 , = 3

We convert this to standard normal as

P( X < x) = P( Z < x - / )

a)

P( X >= 69) = P( Z >= 69 - 67 / 3)

= P( Z >= 0.6667)

= 1 - p( Z < 0.6667)

= 1 - 0.7475

= 0.2525

b)

The central limit theorem is

P( < x) = P( Z < x - / / sqrt(n) )

So, for n = 5

P( >= 69) = P( Z >= 69 - 67 / 3 / sqrt(5) )

= P( Z >= 1.4907)

= 1 - P( Z < 1.4907)

= 1 - 0.9320

= 0.0680

c)

for n = 20

P( >= 69) = P( Z >= 69 - 67 / 3 / sqrt(20) )

= P (Z >= 2.9814)

= 1 - P( Z < 2.9814)

= 1 - 0.9986

= 0.0014