Question

The heights of kindergarten children are approximately normally distributed with the following. (Give your answers correct to four decimal places.) μ = 50 and σ = 2.2 inches (a) If an individual kindergarten child is selected at random, what is the probability that he or she has a height between 47.5 and 52.5 inches? (b) A classroom of 16 of these children is used as a sample. What is the probability that the class mean x is between 47.5 and 52.5 inches? (c) If an individual kindergarten child is selected at random, what is the probability that he or she is taller than 52 inches? (d) A classroom of 16 of these kindergarten children is used as a sample. What is the probability that the class mean x is greater than 52 inches?

Answer 1

mean = 50 , s = 2.2

By using central limit theorem,

z = (x - mean)/sigma

a)

P(47.5 < x < 52.5)
= P((47.5 - 50)/2.2 < z < (52.5 - 50)/2.2)
= P(-1.14 < z < 1.14)
= 0.8721 - 0.1279
= 0.7442

b)
Here, n =16

P(47.5 < x < 52.5)
= P((47.5 - 50)/(2.2/sqrt(16)) < z < (52.5 - 50)/(2.2/sqrt(16)))
= P(-4.55 < z < 4.55)
= 1- 0
= 1

c)

P(x> 52)
= 1 - P(z< (52 - 50)/2.2)
= 1 - P(z < 0.91)
= 1 - 0.8183
= 0.1817

d)

Here, n =16
P(x> 52)
= 1 - P(z< (52 - 50)/(2.2/sqrt(16))
= 1 - P(z < 3.64)
= 1 - 0.9999
= 0.0001