Question

The heights of kindergarten children are approximately normally distributed with the following. (Give your answers correct...

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?

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