# Estimation & The CLT?

The heights of boys at sixth form college are assumed to be normally distributed wit ha mean of 162cm and a standard deviation of 3.5cm.

a) If a sixth form boy is selected at randome what is the probability that his height is greater than 163cm?

b) A sixth form set has 10 boys. Calculate the probability that the mean height of the boys in the set is greater than 163cm.

c) What is the probability that all 10 boys have heights greater than 163cm?

Relevance

a)

μ = 162

σ = 3.5

standardize x to z = (x - μ) / σ

P(x > 163) = P( z > (163-162) / 3.5)

= P(z > 0.2857) = .3859

(From Normal probability table)

b)

Mean μ = 162

Standard deviation σ = 3.5

Standard error σ / √ n = 3.5 / √ 10 = 1.1067972

standardize xbar to z = (xbar - μ) / (σ / √ n )

P(xbar > 163) = P( z > (163-162) / 1.106797)

= P(z > 0.9035) = .1841

(from normal probability table)

c)

From (a) , for one boy, the probability is .3859

For all 10 boys , the probability of height > 163cm = (.3859)^10 = .0000732