Question

Forty are selected from a population with the mean 82 and variance 144.

a) Describe the sampling distribution of the sample mean.

b) Find the sample mean that have the 55th percentile.

c) Find the probability of the sample mean greater than 87.

Answer 1

a)

sampling distribution of sample mean is approx normal with mean =82 , std.dev = 12/sqrt(40) = 1.8974

b)

z value at 55% = 0.13

z = (x - mean)/s

0.13 = (x - 82)/1.8974

x = 1.8974 * 0.13 + 82

x = 82.25

c)

Here, μ = 82, σ = 1.8974 and x = 87. We need to compute P(X >= 87). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (87 - 82)/1.8974 = 2.64

Therefore,
P(X >= 87) = P(z <= (87 - 82)/1.8974)
= P(z >= 2.64)
= 1 - 0.9959 = 0.0041