The central limit theorem

Question

A population of values has a normal distribution with μ=41.8μ=41.8 and σ=59.3σ=59.3. You intend to draw a random sample of size n=231n=231.

Find the probability that a single randomly selected value is between 30.9 and 50.



Find the probability that a sample of size n=231n=231 is randomly selected with a mean between 30.9 and 50

Answer 1

x = value, want to find P(30.9 < x < 50), where x is normally distributed with mean = 41.8 and SD = 59.3.

transform x to standard normal z variable = (x - mean)/SD = (x - 41.8)/59.3.

so P(30.9 < x < 50) = P((30.9-41.8)/59.3 < z < (50-41.8)/59.3) =

P(-0.184 < z < 0.138) =

P(z < 0.138) - P(z < -0.184)

Look up values from standard normal probabilities table