In a population μY = 100 and σ Y2 = 43. In a random sample of size n = 100, what is Pr (Ȳ < 101)?
The sample variance = (σ Y2 / n) = 43/100 = 0.43
Therefore, the Standard Error (SE) = sqrt(0.43) = 0.6557.
Normalizing this to a Standard Normal Distribution,
Z = ((101 – μY) / SE)
Z = ((101 – 100) / 0.6557) = 1.525
The Z value is the number of Standard Errors away from the mean that will yield the desired Y value of 101.
This is a one sided hypothesis since we are interested in the probability of Y being < 101.
In EXCEL, the probability that Y is < 101 is:
=NORMDIST(Z-Value, Mean of 0, Standard Deviation of 1, 1 for Cumulative)
=NORMDIST(1.525, 0, 1, 1)
=0.9364