First, let's make clear the numbers:
m = 128 mm
σ = 8 mm
n = 31
X - m = 0.9 mm
In order to find the probability requested, you need to compute the z-scores:
z = (X - m) / σ
Therefore:
z₁ = -0.9 / 8
= -0.1125
z₂ = 0.9 / 8
= 0.1125
Now, consider that:
p(z < z₁; z > z₂) = p(z < z₁) + p(z > z₂)
= p(z < z₁) + [1 - p(z < z₂)]
Look at a normal standard table to find the probabilities connected to the calculated z-scores (rounded to two decimal places):
p(z < -0.11) = 0.45620
p(z < +0.11) = 0.54380
Now, you can compute the total probability:
p(z < z₁; z > z₂) = 0.45620 + ( 1 - 0.54380)
= 0.9124
Therefore, the probability that the sample mean would differ from the population mean by more than 0.9 millimeters is p = 0.9124 which means 91.24%.
