Question

The state of Texas has a mean annual rainfall of 25.2 inches, whereas the state of New Jersey has a Mean annual rainfall of 44.5 inches. Assume the standard deviation for Texas is 4.9 inches and for New Jersey is 3.1 inches.

Find the probabilty that, for a sample of 40 years of rainfall for New Jersey. the mean annual rainfall is at least 45 inches.

Answer 1

Solution:
Given in the question
Mean annual rainfall for Texas = 25.2 inches
Mean annual rainfall for New Jersey = 44.5 inches
Standard deviation for Texas = 4.9 inches
Standard deviation for New Jersey = 3.1 inches
No. of sample years of rainfall for New Jersey= 40
We need to calculate P(X>=45)
Here we will use Z test because number of sample is large enough and population standard deviation is also known. So Z score can be calculated as
Z-Score = (X - mean)/SD/sqrt(n) = (45 - 44.5)/3.1/sqrt(40) = 1.02
From Z table we found p-value
P(X>=45) = 1 - 0.8461 = 0.1539
So there is 15.39% probability that the mean annual rainfall is at least 45 inches.