The annual rainfall in a certain region is approximately normally distributed with

Question

The annual rainfall in a certain region is approximately normally distributed with mean 41.6 inches and standard deviation 5.4 inches. Round answers to the nearest tenth of a percent.

a) What percentage of years will have an annual rainfall of less than 44 inches? _____%

b) What percentage of years will have an annual rainfall of more than 38 inches? _____ %

c) What percentage of years will have an annual rainfall of between 37 inches and 42 inches? _____%

Answer 1

This is a classical Z-score or "find a particular percentile" in the normal distribution problem.

So, what is being asked in each part and how do you find it?

a.) p(X < x).

1. Use Excel's norm.dist() function to calculate this probability directory or
2. convert x to a Z-score and use Excel's norm.s.dist() function or a Z table to find p(Z < z) or
3. convert x to a Z-score and look up the Z-score in a Z-table

norm.dist(x, mean, sd, True)

norm.dist(44, 41.6, 5.4, True). the 'True' tells excel to work with the cumulative normal distribution.

or

Z = (44 - 41.6)/5.4 = .44

p(Z <= .44) = .67

b.) p(X > x). Use the above mentioned tools to calculated the p(X <= x) then calculate 1- p(X <= x). This will give you p(X > x)

c.) This is the "find the interval" part of the problem. Find p(X <= x_1), p(X <= x_2), then subtract the smaller from the larger probability. This will give you the probability of x's being in the interval state.

p(37 <= x <= 42) = p(X <= 42) - p(X <= 37)

Note: in a continuous distribution such as the normal distribution, you may assume continuity, so p(x > x) = p(x >= x), p(x < x) = p(x <= x)