Question

The annual rainfall in a certain region is approximately normally distributed with mean 42.6 inches and standard deviation 5.8 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 40 inches? % c) What percentage of years will have an annual rainfall of between 38 inches and 43 inches?

Answer 1

Solution:

We are given that: the annual rainfall in a certain region is approximately normally distributed with mean 42.6 inches and standard deviation 5.8 inches.

That is: µ = 42.6 and σ = 5.8

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

P( X < 44) = ..........?

z score formula:

z = ( x - µ ) / σ

z = ( 44 - 42.6 ) / 5.8 = 1.4 / 5.8 = 0.24

Thus we get:

P( X < 44) = P( Z < 0.24)

Look in z table for z = 0.2 and 0.04 and find area.

From z table, we get: P( Z < 0.24) = 0.5948

Thus

P( X < 44) = P( Z < 0.24)

P( X < 44) = 0.5948

P( X < 44) = 59.48%

P( X < 44) = 59.5%

Part b) What percentage of years will have an annual rainfall of more than 40 inches?

P( X > 40) = ...........?

z = ( x - µ ) / σ

z = ( 40 - 42.6 ) / 5.8 = -2.6 / 5.8 = -0.45

Thus we get:

P( X > 40) = P( Z > -0.45)

P( X > 40) = 1 - P( Z < -0.45)

Look in z table for z = -0.4 and 0.05 and find area.

From z table, we get: P( Z < -0.45) = 0.3264

Thus

P( X > 40) = 1 - P( Z < -0.45)

P( X > 40) = 1 - 0.3264

P( X > 40) = 0.6736

P( X > 40) = 67.36%

P( X > 40) = 67.4%

Part c) What percentage of years will have an annual rainfall of between 38 inches and 43 inches?

P( 38 < X < 43) = .........?

z = ( x - µ ) / σ

z = ( 38 - 42.6 ) / 5.8 = -4.6 / 5.8 = -0.79

and

z = ( 43 - 42.6 ) / 5.8 = 0.4 / 5.8 = 0.07

Thus we get:

P( 38 < X < 43) = P( -0.79 < Z < 0.07)

P( 38 < X < 43) = P( Z < 0.07) - P( Z < -0.79)

Look in z table for z = 0.0 and 0.07 as well as for z = -0.7 and 0.09 and find corresponding area.

From z table we get:

P( Z < 0.07) = 0.5279

P( Z < -0.79) = 0.2148

Thus

P( 38 < X < 43) = P( Z < 0.07) - P( Z < -0.79)

P( 38 < X < 43) = 0.5279 - 0.2148

P( 38 < X < 43) = 0.3131

P( 38 < X < 43) = 31.31%

P( 38 < X < 43) = 31.3%