Question

The annual rainfall in a certain region is approximately normally distributed with mean 40.9 inches

and standard deviation 5.3 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 43 inches?
%

Answer 1

Explanation:

a) To find the percentage of years with an annual rainfall of less than 44 inches, we need to find the area under the normal distribution curve to the left of 44 inches.

Using a standard normal distribution table or calculator, we can find the z-score:

z = (44 - 40.9) / 5.3 = 0.585

The area to the left of this z-score is approximately 0.72, or 72%.

Therefore, approximately 72% of years will have an annual rainfall of less than 44 inches.

b) To find the percentage of years with an annual rainfall of more than 38 inches, we need to find the area under the normal distribution curve to the right of 38 inches.

Using a standard normal distribution table or calculator, we can find the z-score:

z = (38 - 40.9) / 5.3 = -0.736

The area to the right of this z-score is approximately 0.77, or 77%.

Therefore, approximately 77% of years will have an annual rainfall of more than 38 inches.

c) To find the percentage of years with an annual rainfall between 37 inches and 43 inches, we need to find the area under the normal distribution curve between the z-scores for 37 inches and 43 inches.

Using a standard normal distribution table or calculator, we can find the z-scores:

z1 = (37 - 40.9) / 5.3 = -0.736

z2 = (43 - 40.9) / 5.3 = 0.394

The area between these z-scores is approximately 0.53, or 53%.

Therefore, approximately 53% of years will have an annual rainfall between 37 inches and 43 inches.

Answer 2

a) To find the percentage of years with an annual rainfall of less than 44 inches, we need to find the area under the normal curve to the left of 44. Using a standard normal distribution table or a calculator, we find the z-score corresponding to 44 inches:

z = (44 - 40.9) / 5.3 = 0.585

The area to the left of z = 0.585 is approximately 0.7202. Therefore, about 72.0% of years will have an annual rainfall of less than 44 inches.

b) To find the percentage of years with an annual rainfall of more than 38 inches, we need to find the area under the normal curve to the right of 38. Using a standard normal distribution table or a calculator, we find the z-score corresponding to 38 inches:

z = (38 - 40.9) / 5.3 = -0.717

The area to the right of z = -0.717 is approximately 0.4713. Therefore, about 47.1% of years will have an annual rainfall of more than 38 inches.

c) To find the percentage of years with an annual rainfall between 37 inches and 43 inches, we need to find the area under the normal curve between the corresponding z-scores:

z1 = (37 - 40.9) / 5.3 = -0.736

z2 = (43 - 40.9) / 5.3 = 0.396

Using a standard normal distribution table or a calculator, we find the area between z = -0.736 and z = 0.396 is approximately 0.6181. Therefore, about 61.8% of years will have an annual rainfall between 37 inches and 43 inches.

Explanation: