Assume that the average annual salary for a worker in the United States is $32,000 and that the annual salaries for Americans are normally distributed with a standard deviation equal to $7,000. Find the following:


(A) What percentage of Americans earn below $24,000?
(B) What percentage of Americans earn above $36,000?

Please show all of your work.
(Points : 6)

(A) 24,000 is 8/7 of one standard deviation below the mean.

(B) 36,000 is 4/7 of one standard deviation above the mean.

You need a "normal distribution" table or computational tool to compute the percentages below 24,000 and above 36,000.

There is one such calcualtor tool at
http://psych.colorado.edu/~mcclella/java/normal/accurateNormal.html

For (A), I get 12.65%
For (B), I get 28.4%

To find the percentage of Americans earning below $24,000 and above $36,000, we can use the concept of the standard normal distribution.

(A) To find the percentage of Americans earning below $24,000, we need to find the z-score for $24,000 and then use the z-table to determine the percentage of data below that z-score.

First, let's calculate the z-score:

z = (X - μ) / σ

Where:
X = $24,000 (value we want to find the percentage below)
μ = $32,000 (average annual salary)
σ = $7,000 (standard deviation)

z = (24,000 - 32,000) / 7,000
z = -8,000 / 7,000
z ≈ -1.14

Next, we need to find the percentage of data below the z-score of -1.14 on the z-table. Looking up the z-score in the table, we find that the percentage is approximately 0.1271 or 12.71%.

Therefore, approximately 12.71% of Americans earn below $24,000.

(B) To find the percentage of Americans earning above $36,000, we can use the same process. Let's calculate the z-score:

z = (X - μ) / σ

Where:
X = $36,000 (value we want to find the percentage above)
μ = $32,000 (average annual salary)
σ = $7,000 (standard deviation)

z = (36,000 - 32,000) / 7,000
z = 4,000 / 7,000
z ≈ 0.57

Next, we need to find the percentage of data above the z-score of 0.57 on the z-table. Looking up the z-score in the table, we find that the percentage is approximately 0.2852 or 28.52%.

Therefore, approximately 28.52% of Americans earn above $36,000.

To summarize:
(A) Approximately 12.71% of Americans earn below $24,000.
(B) Approximately 28.52% of Americans earn above $36,000.