Assume that the average annual salary for a worker in the United States is 300,000- 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 $22,000?
B. What percentage of Americans earn above 42,000?
I'm going to assume that you mean the average annual salary is $30k, not $300k - because $300k sounds a bit excessive to me, even for Americans! If so, the proportion of Americans who earn below $22k should be the area to the left of ($22k-$30k)/$7k = -1.143 in a set of Normal tables, which is 12.7%. The same sort of calculation for b) gives the area to the right of ($42k-$30k)/$7k = 1.714, which is the same as 1 - (area to the left of 1.714) = 1 - 0.957 = 4.3%.
To find the percentage of Americans who earn below $22,000 and above $42,000, we can use the concept of z-scores. A z-score allows us to compare a value from a normal distribution to the mean in terms of standard deviations.
First, let's calculate the z-score for each salary range:
A. To find the percentage of Americans who earn below $22,000:
Step 1: Calculate the z-score using the formula:
z = (X - μ) / σ
where X is the value ($22,000), μ is the mean ($300,000), and σ is the standard deviation ($7,000).
Plug in the values:
z = (22,000 - 300,000) / 7,000
= -278,000 / 7,000
= -39.71
Step 2: Find the corresponding percentage using a standard normal distribution table or a calculator.
Using the z-score table or calculator, we can find that the percentage of Americans who earn below $22,000 is approximately 0%. This indicates that no one in the population falls below $22,000.
B. To find the percentage of Americans who earn above $42,000:
Step 1: Calculate the z-score in a similar way:
z = (X - μ) / σ
where X is the value ($42,000), μ is the mean ($300,000), and σ is the standard deviation ($7,000).
Plug in the values:
z = (42,000 - 300,000) / 7,000
= -258,000 / 7,000
= -36.86
Step 2: Use the z-score to find the percentage using the z-score table or calculator.
Using the z-score table or calculator, we can find that the percentage of Americans who earn above $42,000 is approximately 0%. This indicates that no one in the population earns more than $42,000.
It's worth noting that these results seem unusual. The given data indicates a high mean salary and a low percentage of people earning below $22,000 or above $42,000. Double-check the data and calculations to ensure accuracy.