The salaries at a corporation are normally distributed with an average salary of $29,000 and a standard deviation of $4,000. Sixteen percent of the employees make more than Sara. What is Sara's salary? Round your answer to the nearest dollar.
you will need a normal distribution table or chart.
I use this online applet
http://davidmlane.com/hyperstat/z_table.html
Use the second part at the bottom of the page
for mean enter 29000
for Sd enter 4000
in 'shaded' area, enter .16 and click on "above"
or
in 'shaded' area enter .84 and click on "below"
in either case you get
$32977.83 or $32978 to the nearest dollar
To find Sara's salary, we need to find the value that corresponds to the 84th percentile of the normal distribution.
First, we subtract 84% (100% - 16%) from 100% to find the area to the left of the desired value: 100% - 84% = 16%.
Next, we need to find the z-score that corresponds to this area using the z-table.
To find the z-score, we can use the formula:
z = (x - μ) / σ
where:
x = Sara's salary
μ = mean salary = $29,000
σ = standard deviation = $4,000
Plugging in the values, we have:
z = (x - 29,000) / 4,000
Using the z-table, we find that the z-score corresponding to an area of 16% is approximately -0.994.
Now, we can solve for Sara's salary:
-0.994 = (x - 29,000) / 4,000
Multiply both sides of the equation by 4,000:
-0.994 * 4,000 = x - 29,000
-3,976 = x - 29,000
Add 29,000 to both sides of the equation:
-3,976 + 29,000 = x
25,024 = x
Therefore, Sara's salary is approximately $25,024.
To find Sara's salary, we need to determine the value that corresponds to the 16th percentile of the salary distribution.
Step 1: Convert the given average salary and standard deviation into z-scores.
The z-score formula is: z = (x - μ) / σ
where z is the z-score, x is the value we want to find the percentile for, μ is the mean, and σ is the standard deviation.
For Sara's salary, let x be her salary, μ (mean) be $29,000, and σ (standard deviation) be $4,000.
Step 2: Find the z-score corresponding to the 16th percentile.
The percentage above a certain z-score can be found in the standard normal table. However, since we are given that 16% of the employees make more than Sara, we need to find the z-score corresponding to the 84th percentile instead (100% - 16% = 84%).
From the standard normal table, the z-score that corresponds to the 84th percentile is approximately 1.04.
Step 3: Solve for Sara's salary using the z-score formula.
1.04 = (x - 29,000) / 4,000
Rearrange the formula to solve for x (Sara's salary):
x - 29,000 = 4,000 * 1.04
x - 29,000 = 4,160
x = 4,160 + 29,000
x = 33,160
Therefore, Sara's salary is approximately $33,160.