Assume that the salaries of elementary school teachers in the U.S. are normally distributed with a mean of $37,000 and standard deviation of $3500. The cutoff salary for teachers in the bottom 12% is
To find the cutoff salary for teachers in the bottom 12%, we need to determine the z-score associated with this percentage value and then convert it back to the corresponding salary.
Step 1: Find the z-score.
The z-score formula is:
z = (x - μ) / σ
Where:
z = z-score
x = data value (salary)
μ = mean
σ = standard deviation
We need to find the z-score associated with the bottom 12%, which is the area to the left of the cutoff salary. Since the normal distribution is symmetrical, we can find the z-score that corresponds to the bottom 12% by subtracting it from 1 (to account for the area to the right of the cutoff).
Bottom 12% = 0.12
Z-score = 1 - 0.12 = 0.88 (rounded to two decimal places)
Step 2: Convert the z-score back to the salary value.
Using the z-score formula, rearranging it to solve for x (salary):
x = z * σ + μ
x = 0.88 * $3500 + $37,000
After substituting the values into the formula, we get the cutoff salary for teachers in the bottom 12%:
x = $3080 + $37,000
x ≈ $40080
Therefore, the cutoff salary for teachers in the bottom 12% is approximately $40,080.
To find the cutoff salary for teachers in the bottom 12%, we need to calculate the z-score associated with that percentile and then use it to find the corresponding salary value.
Here's how you can do it step by step:
1. Determine the z-score: The z-score represents the number of standard deviations away from the mean a particular value is. We can use the z-score formula: z = (x - μ) / σ, where x is the observed value, μ is the mean, and σ is the standard deviation.
2. Locate the z-score on the standard normal distribution table: The standard normal distribution table provides the corresponding percentile or area under the curve for each z-score.
3. Find the salary corresponding to the given percentile: Once you know the z-score associated with a certain percentile, you can use it to find the salary value by rearranging the z-score formula: x = μ + (z * σ), where x is the salary, μ is the mean, σ is the standard deviation, and z is the z-score.
Now let's calculate the cutoff salary for teachers in the bottom 12%:
1. First, find the z-score corresponding to the bottom 12%:
- Since we are looking for the cutoff salary in the bottom 12%, this corresponds to the percentile 100% - 12% = 88%.
- In the standard normal distribution table, the z-score associated with the percentile 88% is approximately 1.18 (you can either use a z-table or a calculator with a built-in normal distribution function to find this value).
2. Calculate the salary corresponding to the z-score:
- Using the formula x = μ + (z * σ), where μ = $37,000 (mean) and σ = $3,500 (standard deviation), we can substitute the known values to find the cutoff salary.
- x = $37,000 + (1.18 * $3,500)
- x ≈ $37,000 + $4,130
- x ≈ $41,130
Therefore, the cutoff salary for teachers in the bottom 12% is approximately $41,130.
Working this kind of problem will require tables or some other form of "lookup".
Here is the perfect method to replace good ol' tables in back of textbooks
http://davidmlane.com/hyperstat/z_table.html
click on "value from area"
enter .12 in "area"
enter 37000 in "mean"
enter 3500 in SD
click on "below" and
lo and behold ----- cutoff salary is $32887