To find the cutoff salary for teachers in the bottom 10%, we need to calculate the z-score associated with the 10th percentile and then convert it back to the original salary using the mean and standard deviation.
1. First, let's find the z-score using the z-score formula:
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.
2. In this case, we want to find the z-score for the 10th percentile. Since the normal distribution is symmetrical, we can find the z-score for the 90th percentile (which is the complement of the 10th percentile) and then negate it.
3. Using a standard normal distribution table or calculator, you can find that the z-score for the 90th percentile is approximately 1.28. Negating it gives us -1.28.
4. Now we can use the z-score formula to find the cutoff salary:
-1.28 = (x - 34,000) / 2000
Rearranging the formula, we get:
x - 34,000 = -1.28 * 2000
Simplifying the equation:
x - 34,000 = -2560
x = 34,000 - 2560
x = 31,440
Therefore, the cutoff salary for teachers in the bottom 10% is $31,440.