Assume that the salaries of elementary school teachers in the United States are normally distributed with a mean of $34,000 and a standard deviation of $2000. What is the cutoff salary for teachers in the bottom 10%?
Answer: 31,440
Assume that the salaries of elementary school teachers in the United States are normally distributed with a mean of $34,000 and a standard deviation of $2000. What is the cutoff salary for teachers in the bottom 10%?
Well, let's calculate the cutoff salary for teachers in the bottom 10%.
First, we need to find the z-score corresponding to the bottom 10% of the distribution. In other words, we want to find the z-score that separates the bottom 10% from the rest of the distribution.
Using a standard normal distribution table, we find that the z-score for the bottom 10% is approximately -1.28.
Now, we can use the formula for z-score to find the corresponding value in terms of salary:
z = (X - μ) / σ
Rearranging the formula, we have:
X = z * σ + μ
Plugging in the values:
X = -1.28 * 2000 + 34000
Calculating:
X ≈ 31440
So, the cutoff salary for teachers in the bottom 10% is approximately $31,440.
And if you're wondering how they can survive on such a low salary, well, they probably have a side gig as professional balloon animal artists. Talk about multi-talented!
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.
Find a Percentile to Z-Score Calculator. You can Google one, and likely there's a table of this in your text.
That'll tell you how many SD below the mean you have to go.
I make it about -1.28 standard deviations, so that'll be about 34,000 - (1.28 * 2000)