Find P9 in a normal distribution

Question

A distribution of values is normal with a mean of 45.9 and a standard deviation of 53.5.

Find P₉, which is the score separating the bottom 9% from the top 91%.

Enter your answer as a number accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Answer 1

Hi Bailey,

In this case, you want to find the score, denoted as P9, which separates the bottom 9% from the top 91% in a normal distribution with a mean (μ) of 45.9 and a standard deviation (σ) of 53.5.

Final answer: the correct value for P9 is approximately -80.36.

To find P9, you can follow these steps:

1. Use a z-table or calculator to find the z-score that corresponds to the 9th percentile (0.09). The closest z-score for 0.09 is approximately -2.36.
2. Now, use the z-score formula:
3. z = (x - μ) / σ
4. Where:
5. z is the z-score (-2.36 in this case).
6. x is the value we want to find (P9).
7. μ is the mean (45.9).
8. σ is the standard deviation (53.5).
9. Plug in the values:
10. -2.36 = (P9 - 45.9) / 53.5
11. Solve for P9 by multiplying both sides by 53.5 and adding 45.9:
12. -2.36 * 53.5 = P9 - 45.9
13. -126.26 = P9 - 45.9
14. P9 = -126.26 + 45.9
15. P9 ≈ -80.36

So, the correct value for P9 is approximately -80.36. This score separates the bottom 9% from the top 91% in the given normal distribution.

As mentioned, I would be happy to provide our first session free to review answers to as many questions as you have! :)

Thank you,

Benjamin M.

Answer 2

Hi Bailey,

You have posted many similar problems. I want to make sure you are actually getting the material. You need to:

1. Memorize the formula z=x-mu/sigma
2. Learn how to read a z-score table. I cannot link to one because Wyzant reviews and delays posts with links, but you can find one readily online.

In this case, find the value within the z-table closest to the percentile you are looking for--9% or 0.09. Then look at the vertical column and horizontal row for your z-score, the value you will sub in for z in the equation. In this case, the closest internal probability is 0.091 and z=-2.36. So:

z=-2.36

x=P9, the score you seek

mu=45.9

sigma=53.5

Note: data with standard deviations this high is usually not good in the real world. It reflects too much variability in the sample and therefore too little predictability.

-2.36=(x-45.9)/53.5

Multiply both sides by 53.5

-126.26=x-45.9

x= -80.36

Again, very unlikely in the real world. That standard deviation seems far too high.

Please try some problems like these on your own and follow my earlier advice--remember z=x-mu/sigma and have a z-table on hand.