Answer:
The 10th percentile is 0.0784.
The 90th percentile is 0.1616.
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
For a proportion p in a sample of size n, we have that ![\mu = p, \sigma = \sqrt{\frac{p(1-p)}{n}}](https://tex.z-dn.net/?f=%5Cmu%20%3D%20p%2C%20%5Csigma%20%3D%20%5Csqrt%7B%5Cfrac%7Bp%281-p%29%7D%7Bn%7D%7D)
In this problem, we have that:
![\mu = 0.12, \sigma = \sqrt{\frac{0.12*0.88}{100}} = 0.0325](https://tex.z-dn.net/?f=%5Cmu%20%3D%200.12%2C%20%5Csigma%20%3D%20%5Csqrt%7B%5Cfrac%7B0.12%2A0.88%7D%7B100%7D%7D%20%3D%200.0325)
10th percentile:
X when Z has a pvalue of 0.1. So X when Z = -1.28.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![-1.28 = \frac{X - 0.12}{0.0325}](https://tex.z-dn.net/?f=-1.28%20%3D%20%5Cfrac%7BX%20-%200.12%7D%7B0.0325%7D)
![X - 0.12 = -1.28*0.0325](https://tex.z-dn.net/?f=X%20-%200.12%20%3D%20-1.28%2A0.0325)
![X = 0.0784](https://tex.z-dn.net/?f=X%20%3D%200.0784)
The 10th percentile is 0.0784.
90th percentile:
X when Z has a pvalue of 0.9. So X when Z = 1.28.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![1.28 = \frac{X - 0.12}{0.0325}](https://tex.z-dn.net/?f=1.28%20%3D%20%5Cfrac%7BX%20-%200.12%7D%7B0.0325%7D)
![X - 0.12 = 1.28*0.0325](https://tex.z-dn.net/?f=X%20-%200.12%20%3D%201.28%2A0.0325)
![X = 0.1616](https://tex.z-dn.net/?f=X%20%3D%200.1616)
The 90th percentile is 0.1616.