Respuesta :
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 [tex]\mu = p, \sigma = \sqrt{\frac{p(1-p)}{n}}[/tex]
In this problem, we have that:
[tex]\mu = 0.12, \sigma = \sqrt{\frac{0.12*0.88}{100}} = 0.0325[/tex]
10th percentile:
X when Z has a pvalue of 0.1. So X when Z = -1.28.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.28 = \frac{X - 0.12}{0.0325}[/tex]
[tex]X - 0.12 = -1.28*0.0325[/tex]
[tex]X = 0.0784[/tex]
The 10th percentile is 0.0784.
90th percentile:
X when Z has a pvalue of 0.9. So X when Z = 1.28.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.28 = \frac{X - 0.12}{0.0325}[/tex]
[tex]X - 0.12 = 1.28*0.0325[/tex]
[tex]X = 0.1616[/tex]
The 90th percentile is 0.1616.
