Respuesta :
Answer:
a) The credit score that defines the upper 5% is 764.50.
b) Seventy-five percent of the customers will have a credit score higher than 532.5.
c) Range 472 to 728.
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.
In this problem, we have that:
[tex]\mu = 600, \sigma = 100[/tex]
a) Find the credit score that defines the upper 5 percent.
Value of X when Z has a pvalue of 1-0.05 = 0.95. So X when Z = 1.645.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.645 = \frac{X - 600}{100}[/tex]
[tex]X - 600 = 1.645*100[/tex]
[tex]X = 764.50[/tex]
The credit score that defines the upper 5% is 764.50.
(b) Seventy-five percent of the customers will have a credit score higher than what value?
100 - 75 = 25
This the 25th percentile, which is the value of X when Z has a pvalue of 0.25. So it ix X when Z = -0.675.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.675 = \frac{X - 600}{100}[/tex]
[tex]X - 600 = -0.675*100[/tex]
[tex]X = 532.5[/tex]
Seventy-five percent of the customers will have a credit score higher than 532.5.
(c) Within what range would the middle 80 percent of credit scores lie?
50 - 80/2 = 10th percentile to 50 + 80/2 = 90th percentile.
10th percentile
value of 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 - 600}{100}[/tex]
[tex]X - 600 = -1.28*100[/tex]
[tex]X = 472[/tex]
90th percentile
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.28 = \frac{X - 600}{100}[/tex]
[tex]X - 600 = 1.28*100[/tex]
[tex]X = 728[/tex]
Range 472 to 728.
