Respuesta :
Answer:
a) 8.2962
b) 6.3956
c) 3.845
Step-by-step explanation:
Z-score:
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 question, we have that:
[tex]\mu = 5.6, \sigma = 2.6[/tex]
(a) What response represents the 85th percentile?
This is X when Z has a pvalue of 0.85. So X when Z = 1.037.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.037 = \frac{X - 5.6}{2.6}[/tex]
[tex]X - 5.6 = 2.6*1.037[/tex]
[tex]X = 8.2962[/tex]
(b) What response represents the 62nd percentile?
This is X when Z has a pvalue of 0.62. So X when Z = 0.306.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.306 = \frac{X - 5.6}{2.6}[/tex]
[tex]X - 5.6 = 2.6*0.306[/tex]
[tex]X = 6.3956[/tex]
(c) What response represents the first quartile?
The first quartile is the 100/4 = 25th percentile. So this is X when Z has a pvalue of 0.25, so X when Z = -0.675.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.675 = \frac{X - 5.6}{2.6}[/tex]
[tex]X - 5.6 = 2.6*(-0.675)[/tex]
[tex]X = 3.845[/tex]
