Respuesta :
Answer:
[tex]\mu = 11[/tex]
[tex]\sigma = 2.08[/tex]
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.
Middle 85%.
Values of X when Z has a pvalue of 0.5 - 0.85/2 = 0.075 to 0.5 + 0.85/2 = 0.925
Above the interval (8,14)
This means that when Z has a pvalue of 0.075, X = 8. So when [tex]Z = -1.44, X = 8[/tex]. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.44 = \frac{8 - \mu}{\sigma}[/tex]
[tex]8 - \mu = -1.44\sigma[/tex]
[tex]\mu = 8 + 1.44\sigma[/tex]
Also, when X = 14, Z has a pvalue of 0.925, so when [tex]X = 8, Z = 1.44[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.44 = \frac{14 - \mu}{\sigma}[/tex]
[tex]14 - \mu = 1.44\sigma[/tex]
[tex]1.44\sigma = 14 - \mu[/tex]
Replacing in the first equation
[tex]\mu = 8 + 1.44\sigma[/tex]
[tex]\mu = 8 + 14 - \mu[/tex]
[tex]2\mu = 22[/tex]
[tex]\mu = \frac{22}{2}[/tex]
[tex]\mu = 11[/tex]
Standard deviation:
[tex]1.44\sigma = 14 - \mu[/tex]
[tex]1.44\sigma = 14 - 11[/tex]
[tex]\sigma = \frac{3}{1.44}[/tex]
[tex]\sigma = 2.08[/tex]
