Respuesta :
Answer:
Lower limit: 113.28
Upper limit: 126.72
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 = 120, \sigma = 8[/tex]
Middle 60%
So it goes from X when Z has a pvalue of 0.5 - 0.6/2 = 0.2 to X when Z has a pvalue of 0.5 + 0.6/2 = 0.8
Lower limit
X when Z has a pvalue of 0.20. So X when [tex]Z = -0.84[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.84 = \frac{X - 120}{8}[/tex]
[tex]X - 120 = -0.84*8[/tex]
[tex]X = 113.28[/tex]
Upper limit
X when Z has a pvalue of 0.80. So X when [tex]Z = 0.84[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.84 = \frac{X - 120}{8}[/tex]
[tex]X - 120 = 0.84*8[/tex]
[tex]X = 126.72[/tex]
