Respuesta :
Answer:
P(57 < X < 69) = 0.1513
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 = 45, \sigma = 14[/tex]
Find P(57 < X < 69):
This is the pvalue of Z when X = 69 subtracted by the pvalue of Z when X = 57. So
X = 69
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{69 - 45}{14}[/tex]
[tex]Z = 1.71[/tex]
[tex]Z = 1.71[/tex] has a pvalue of 0.9564
X = 57
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{57 - 45}{14}[/tex]
[tex]Z = 0.86[/tex]
[tex]Z = 0.86[/tex] has a pvalue of 0.8051
0.9564 - 0.8051 = 0.1513
P(57 < X < 69) = 0.1513
