Respuesta :
Answer:
a) 40.17% probability of a value between 75.0 and 90.0.
b) 35.94% probability of a value 75.0 or less.
c) 20.22% probability of a value between 55.0 and 70.0.
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 = 80, \sigma = 14[/tex]
a. Compute the probability of a value between 75.0 and 90.0.
This is the pvalue of Z when X = 90 subtracted by the pvalue of Z when X = 75.
X = 90
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{90 - 80}{14}[/tex]
[tex]Z = 0.71[/tex]
[tex]Z = 0.71[/tex] has a pvalue of 0.7611
X = 75
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{75 - 80}{14}[/tex]
[tex]Z = -0.36[/tex]
[tex]Z = -0.36[/tex] has a pvalue of 0.3594
0.7611 - 0.3594 = 0.4017
40.17% probability of a value between 75.0 and 90.0.
b. Compute the probability of a value 75.0 or less.
This is the pvalue of Z when X = 75. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{75 - 80}{14}[/tex]
[tex]Z = -0.36[/tex]
[tex]Z = -0.36[/tex] has a pvalue of 0.3594
35.94% probability of a value 75.0 or less.
c. Compute the probability of a value between 55.0 and 70.0.
This is the pvalue of Z when X = 70 subtracted by the pvalue of Z when X = 55.
X = 70
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{70 - 80}{14}[/tex]
[tex]Z = -0.71[/tex]
[tex]Z = -0.71[/tex] has a pvalue of 0.2389
X = 55
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{55 - 80}{14}[/tex]
[tex]Z = -1.79[/tex]
[tex]Z = -1.791[/tex] has a pvalue of 0.0367
0.2389 - 0.0367 = 0.2022
20.22% probability of a value between 55.0 and 70.0.
