Respuesta :
Answer:
There is a 16.62% probability that it will take between 72 and 77 minutes to complete the test.
Step-by-step explanation:
Problems of normally distributed samples can be 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]
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.
The length of time needed to complete a certain test is normally distributed with mean 74 minutes and standard deviation 12 minutes, so [tex]\mu = 74, \sigma = 12[/tex].
Find the probability that it will take between 72 and 77 minutes to complete the test.
We have to subtract the pvalue of Z when X = 77 by the pvalue of Z when X = 72.
So
X = 77
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{77 - 74}{12}[/tex]
[tex]Z = 0.25[/tex]
[tex]Z = 0.25[/tex] has a pvalue of 0.5987.
X = 72
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{72 - 74}{12}[/tex]
[tex]Z = -0.17[/tex]
[tex]Z = -0.17[/tex] has a pvalue of 0.4325.
So, there is a 0.5987 - 0.4325 = 0.1662 = 16.62% probability that it will take between 72 and 77 minutes to complete the test.
