Respuesta :
Answer:
69.15% probability the project will be finished in 62 weeks or less.
93.32% probability the project will be finished in 66 weeks or less.
10.56% probability the project will take longer than 65 weeks.
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]
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 = 60, \sigma = 4[/tex]
What is the probability the project will be finished in 62 weeks or less?
This is the pvalue of Z when X = 62.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{62 - 60}{4}[/tex]
[tex]Z = 0.5[/tex]
[tex]Z = 0.5[/tex] has a pvalue of 0.6915.
So there is a 69.15% probability the project will be finished in 62 weeks or less.
What is the probability the project will be finished in 66 weeks or less?
This is the pvalue of Z when X = 66.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{66 - 60}{4}[/tex]
[tex]Z = 1.5[/tex]
[tex]Z = 1.5[/tex] has a pvalue of 0.9332.
So there is a 93.32% probability the project will be finished in 66 weeks or less.
What is the probability the project will take longer than 65 weeks?
This is 1 subtracted by the pvalue of Z when X = 65.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{65 - 60}{4}[/tex]
[tex]Z = 1.25[/tex]
[tex]Z = 1.25[/tex] has a pvalue of 0.8944.
So there is a 1-0.8944 = 0.1056 = 10.56% probability the project will take longer than 65 weeks.
