Respuesta :
Using the normal distribution, it is found that:
a) The minimum percentage of commute times within 2 standard deviations of the mean is of 95.44%.
b) The minimum percentage of commute times within 2.5 standard deviations of the mean is of 98.76%.
c) The minimum percentage of commute times within 3 standard deviations of the mean is of 99.74%.
Normal Probability Distribution
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It 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, which is the percentile of X.
In this problem:
- The mean is of [tex]\mu = 27.8[/tex].
- The standard deviation is of [tex]\sigma = 6.6[/tex].
Item a:
It asks the minimum percentage of commute times within 2 standard deviations of the mean. We consider that:
- Z = 2 has a p-value of 0.9772.
- Z = -2 has a p-value of 0.0228.
0.9772 - 0.0228 = 0.9544 = 95.44%.
The minimum percentage of commute times within 2 standard deviations of the mean is of 95.44%.
Item b:
Within 2.5 standard deviations, hence:
- Z = 2.5 has a p-value of 0.9938.
- Z = -2.5 has a p-value of 0.0062.
0.9938 - 0.0062 = 0.9876 = 98.76%.
The minimum percentage of commute times within 2.5 standard deviations of the mean is of 98.76%.
Item c:
It asks the minimum percentage between 8 and 47.6 minutes, which is within 3 standard deviations of the mean, hence:
- Z = 3 has a p-value of 0.9987.
- Z = -3 has a p-value of 0.0013.
0.9987 - 0.0013 = 0.9974.
The minimum percentage of commute times within 3 standard deviations of the mean is of 99.74%.
More can be learned about the normal distribution at https://brainly.com/question/24663213
