Respuesta :
Answer:
a) 22.96% probability that a new college graduate in business will earn a starting salary of at least $65,000
b) 11.12% probability that a new college graduate in health sciences will earn a starting salary of at least $65,000
c) 14.69% probability that a new college graduate in health sciences will earn a starting salary of less than $40,000
d) He would have to earn $88,776.
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:
Health sciences:
[tex]\mu = 51541, \sigma = 11000[/tex]
Business:
[tex]\mu = 53901, \sigma = 15000[/tex]
a. What is the probability that a new college graduate in business will earn a starting salary of at least $65,000?
This is 1 subtracted by the pvalue of Z when X = 65000. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{65000 - 53901}{15000}[/tex]
[tex]Z = 0.74[/tex]
[tex]Z = 0.74[/tex] has a pvalue of 0.7704
1 - 0.7704 = 0.2296
22.96% probability that a new college graduate in business will earn a starting salary of at least $65,000
b. What is the probability that a new college graduate in health sciences will earn a starting salary of at least $65,000?
This is 1 subtracted by the pvalue of Z when X = 65000. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{65000 - 51541}{11000}[/tex]
[tex]Z = 1.22[/tex]
[tex]Z = 1.22[/tex] has a pvalue of 0.8888
1 - 0.8888 = 0.1112
11.12% probability that a new college graduate in health sciences will earn a starting salary of at least $65,000
c. What is the probability that a new college graduate in health sciences will earn a starting salary of less than $40,000?
This is the pvalue of Z when X = 40000. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{40000 - 51541}{11000}[/tex]
[tex]Z = -1.05[/tex]
[tex]Z = -1.05[/tex] has a pvalue of 0.1469
14.69% probability that a new college graduate in health sciences will earn a starting salary of less than $40,000
d. How much would a new college graduate in business have to earn in order to have a starting salary higher than 99% of all starting salaries of new college graduates in the health sciences?
This is X when Z has a pvalue of 0.99. So it is X when Z = 2.325.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]2.325 = \frac{X - 53901}{15000}[/tex]
[tex]X - 53901 = 2.325*15000[/tex]
[tex]X = 88776[/tex]
He would have to earn $88,776.
