Answer :
Answer:
4.56% of salaries are less than $68,000 or more than $92,000. A percentage lower than 5% is unusual, so it is unusual to find a professor earning less than $68,000 or more than $92,000
Step-by-step explanation:
Problems of normally distributed(bell-shaped) 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 = 80000, \sigma = 6000[/tex]
What can be said about the percentage of salaries that are less than $68,000 or more than $92,000?
Less than 68,000
pvalue of Z when X = 68000
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{68000 - 80000}{6000}[/tex]
[tex]Z = -2[/tex]
[tex]Z = -2[/tex] has a pvalue of 0.0228
2.28% of salaries are less than 68,000.
More than 92,000
1 subtracted by the pvalue of Z when X = 92000
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{68000 - 80000}{6000}[/tex]
[tex]Z = 2[/tex]
[tex]Z = 2[/tex] has a pvalue of 0.9772
1 - 0.9772 = 0.0228
2.28% of salaries are more than 92,000.
Less than 68,000 or more than 92,000:
2*2.28 = 4.56% of salaries are less than $68,000 or more than $92,000. A percentage lower than 5% is unusual, so it is unusual to find a professor earning less than $68,000 or more than $92,000