Respuesta :
Answer:
a) 30.85% of people earn less than $40,000
b) 37.21% of people earn between $45,000 and $65,000.
c) 15.87% of people earn more than $70,000
Step-by-step explanation:
Problems of normally distributed 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 = 50000, \sigma = 20000[/tex]
a.What percent of people earn 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 - 50000}{20000}[/tex]
[tex]Z = -0.5[/tex]
[tex]Z = -0.5[/tex] has a pvalue of 0.3085.
30.85% of people earn less than $40,000
b.What percent of people earn between $45,000 and $65,000?
This is the pvalue of Z when X = 65000 subtracted by the pvalue of Z when X = 45000. So
X = 65000
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{65000 - 50000}{20000}[/tex]
[tex]Z = 0.75[/tex]
[tex]Z = 0.75[/tex] has a pvalue of 0.7734.
X = 45000
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{45000 - 50000}{20000}[/tex]
[tex]Z = -0.25[/tex]
[tex]Z = -0.25[/tex] has a pvalue of 0.4013.
0.7734 - 0.4013 = 0.3721
37.21% of people earn between $45,000 and $65,000.
c.What percent of people earn more than $70,000?
This is 1 subtracted by the pvalue of Z when X = 70000. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{70000 - 50000}{20000}[/tex]
[tex]Z = 1[/tex]
[tex]Z = 1[/tex] has a pvalue of 0.8413.
1 - 0.8413 = 0.1587
15.87% of people earn more than $70,000
