Answer :
a) The z-score for a sales associate who makes $36,000 is 1.4
b) The percentage of sales associates with salaries between $26,250 and $38,750 is 84%
c) The percentage of sales associates with salaries between $27,500 and $37,500 is 95.4%
d) The percentage of sales associates with salaries less than $27,500 is 2.3%
e) Yes, the salary of $42000 should be considered as outlier.
Given,
Mean, [tex]\mu[/tex]=32500
Standard deviation, [tex]\sigma[/tex]=2500
a)
[tex]z=\frac{x-\mu}{\sigma}\\\\z=\frac{36000-32500}{2500}=1.40[/tex]
The sales associate who makes 36000 is 1.4 standard deviation above the mean salary.
b)
As per chebyshev's theorem, [tex](1-\frac{1}{k^2})[/tex] of the observations fall within k standard deviation of the mean, i.e. within [tex]\mu\pm k\sigma[/tex]
[tex]z_1=\frac{x_1-\mu}{\sigma}\\\\z_1=\frac{26250-32500}{2500}=-2.50\\\\z_2=\frac{x_2-\mu}{\sigma}\\\\z_2=\frac{38750-32500}{2500}=2.50[/tex]
Hence, we need to find proportion of observations within [tex]\mu\pm2.5\sigma[/tex]
k=2.5
[tex](1-\frac{1}{k^2})=1-\frac{1}{2.5^2}=0.84=84\%[/tex]
Hence, 84% of sales associates have salaries between 26250 and 38750.
c)
As per empirical rule for a bell shaped distribution:
- 68.3% of the observations fall within 1 standard deviation of the mean i.e. within [tex]\mu\pm 1\sigma[/tex]
- 95.4% of the observations fall within 2 standard deviation of the mean i.e. within [tex]\mu\pm 2\sigma[/tex]
- 99.7% of the observations fall within 3 standard deviation of the mean i.e. within [tex]\mu\pm 3\sigma[/tex]
[tex]z_1=\frac{x_1-\mu}{\sigma}\\\\z_1=\frac{27500-32500}{2500}=-2\\\\z_2=\frac{x_2-\mu}{\sigma}\\\\z_2=\frac{37500-32500}{2500}=2[/tex]
Hence, we need to find the proportion of observations within \mu \pm 2\sigma
As per empirical rule, 95.4% of the observations fall within 2 standard deviation of the mean .i.e within [tex]\mu \pm 2\simga[/tex] .Therefore, percentage of sales associates with salaries between 27500 and 37500=95.4%
d)
[tex]z_1=\frac{x_1-\mu}{\sigma}\\\\z_1=\frac{27500-32500}{2500}=-2[/tex]
Percentage of sales associates with salaries less than 27500 will be equal to the area under the normal curve to the left z=-2
As per empirical rule 95.4% of the observations fall within 2 standard deviation of the mean
Therefore, the percentage of sales associates with salaries less than 27500 : [tex]\frac{100-95.4}{2}=2.3\%[/tex]
e)
[tex]z_1=\frac{x_1-\mu}{\sigma}\\\\z_1=\frac{42000-32500}{2500}=3.8[/tex]
A z-value greater than 3 is considered as an outlier.
Hence, the salary of 42000 should be considered as outlier.
To learn more about chebyshev's theorem refer here
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