Question

you are applying for a job at two companies. company a offers starting salaries with \\( \mu = \\$ 35,000 \\) and \\( \sigma = \\$ 3,000 \\). company b offers starting salaries with \\( \mu = \\$ 35,000 \\) and \\( \sigma = \\$ 7,000 \\). from which company are you more likely to get an offer of \\( \\$ 41,000 \\) or more?

choose the correct answer below.

a. company a, because data values that lie more than two standard deviations from the mean are considered unusual.
b. company b, because data values that lie within one standard deviation from the mean are not considered unusual.
c. no difference, because data values that lie more than three standard deviations from the mean are considered very unusual.

Explanation:

Step1: Calculate the number of standard deviations from the mean for Company A

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\). For Company A, \(\mu = 35000\), \(\sigma=3000\), and \(x = 41000\).

$$z_A=\frac{41000 - 35000}{3000}=\frac{6000}{3000}=2$$

Step2: Calculate the number of standard deviations from the mean for Company B

For Company B, \(\mu = 35000\), \(\sigma = 7000\), and \(x = 41000\).

$$z_B=\frac{41000-35000}{7000}=\frac{6000}{7000}\approx0.86$$

Data values that lie more than two standard deviations from the mean are considered unusual. For Company A, the value of \(41000\) is \(2\) standard deviations from the mean. For Company B, the value of \(41000\) is approximately \(0.86\) standard deviations from the mean. Since \(0.86<2\), getting an offer of \(41000\) or more is more likely from Company B (because a value that is closer to the mean (in terms of standard deviations) is more likely to occur). But let's check the options again.

Option A: Company A, because data values that lie more than two standard deviations from the mean are considered unusual. But we want the company where \(41000\) is less “unusual” (more likely).
Option B: Company B, because data values that lie within one standard deviation from the mean are not considered unusual. \(z_B\approx0.86<1\), so the value of \(41000\) for Company B is within one standard deviation.
Option C: No difference, because data values that lie more than three standard deviations from the mean are considered very unusual. But our calculations show a difference.

Answer:

B. Company B, because data values that lie within one standard deviation from the mean are not considered unusual.