Question

1. the table shows the starting salary per year (in thousands of dollars) rounded to the nearest thousand, for a random sample of graduates majoring in business and graduates majoring in communications at a state university.

group name	( n )	mean	sd	min	( q_1 )	med	( q_3 )	max
communications	15	56.467	6.632	44	52	57	63	65

a. who has a higher starting salary, a business graduate with a ( z )-score of -1.2 or a communications graduate with a ( z )-score of 0.7? explain.

b. lindsey is one of the business graduates who earned a starting salary of $72,000. luke is a communications graduate with a starting salary of $65,000. which of these two graduates has the higher starting salary relative to graduates in their respective majors? explain.

c. for which group of graduates, business or communications, would it be more surprising to receive a starting salary of $60,000? explain.

Explanation:

Part (a)

Step 1: Recall the z - score formula

The z - score formula is $z=\frac{x - \mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. We can solve for $x$ using $x=\mu + z\sigma$.

Step 2: Calculate the salary for the business graduate

For business graduates: $\mu = 69.929$, $\sigma = 5.954$, $z=- 1.2$
$x_{business}=69.929+(-1.2)\times5.954=69.929 - 7.1448 = 62.7842$ (in thousands of dollars)

Step 3: Calculate the salary for the communications graduate

For communications graduates: $\mu = 56.467$, $\sigma = 6.632$, $z = 0.7$
$x_{communications}=56.467+0.7\times6.632=56.467 + 4.6424=61.1094$ (in thousands of dollars)

Step 4: Compare the two salaries

Since $62.7842>61.1094$, the business graduate with a z - score of - 1.2 has a higher starting salary.

Step 1: Recall the z - score formula

We use the z - score formula $z=\frac{x - \mu}{\sigma}$ to find the z - scores for Lindsey and Luke.

Step 2: Calculate Lindsey's z - score (business graduate)

Lindsey's salary $x = 72$ (in thousands of dollars), $\mu_{business}=69.929$, $\sigma_{business}=5.954$
$z_{Lindsey}=\frac{72 - 69.929}{5.954}=\frac{2.071}{5.954}\approx0.348$

Step 3: Calculate Luke's z - score (communications graduate)

Luke's salary $x = 65$ (in thousands of dollars), $\mu_{communications}=56.467$, $\sigma_{communications}=6.632$
$z_{Luke}=\frac{65 - 56.467}{6.632}=\frac{8.533}{6.632}\approx1.287$

Step 4: Compare the z - scores

Since $|z_{Luke}|>|z_{Lindsey}|$, Luke's salary is further from the mean of his major's distribution. So Luke has a higher starting salary relative to graduates in their respective majors.

Step 1: Recall the z - score formula

We use the z - score formula $z=\frac{x - \mu}{\sigma}$ to find the z - scores for a salary of $x = 60$ (in thousands of dollars) for both business and communications graduates.

Step 2: Calculate the z - score for business graduates

For business: $\mu = 69.929$, $\sigma = 5.954$
$z_{business}=\frac{60 - 69.929}{5.954}=\frac{- 9.929}{5.954}\approx - 1.668$

Step 3: Calculate the z - score for communications graduates

For communications: $\mu = 56.467$, $\sigma = 6.632$
$z_{communications}=\frac{60 - 56.467}{6.632}=\frac{3.533}{6.632}\approx0.533$

Step 4: Compare the z - scores

The magnitude of the z - score for business graduates ($| - 1.668|=1.668$) is larger than the magnitude of the z - score for communications graduates ($|0.533| = 0.533$). A larger magnitude of z - score means the value is more unusual (farther from the mean). So it would be more surprising to receive a starting salary of $60,000 for business graduates.

Answer:

The business graduate with a z - score of - 1.2 has a higher starting salary.

Part (b)