Question

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	q₁	med	q₃	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 the formula $x=\mu+z\sigma$.

Step 2: Calculate the salary for the business graduate

For the business graduate: $\mu = 69.929$ (in thousands of dollars), $\sigma = 5.954$, and $z=- 1.2$.
Using $x=\mu+z\sigma$, we have $x_{business}=69.929+(-1.2)\times5.954$.
First, calculate $(-1.2)\times5.954=-7.1448$.
Then, $x_{business}=69.929 - 7.1448 = 62.7842$ (in thousands of dollars).

Step 3: Calculate the salary for the communications graduate

For the communications graduate: $\mu = 56.467$ (in thousands of dollars), $\sigma = 6.632$, and $z = 0.7$.
Using $x=\mu+z\sigma$, we have $x_{communications}=56.467+0.7\times6.632$.
First, calculate $0.7\times6.632 = 4.6424$.
Then, $x_{communications}=56.467 + 4.6424=61.1094$ (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, as the z - score measures the number of standard deviations a data point is from the mean. A higher z - score (in terms of magnitude or value) indicates that the data point is relatively further from the mean in the positive direction.

Step 2: Calculate Lindsey's z - score

Lindsey is a business graduate with $x = 72$ (in thousands of dollars), $\mu = 69.929$, and $\sigma = 5.954$.
$z_{Lindsey}=\frac{72 - 69.929}{5.954}=\frac{2.071}{5.954}\approx0.348$.

Step 3: Calculate Luke's z - score

Luke is a communications graduate with $x = 65$ (in thousands of dollars), $\mu = 56.467$, and $\sigma = 6.632$.
$z_{Luke}=\frac{65 - 56.467}{6.632}=\frac{8.533}{6.632}\approx1.287$.

Step 4: Compare the z - scores

Since $1.287>0.348$, Luke's salary is further above the mean of his major's salary 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. A z - score that is further from 0 (in either positive or negative direction) indicates that the value is more unusual (surprising) in the distribution.

Step 2: Calculate the z - score for business graduates

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

Step 3: Calculate the z - score for communications graduates

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

Step 4: Compare the z - scores

The magnitude of the z - score for business graduates ($| - 1.668|=1.668$) is greater than the magnitude of the z - score for communications graduates ($|0.533| = 0.533$). A more extreme z - score (further from 0) means the value is more surprising in the distribution. So it would be more surprising for a business graduate to receive a starting salary of $60,000$.

Answer:

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

Part (b)