Comparing the relative sizes of data values based on thelr z-scores
Jane and Salma each graduated from college last year and got entry-level positions, Last year, the distributions of entry-level salaries were clearly bell-shaped
for mathematicians and for pharmacists. That year, the entry-level salaries of mathematicians had a population mean of , 500 and a standard deviation of
, and the entry-level salaries of pharmacists had a population mean of and a standard deviation of .
Last year, Jane got an entry-level position as a mathematician with a salary of , and Salma got an entry-level position as a pharmacist with a salary of
.
(a) Find the z-scores of Jane’s entry-level salary as a mathematician and Salma’s entry-level salary as a pharmacist. Round
your answers to two decimal places.
z-score of Jane’s salary:
z-score of Salma’s salary:
(b) Relative to her population, who got the higher salary?
Choose the best answer based on the z-scores of the two salaries.
O Jane
O Salma
O It is unclear who got the higher salary relative to her population
(5)

Ask by Frazier Tran. in the United States

Mar 20,2025

Question

Comparing the relative sizes of data values based on thelr z-scores
Jane and Salma each graduated from college last year and got entry-level positions, Last year, the distributions of entry-level salaries were clearly bell-shaped
for mathematicians and for pharmacists. That year, the entry-level salaries of mathematicians had a population mean of , 500 and a standard deviation of
, and the entry-level salaries of pharmacists had a population mean of and a standard deviation of .
Last year, Jane got an entry-level position as a mathematician with a salary of , and Salma got an entry-level position as a pharmacist with a salary of
.
(a) Find the z-scores of Jane’s entry-level salary as a mathematician and Salma’s entry-level salary as a pharmacist. Round
your answers to two decimal places.
z-score of Jane’s salary:
z-score of Salma’s salary:
(b) Relative to her population, who got the higher salary?
Choose the best answer based on the z-scores of the two salaries.
O Jane
O Salma
O It is unclear who got the higher salary relative to her population
(5)

Ask by Frazier Tran. in the United States

Mar 20,2025

UpStudy AI · Accepted Answer

**Step 1. Compute the z-score of Jane's salary as a mathematician**

The formula for a z-score is given by

$$
z = \frac{X - \mu}{\sigma}
$$

where  
$$ X $$ is the observed value,  
$$ \mu $$ is the population mean, and  
$$ \sigma $$ is the population standard deviation.

For Jane,  
$$ X = \$64,\!488 $$,  
$$ \mu = \$57,\!500 $$, and  
$$ \sigma = \$2950 $$.

So, the z-score is

$$
z = \frac{64,\!488 - 57,\!500}{2950} = \frac{6,\!988}{2950} \approx 2.37.
$$

**Step 2. Compute the z-score of Salma's salary as a pharmacist**

For Salma,  
$$ X = \$60,\!205 $$,  
$$ \mu = \$69,\!000 $$, and  
$$ \sigma = \$2400 $$.

Thus, her z-score is

$$
z = \frac{60,\!205 - 69,\!000}{2400} = \frac{-8,\!795}{2400} \approx -3.66.
$$

**Step 3. Compare the z-scores to determine the relative standing**

The z-score tells us how many standard deviations the value $$ X $$ is away from the mean $$ \mu $$.

- Jane's z-score is $$ 2.37 $$, which means her salary is $$ 2.37 $$ standard deviations above the mathematicians' mean.  
- Salma's z-score is $$ -3.66 $$, which means her salary is $$ 3.66 $$ standard deviations below the pharmacists' mean.

Since a higher z-score indicates a salary that is farther above the mean relative to the population, Jane's salary is relatively much higher compared to her population.

**Answer**

- z-score of Jane's salary: $$2.37$$  
- z-score of Salma's salary: $$-3.66$$

Relative to her population, Jane got the higher salary.
Jane's z-score is 2.37, and Salma's z-score is -3.66. Therefore, Jane received a higher salary relative to her population.

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