Questions: The year Teresa graduated college, the starting salaries of all entry-level software engineers in the United States had a mean of 92,000 and a standard deviation of 1080. That year, Teresa got a job as an entry-level software engineer working for a firm in the United States. Her starting salary was 89,000. (a) Find the z-score of Teresa's starting salary relative to the starting salaries among all the entry-level software engineers in the United States for the year Teresa graduated. Round your answer to two decimal places. [ z=square ] (b) Fill in the blanks to interpret the z-score of Teresa's starting salary. Make sure to express your answer in terms of a positive number of standard deviations. Teresa's starting salary was standard deviations (Choose one) V the mean starting salary of all entry-level software engineers in the United States.

The year Teresa graduated college, the starting salaries of all entry-level software engineers in the United States had a mean of 92,000 and a standard deviation of 1080. That year, Teresa got a job as an entry-level software engineer working for a firm in the United States. Her starting salary was 89,000. (a) Find the z-score of Teresa's starting salary relative to the starting salaries among all the entry-level software engineers in the United States for the year Teresa graduated. Round your answer to two decimal places. [ z=square ] (b) Fill in the blanks to interpret the z-score of Teresa's starting salary. Make sure to express your answer in terms of a positive number of standard deviations. Teresa's starting salary was standard deviations (Choose one) V the mean starting salary of all entry-level software engineers in the United States.
Transcript text: The year Teresa graduated college, the starting salaries of all entry-level software engineers in the United States had a mean of $\$ 92,000$ and a standard deviation of $\$ 1080$. That year, Teresa got a job as an entry-level software engineer working for a firm in the United States. Her starting salary was $\$ 89,000$. (a) Find the $z$-score of Teresa's starting salary relative to the starting salaries among all the entry-level software engineers in the United States for the year Teresa graduated. Round your answer to two decimal places. \[ z=\square \] (b) Fill in the blanks to interpret the $z$-score of Teresa's starting salary. Make sure to express your answer in terms of a positive number of standard deviations. Teresa's starting salary was standard deviations (Choose one) $\mathbf{V}$ the mean starting salary of all entrylevel software engineers in the United States.
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Solution

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Solution Steps

To find the $z$-score of Teresa's starting salary, we use the formula for the $z$-score: \( z = \frac{X - \mu}{\sigma} \), where \( X \) is Teresa's salary, \( \mu \) is the mean salary, and \( \sigma \) is the standard deviation. For part (b), interpret the $z$-score by determining how many standard deviations Teresa's salary is from the mean and whether it is above or below the mean.

Step 1: Calculate the $z$-score

To find the $z$-score of Teresa's starting salary, we use the formula:

\[ z = \frac{X - \mu}{\sigma} \]

where:

  • \( X = 89000 \) (Teresa's salary),
  • \( \mu = 92000 \) (mean salary),
  • \( \sigma = 1080 \) (standard deviation).

Substituting the values, we get:

\[ z = \frac{89000 - 92000}{1080} = \frac{-3000}{1080} \approx -2.778 \]

Step 2: Interpret the $z$-score

The $z$-score of \(-2.778\) indicates that Teresa's starting salary is approximately \(2.778\) standard deviations below the mean starting salary of all entry-level software engineers in the United States.

Final Answer

(a) The $z$-score of Teresa's starting salary is \(\boxed{-2.78}\).

(b) Teresa's starting salary was \(\boxed{2.78}\) standard deviations below the mean starting salary of all entry-level software engineers in the United States.

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