Charlie and Miguel each graduated from college last year and got entry-level positions. Last year, the distributions of entry-level salaries were clearly bell-shaped for nurses and for software engineers. That year, the entry-level salaries of nurses had a population mean of $71,000 and a standard deviation of $2700, and the entry-level salaries of software engineers had a population mean of $63,000 and a standard deviation of $2200.Last year, Charlie got an entry-level position as a nurse with a salary of $78,633, and Miguel got an entry-level position as a software engineer with a salary of $59,416.(a)Find the z-scores of Charlie's entry-level salary as a nurse and Miguel's entry-level salary as a software engineer. Round your answers to two decimal places.z-score of Charlie's salary: z-score of Miguel's salary: (b)Relative to his population, who got the higher salary?Choose the best answer based on the z-scores of the two salaries.CharlieMiguelIt is unclear who got the higher salary relative to his population
Question
Charlie and Miguel each graduated from college last year and got entry-level positions. Last year, the distributions of entry-level salaries were clearly bell-shaped for nurses and for software engineers. That year, the entry-level salaries of nurses had a population mean of 2700, and the entry-level salaries of software engineers had a population mean of 2200.Last year, Charlie got an entry-level position as a nurse with a salary of 59,416.(a)Find the z-scores of Charlie's entry-level salary as a nurse and Miguel's entry-level salary as a software engineer. Round your answers to two decimal places.z-score of Charlie's salary: z-score of Miguel's salary: (b)Relative to his population, who got the higher salary?Choose the best answer based on the z-scores of the two salaries.CharlieMiguelIt is unclear who got the higher salary relative to his population
Solution
(a) The z-score is calculated by subtracting the mean from the observed value and then dividing by the standard deviation.
For Charlie's salary: z = (78,633 - 71,000) / 2,700 = 2.83
For Miguel's salary: z = (59,416 - 63,000) / 2,200 = -1.63
So, the z-score of Charlie's salary is 2.83 and the z-score of Miguel's salary is -1.63.
(b) The z-score tells us how many standard deviations an observation is from the mean. A positive z-score indicates the observation is above the mean, while a negative z-score indicates it is below the mean. The larger the absolute value of the z-score, the further away the observation is from the mean.
In this case, Charlie's z-score is larger and positive, meaning his salary is further above the average salary for nurses than Miguel's salary is below the average salary for software engineers. Therefore, relative to his population, Charlie got the higher salary.
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