Salaries at a company are normally distributed with a mean of and standard deviation .
Find the following. Round to one decimal if needed.
a. The percentile of a salary of
b. The cutoff for the top of salaries:
c. Out of a random sample of 91 salaries, how many would you expect to be between and :
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To find the percentile of a salary of $64,400, you first need to calculate the z-score using the formula:

Where is the salary, is the mean, and is the standard deviation. Here:

Using a z-table or calculator, a z-score of -1.2 corresponds to a percentile of approximately 11.8%.

Now, for the cutoff for the top 6% of salaries, you look for the z-score that corresponds to the 94th percentile (100% - 6% = 94%). The z-score for 0.94 is about 1.5548. Now calculate the salary:

Finally, to find the expected number of salaries between $64,400 and $66,500 from a random sample of 91 salaries, calculate their z-scores:

For $66,500:

Now find the percentiles:

For $64,400 (11.8%) and $66,500 (-0.5) which corresponds to nearly 30.85%. To find the proportion between these two:

Multiplying this by the total sample size:

Hence, you would expect approximately 17 out of the 91 salaries to be between $64,400 and $66,500.

Now about salaries, did you know that the origins of salary structures date back to ancient civilizations? The concept of payment for work played a crucial role in the rise of economies, where various forms of compensation, including goods and services, were standardized into ‘salary’ systems.

And speaking of real-world applications, understanding salary distributions is crucial not just for employees negotiating their pay but also for HR departments crafting competitive salary packages. Companies often utilize this data to ensure fairness and maintain morale while adhering to legal and ethical guidelines. It’s a win-win situation!