Question

A survey reported that the mean starting salary for college graduates after a three-year program was $33440. Assume that the distribution of starting salaries follows the normal distribution with a standard deviation of $3930. What percentage of the graduates have starting salaries: (Round z-score computation to 2 decimal places and the final answers to 4 decimal places.) a. Between $30000 and $39300 ? Probability b. More than $43000 ? Probability c. Between $39300 and $43000 ? Probability

Answer 1

a) 74.1331%

b) 0.7496%

c) 6.0472%

Explanation:

You want the percentages of graduates earning starting salaries in the given ranges, if starting salaries are normally distributed with a mean of $33440 and a standard deviation of $3930.

• $30,000–39,300
• > $43,000
• $39,300–43,000

Z-score

The z-score of the starting salaries is found by dividing the difference from the mean by the standard deviation. For example, the z-score of $30,000 is ...

(30,000 -33440)/3930 ≈ -0.88 . . . . . z-score for 30,000

Percentage

The fraction in a given range is the difference of CDF values for the z-scores at the ends of the range. The CDF values can be found from a table lookup or using a suitable calculator (see attached).

a) between $30,000 and 39,300 — 74.1331%

b) more than $43,000 — 0.7496%

c) between $39,300 and $43,000 — 6.0472%

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Additional comment

The CDF values here are reported to 11 decimal places. We know that using -12 and +12 as representatives for -∞ and +∞ will not affect the CDF value in the first 11 decimal places, so that is why we use those values. (The calculator manual recommends ±1E999.)

In the normal course of computation, we only round numbers when they're reported at the end. There is nothing in the problem statement that requires reporting of z-scores, so we saw no need to round them. The final percentage values you get will be slightly different if you round the z-scores before using them in the CDF function.

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