A sample of assistant professors on the business faculty at the largest college in Ontario revealed the mean annual income to be $62,000 with a standard deviation of $3,000. Using the Empirical Rule, what proportion of faculty earn more than $56,000 but less than $68,000?
A) At least 50%
B) Approximately 68%
C) At least 75%
D) Approximately 95%
E) Almost all

Question

Quizplus · Accepted Answer

According to the Empirical Rule, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations of the mean, and almost all, or 99.7%, falls within three standard deviations of the mean.

To determine the proportion of faculty earning between $56,000 and $68,000, we need to calculate the z-scores of these values:
z-score for $56,000: (56,000 - 62,000) / 3,000 = -2.00 
z-score for $68,000: (68,000 - 62,000) / 3,000 = 2.00

Using a standard normal distribution table or calculator, we can find that the proportion of data falling between these z-scores is approximately 0.9545. Therefore, the proportion of faculty earning between $56,000 and $68,000 is approximately 95%. Thus, the best choice is D, which states "Approximately 95%."

A Sample of Assistant Professors on the Business Faculty at the Largest