CFA Level 1: Normal Distribution of Incomes

Middle Management Employees' Incomes

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Question

The distribution of the annual incomes of a group of middle management employees approximated a normal distribution with a mean of $37,200 and a standard deviation of $800. About 68 percent of the incomes lie between what two incomes?

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Explanations

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A. B. C. D. E.

A

68/2 = 0.34. The z value for 0.34 is 1. x = u +/- z*sigma. So x = 37200 +/- 1*800. e get x = 38,000 and 36,400.

To determine the range within which 68 percent of the incomes lie, we can use the concept of the empirical rule, also known as the 68-95-99.7 rule. According to this rule, for a normal distribution:

  • Approximately 68 percent of the data falls within one standard deviation of the mean.
  • Approximately 95 percent of the data falls within two standard deviations of the mean.
  • Approximately 99.7 percent of the data falls within three standard deviations of the mean.

In this case, we are given a normal distribution with a mean of $37,200 and a standard deviation of $800. Since we are looking for the range within one standard deviation, we can calculate the lower and upper bounds as follows:

Lower bound = Mean - Standard deviation Upper bound = Mean + Standard deviation

Lower bound = $37,200 - $800 = $36,400 Upper bound = $37,200 + $800 = $38,000

Therefore, approximately 68 percent of the incomes lie between $36,400 and $38,000.

The correct answer is option A: $36,400 and $38,000.

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