Questions: The maintenance department at the main campus of a large state university receives daily requests to replace fluorescent lightbulbs. The distribution of the number of daily requests is bell-shaped and has a mean of 42 and a standard deviation of 4. Using the 68-95-99.7 rule, what is the approximate percentage of lightbulb replacement requests numbering between 42 and 46?

Solution Steps

The 68-95-99.7 rule, also known as the empirical rule, states that for a bell-shaped distribution:

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

Given:

• Mean (\(\mu\)) = 42
• Standard deviation (\(\sigma\)) = 4

The range for one standard deviation from the mean is:

• Lower bound: \(\mu = 42\)
• Upper bound: \(\mu + \sigma = 42 + 4 = 46\)

According to the 68-95-99.7 rule, approximately 68% of the data falls within one standard deviation of the mean. Since the range from 42 to 46 is exactly one standard deviation above the mean, the percentage of requests between 42 and 46 is half of 68%, because the rule covers both sides of the mean.

\[ \text{Percentage} = \frac{68}{2} = 34\% \]

Final Answer