The maintenance department at the main campus of a large state university receives daily requests to replace fluorecent lightbulbs. The distribution of the number of daily requests is bell-shaped and has a mean of 58 and a standard deviation of 5. Using the 68-95-99.7 rule, what is the approximate percentage of lightbulb replacement requests numbering between 43 and 58? Do not enter the percent symbol. ans = Submit Question
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![**Educational Website - Maintenance Department Statistics**
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 58 and a standard deviation of 5. Using the 68-95-99.7 rule, what is the approximate percentage of lightbulb replacement requests numbering between 43 and 58?
Do not enter the percent symbol.
Ans = [Input box]
**[Submit Question Button]**
**Explanation of Rule and Distribution:**
The 68-95-99.7 rule, also known as the empirical rule, states that for a bell-shaped distribution (normal distribution):
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
In this case,
- Mean (μ) = 58
- Standard Deviation (σ) = 5
**Calculation:**
To find the percentage of lightbulb replacement requests between 43 and 58:
- Determine how many standard deviations 43 is from the mean:
\( Z = \frac{(X - μ)}{σ} = \frac{(43 - 58)}{5} = -3 \text{(i.e., 3 standard deviations below the mean)}
According to the 68-95-99.7 rule, about 99.7% of values fall within three standard deviations (32 to 88). However, we only need the percentage from 43 to 58:
- Percentage from the mean to 3σ (lower half) : 50%
- Since we're looking for 43 to 58 (one side of the mean):
The percentage approximation for the requested range (23 to 68) is roughly half of 99.7% since we are considering from the mean to -3σ (68/2) = 49.85%.
Therefore, the approximate percentage of lightbulb requests between 43 and 58 is approximately 49.85%.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e82b789-902b-4bed-91c7-23ced64abede%2Fabb1d348-ae67-4a77-84f0-854097d909ea%2Fpxlrbs5_processed.jpeg&w=3840&q=75)
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