To solve this problem using the empirical rule (also known as the 68-95-99.7 rule), let's follow these steps:
1. Understand the Problem:
- We are given that the number of daily requests follows a bell-shaped (normal) distribution with a mean ([tex]\(\mu\)[/tex]) of 56 and a standard deviation ([tex]\(\sigma\)[/tex]) of 5.
- We need to find the percentage of days with the number of requests between 51 and 56.
2. Calculate Z-scores:
- The empirical rule states that for a normal distribution, approximately [tex]\(68\%\)[/tex] of the data falls within one standard deviation ([tex]\(\sigma\)[/tex]) of the mean ([tex]\(\mu\)[/tex]), [tex]\(95\%\)[/tex] within two standard deviations, and [tex]\(99.7\%\)[/tex] within three standard deviations.
- To find how many standard deviations 51 and 56 are away from the mean, we calculate the z-scores:
- [tex]\( Z_{\text{lower}} = \frac{51 - 56}{5} = -1 \)[/tex]
- [tex]\( Z_{\text{upper}} = \frac{56 - 56}{5} = 0 \)[/tex]
3. Determine the Area Under the Normal Curve:
- From the empirical rule, we know:
- About [tex]\(34\%\)[/tex] of the data falls between the mean and one standard deviation below the mean (from 56 to 51).
- About [tex]\(34\%\)[/tex] of the data falls between the mean and one standard deviation above the mean (this is not relevant in this case, as our upper bound is the mean itself).
4. Summarize the Findings:
- Since we calculated the z-scores as [tex]\( -1 \)[/tex] and [tex]\( 0 \)[/tex], this implies that the area between [tex]\( \mu - \sigma \)[/tex] and [tex]\( \mu \)[/tex] (which corresponds to [tex]\(51\)[/tex] and [tex]\(56\)[/tex], respectively) is [tex]\( 34.13\% \)[/tex].
Thus, the approximate percentage of lightbulb replacement requests numbering between 51 and 56 is 34.13%.
So the answer is [tex]\( 34.13 \% \)[/tex].
