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 52 and a standard deviation of 10. Using the 68-95-99.7 rule, what is the approximate percentage of lightbulb replacement requests numbering between 52 and 82?

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ans =
%

The 68-95-99.7 rule (also known as the empirical rule) tells us that for a normal distribution:

- **68%** of the data falls within 1 standard deviation of the mean.
- **95%** of the data falls within 2 standard deviations of the mean.
- **99.7%** of the data falls within 3 standard deviations of the mean.

Given:
- Mean = 52
- Standard deviation = 10

We are asked to find the percentage of lightbulb replacement requests between 52 and 82.

First, calculate how many standard deviations 82 is from the mean:
\[
\frac{82 - 52}{10} = 3
\]
82 is 3 standard deviations above the mean.

According to the 68-95-99.7 rule, the percentage of values between the mean and 3 standard deviations above the mean is half of the 99.7% (since we're only considering the right side of the mean):
\[
\frac{99.7}{2} = 49.85
\]

Therefore, the approximate percentage of requests between 52 and 82 is **49.85%**.

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. What percentage of lightbulb replacement requests fall between 42 and 62?
2. What percentage of requests are below 42?
3. How would the rule apply for requests between 32 and 72?
4. How do you calculate percentages for non-standard deviations like 1.5 or 2.5?
5. What would be the approximate number of requests above 82?

**Tip**: When applying the empirical rule, always remember that percentages beyond 3 standard deviations are rare events (approximately 0.3%).

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 52 and a standard deviation of 10. Using the 68-95-99.7 rule, what is the approximate percentage of lightbulb replacement requests numbering between 52 and 82?

Calculate the percentage of daily lightbulb replacement requests between 52 and 82 using the empirical rule. This problem involves understanding the mean, standard deviation, and the 68-95-99.7 rule in a normal distribution. Suitable for high school students in Grades 9-12.

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