The distribution of the number of daily requests is bell-shaped and has a mean of 39 and a standard deviation of 11. Using the empirical rule, what is the approximate percentage of lightbulb replacement requests numbering between 17 and 39?

Do not enter the percent symbol.
ans = 
 %

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

1. About 68% of the data lies within 1 standard deviation of the mean.
2. About 95% of the data lies within 2 standard deviations of the mean.
3. About 99.7% of the data lies within 3 standard deviations of the mean.

Given:
- Mean (\(\mu\)) = 39
- Standard deviation (\(\sigma\)) = 11
- Range: Between 17 and 39

Now, let's find how many standard deviations away from the mean the value of 17 is:
\[
z = \frac{39 - 17}{11} = 2
\]
So, 17 is 2 standard deviations below the mean.

According to the empirical rule, about 95% of the data lies within 2 standard deviations of the mean. Since the range 17 to 39 represents half of this (from the mean to 2 standard deviations below it), the percentage of requests in this range is:
\[
\frac{95\%}{2} = 47.5\%
\]

Thus, the approximate percentage is \( \boxed{47.5} \).

Would you like more details, or do you have any other questions?

Here are five related questions:
1. What percentage of requests fall between 28 and 50 using the empirical rule?
2. How can the empirical rule be applied to a dataset with different mean and standard deviation?
3. What is the percentage of requests falling below 17 using the empirical rule?
4. What would the range be for 68% of the requests?
5. How does the empirical rule differ from exact probabilities in a normal distribution?

**Tip:** The empirical rule is most accurate for perfectly normal distributions but gives a good approximation for many real-world data sets.

The distribution of the number of daily requests is bell-shaped and has a mean of 39 and a standard deviation of 11. Using the empirical rule, what is the approximate percentage of lightbulb replacement requests numbering between 17 and 39?

Using the empirical rule, this problem demonstrates how to find the percentage of daily lightbulb replacement requests that fall between 17 and 39, with a mean of 39 and a standard deviation of 11. Learn how to apply the 68-95-99.7 rule to solve this normal distribution problem, suitable for students in Grades 9-12.

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