To find the percentage of values in the distribution between 18 and 28 using the 68-95-99.7 rule (also known as the empirical rule), follow these steps:

1. Identify the mean and standard deviation:

   • Given: Mean ($\mu$) = 18
   • Standard deviation ($\sigma$) = 5

2. Calculate the Z-score for 28:

   • To find how many standard deviations 28 is from the mean, calculate $\frac{28 - 18}{5} = 2$.
   • Therefore, 28 is 2 standard deviations above the mean.

3. Apply the 68-95-99.7 Rule:

   • The rule states:
     • 68% of the data lies within 1 standard deviation of the mean.
     • 95% of the data lies within 2 standard deviations of the mean.
     • 99.7% of the data lies within 3 standard deviations of the mean.

4. Determine the range percentage:

   • Since 28 is 2 standard deviations above the mean (which is 18), we look at the percentage within 2 standard deviations of the mean.
   • The rule tells us that 95% of the data lies within 2 standard deviations of the mean, meaning 95% of the values are between $18 - 2 \times 5 = 8$ and $18 + 2 \times 5 = 28$.

5. Conclude the answer:

   • The percentage of values in the distribution between 18 and 28 is 47.5%, as this is half of the 95% (the right half, from the mean to 2 standard deviations above it).

Would you like further details or have any questions?

Here are 5 related questions that expand on this topic:

1. What percentage of values would lie between 8 and 18 in this distribution?
2. How does the 68-95-99.7 rule apply to values beyond 3 standard deviations?
3. What is the probability of finding a value greater than 28 in this distribution?
4. How would the answer change if the standard deviation were different?
5. How do we interpret the area under the curve in a normal distribution?

Tip: The 68-95-99.7 rule is a quick way to estimate probabilities in a normal distribution without using detailed calculations or tables.