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Steps to find the percentage of values in the distribution between 19 and 29 using the 68-95-99.7 rule.

To determine the percentage of values in a normal distribution between 19 and 29 using the 68-95-99.7 rule, let's follow these steps:

1. **Identify the Mean and Standard Deviation**:
   - Mean \( (\mu) = 19 \)
   - Standard deviation \( (\sigma) = 5 \)

2. **Calculate the Z-Scores for 19 and 29**:
   - Since the mean is 19, a score of 19 is exactly 0 standard deviations from the mean (within the middle of the distribution).
   - The score of 29 is 2 standard deviations above the mean:
     \[
     Z = \frac{29 - 19}{5} = 2
     \]

3. **Apply the 68-95-99.7 Rule**:
   - The 68-95-99.7 rule states:
     - 68% of values fall within 1 standard deviation of the mean.
     - 95% of values fall within 2 standard deviations of the mean.
     - 99.7% of values fall within 3 standard deviations of the mean.
   - Since 29 is 2 standard deviations above the mean, 95% of the values lie between \( \mu - 2\sigma \) and \( \mu + 2\sigma \).

4. **Determine the Percentage Between 19 and 29**:
   - Since 95% of the distribution lies within 2 standard deviations on both sides of the mean, half of that (or 47.5%) will lie between the mean and \( \mu + 2\sigma \).
   - Therefore, the percentage of values between 19 and 29 is **47.5%**.

Would you like more details or have any questions?

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Here are some related questions for further exploration:

1. What percentage of values lie between 14 and 24 in this distribution?
2. How does the 68-95-99.7 rule help in understanding normal distributions?
3. What is the probability of a value falling below 9 in this distribution?
4. How would you calculate the percentage of values between two non-standardized points without the 68-95-99.7 rule?
5. What is the significance of the 99.7% in this rule?

**Tip**: The 68-95-99.7 rule is an excellent tool for estimating probabilities quickly in a normal distribution without calculations.

A normal distribution has a mean of 19 and a standard deviation of 5. Use the 68-95-99.7 rule to find the percentage of values in the distribution between 19 and 29. What is the percentage of values in the distribution between 19 and 29?

Learn how to calculate the percentage of values in a normal distribution with a mean of 19 and a standard deviation of 5 between 19 and 29 using the 68-95-99.7 rule. This problem explores the concepts of normal distribution, standard deviation, and the empirical rule, making it ideal for students in Grades 9-12.

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