Assume the data set described is normally distributed with the given mean and standard deviation, and with *n* total values. Find the approximate number of data values that will fall in the given range.

Mean =360

Standard deviation =19

*n*=540

Range: 303 to 417

In this case, we expect about  data values to fall between 303 and 417.


To solve this problem, we'll use the properties of the normal distribution, specifically the empirical rule (68-95-99.7 rule) and the z-scores.

### Step-by-Step Solution:

1. **Convert the range to z-scores:**

   The formula for the z-score is:

   \[
   z = \frac{x - \mu}{\sigma}
   \]

   Where:
   - \( x \) is the data value.
   - \( \mu \) is the mean.
   - \( \sigma \) is the standard deviation.

   We'll calculate the z-scores for the values 303 and 417.

   For 303:
   \[
   z = \frac{303 - 360}{19} = \frac{-57}{19} = -3.00
   \]

   For 417:
   \[
   z = \frac{417 - 360}{19} = \frac{57}{19} = 3.00
   \]

2. **Find the probability associated with these z-scores:**

   Using the standard normal distribution table (or a calculator):
   - The probability for \( z = -3.00 \) is approximately 0.0013.
   - The probability for \( z = 3.00 \) is approximately 0.9987.

   So, the proportion of data values between 303 and 417 is:

   \[
   P(303 \leq X \leq 417) = 0.9987 - 0.0013 = 0.9974
   \]

   This means about 99.74% of the data falls between 303 and 417.

3. **Multiply by the total number of values:**

   To find the approximate number of data values:

   \[
   \text{Number of data values} = 0.9974 \times 540 = 538.596 \approx 539
   \]

### Conclusion:
Approximately **539** data values will fall between 303 and 417.

Would you like a more detailed breakdown or have any questions?

Here are 5 related questions:
1. How do you calculate the z-score for any value in a normal distribution?
2. What is the empirical rule (68-95-99.7 rule) for normal distributions?
3. How does changing the mean affect the distribution of data values?
4. What are the typical z-scores for 95% of data in a normal distribution?
5. How would you calculate the probability for a z-score that is not in a standard table?

**Tip:** Always verify that the data set follows a normal distribution before applying z-scores.

Assume the data set described is normally distributed with the given mean and standard deviation, and with *n* total values. Find the approximate number of data values that will fall in the given range. Mean = 360, Standard deviation = 19, *n* = 540, Range: 303 to 417.

Learn how to calculate the approximate number of data values falling between 303 and 417 for a normally distributed dataset with mean 360 and standard deviation 19. This solution involves the use of z-scores and the empirical rule. Suitable for students in Grades 10-12.

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