suppose a normally distributed set of data with 5500 observations has a mean of 197 and a standard deviation of 17 use the 68-95-99.7 rule to determine the number of observations in the data set expected to above a value of 163 round your answer to the nearest whole value

The 68-95-99.7 rule, also known as the empirical rule, tells us how data in a normal distribution is distributed within standard deviations of the mean. Here's a breakdown of how to apply this rule:

1. **Given Data**:
   - Mean (\(\mu\)) = 197
   - Standard deviation (\(\sigma\)) = 17
   - Total observations = 5500
   - We are asked to find the number of observations expected to be above 163.

2. **Calculate the z-score** for the value 163:

   \[
   z = \frac{x - \mu}{\sigma} = \frac{163 - 197}{17} = \frac{-34}{17} = -2
   \]

   So, the value 163 is 2 standard deviations below the mean.

3. **Using the empirical rule**:
   - Approximately **95%** of the data falls within 2 standard deviations of the mean (from \(\mu - 2\sigma\) to \(\mu + 2\sigma\)).
   - Therefore, the remaining 5% of the data is outside this range.
     - Since the normal distribution is symmetric, **2.5%** of the data lies below 2 standard deviations below the mean.
     - **2.5%** of the data lies above 2 standard deviations above the mean.

4. **Finding the percentage of data above 163**:
   - If 2.5% of the data is below 163, then the remaining **97.5%** of the data lies above 163.

5. **Number of observations above 163**:
   \[
   0.975 \times 5500 = 5362.5
   \]

   Rounding to the nearest whole number, the expected number of observations above 163 is **5363**.

Would you like a further breakdown of any steps?

### Related Questions:
1. What is the z-score of a value greater than 197?
2. How would the empirical rule apply if the standard deviation was 20 instead of 17?
3. What percentage of data would lie above 2 standard deviations?
4. How many observations would be expected below a value of 163?
5. How can the 68-95-99.7 rule be used to find values above or below 1 standard deviation from the mean?

**Tip**: Z-scores help to standardize values from different normal distributions to easily compare them.

Suppose a normally distributed set of data with 5500 observations has a mean of 197 and a standard deviation of 17. Use the 68-95-99.7 rule to determine the number of observations in the data set expected to be above a value of 163. Round your answer to the nearest whole value.

Use the 68-95-99.7 rule (Empirical Rule) to find how many observations from a normally distributed data set with 5500 points, a mean of 197, and a standard deviation of 17, lie above a value of 163. Includes a step-by-step calculation using the z-score formula and rounding to the nearest whole number. Suitable for high school students in Grades 10-12.

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