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.