The 68-95-99.7 rule, also known as the empirical rule, is useful for understanding how data in a normal distribution is spread relative to the mean. The rule states:

• 68% of the data falls within 1 standard deviation of the mean.
• 95% of the data falls within 2 standard deviations of the mean.
• 99.7% of the data falls within 3 standard deviations of the mean.

Problem Setup:

• Mean ($\mu$) = 180
• Standard deviation ($\sigma$) = 12
• Number of observations = 3700
• We need to find how many observations lie between 168 and 204.

1. Calculate the distance from the mean:

   • 168 is 1 standard deviation below the mean: $168 = 180 - 12$
   • 204 is 2 standard deviations above the mean: $204 = 180 + 12 \times 2$

2. Apply the empirical rule:

   • The interval from 1 standard deviation below ($168$) to 1 standard deviation above ($192$) contains 68% of the data.
   • The interval from 2 standard deviations below ($156$) to 2 standard deviations above ($204$) contains 95% of the data.

Therefore, the percentage of observations between 168 and 204 includes:

• The 68% from the range $168$ to $192$.
• Half of the additional 27% (since 95% - 68% = 27%) corresponds to the range $192$ to $204$.

So, the percentage of observations between 168 and 204 is: $68\% + \frac{27\%}{2} = 68\% + 13.5\% = 81.5\%$

3. Calculate the expected number of observations: $81.5\% \times 3700 = 0.815 \times 3700 = 3015.5$

Thus, the expected number of observations between 168 and 204 is approximately 3016 observations.

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Would you like further details or have any questions?

5 Relative Questions:

1. What would be the expected number of observations between 168 and 192?
2. How many observations are expected to fall below 168?
3. What is the number of observations expected above 204?
4. How would the result change if the standard deviation was increased to 15?
5. How does the empirical rule help in understanding real-world data distributions?

Tip:

The empirical rule is only an approximation, so it works best for data that is symmetrically and normally distributed. For more accurate results, you can use z-scores and standard normal tables.