The 68-95-99.7 rule (also known as the empirical rule) applies to normally distributed data and helps approximate the percentage of observations within certain standard deviation ranges from the mean. Here's how it applies:

1. Mean (μ) = 159
2. Standard deviation (σ) = 13

According to the empirical rule:

• 68% of the data lies within 1 standard deviation of the mean (between $\mu - 1\sigma$ and $\mu + 1\sigma$),
• 95% of the data lies within 2 standard deviations,
• 99.7% lies within 3 standard deviations.

Step 1: Convert the given values into standard deviations from the mean

We are asked to find the number of observations between 146 and 185.

• Lower bound (146):
  $z = \frac{146 - 159}{13} = \frac{-13}{13} = -1$
  So, 146 is 1 standard deviation below the mean.

• Upper bound (185):
  $z = \frac{185 - 159}{13} = \frac{26}{13} = 2$
  So, 185 is 2 standard deviations above the mean.

Step 2: Determine the percentage of data between -1σ and +2σ

The empirical rule tells us:

• Between -1σ and +1σ (within 1 standard deviation) lies 68% of the data.
• Between -1σ and +2σ includes all of the data between -1σ and +1σ, plus half of the data in the second standard deviation.

From -1σ to +2σ:

• 68% of the data lies between -1σ and +1σ.
• An additional 13.5% (half of the 27% that lies between +1σ and +2σ) falls in the range from +1σ to +2σ.

Thus, the total percentage between -1σ and +2σ is $68\% + 13.5\% = 81.5\%$.

Step 3: Find the number of observations

To find the number of observations within this range, multiply the total percentage by the total number of observations: $\text{Number of observations} = 81.5\% \times 6000 = 0.815 \times 6000 = 4890$

Thus, 4,890 observations are expected to lie between the values of 146 and 185.

Would you like more details on this process or have any further questions?

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Here are 5 related questions to expand your understanding:

1. How does the empirical rule apply to non-normally distributed data?
2. What percentage of data lies between 2 and 3 standard deviations from the mean?
3. How would the number of observations change if the standard deviation were 10 instead of 13?
4. What is the z-score, and how is it used to find probabilities in a normal distribution?
5. How can the empirical rule be used to estimate outliers?

Tip: The empirical rule is a great way to quickly estimate data ranges, but for more precise calculations, especially outside the 68-95-99.7 range, using z-scores and statistical tables is recommended.