To solve this problem using the 68-95-99.7 rule (also known as the empirical rule), we first need to interpret what the rule tells us about a normal distribution:

1. 68% of data falls within 1 standard deviation (σ) of the mean (μ),
2. 95% of data falls within 2 standard deviations of the mean,
3. 99.7% of data falls within 3 standard deviations of the mean.

Step 1: Identify the key components

• Mean ($\mu$) = 159
• Standard deviation ($\sigma$) = 13
• Number of observations = 6000

We are asked to find the percentage of observations between 147 and 185.

Step 2: Determine how many standard deviations these values are from the mean

• 147 is how far below the mean? $\frac{159 - 147}{13} = \frac{12}{13} \approx 0.92 \sigma \quad \text{(approximately 0.92 standard deviations below the mean)}$

• 185 is how far above the mean? $\frac{185 - 159}{13} = \frac{26}{13} \approx 2 \sigma \quad \text{(exactly 2 standard deviations above the mean)}$

Step 3: Use the 68-95-99.7 rule

• From the empirical rule, we know that:
  • Between 1 standard deviation below and above the mean, 68% of the data falls.
  • Between 2 standard deviations below and above the mean, 95% of the data falls.

We need to estimate the percentage of observations that fall between 0.92 standard deviations below the mean and 2 standard deviations above the mean.

• From 0 to 0.92 σ: This is slightly less than 1 standard deviation below the mean. Based on the symmetry of the normal distribution, about 32% of the data lies outside 1 standard deviation (i.e., 16% on each side), so a bit more than 34% lies between the mean and 0.92σ. We can approximate this as 33%.
• From 0.92σ to 2σ: Since 95% of the data lies within 2 standard deviations of the mean, about 47.5% of that data is between the mean and 2σ.

Thus, the percentage of observations between 147 and 185 is roughly: $33\% + 47.5\% = 80.5\%$

Step 4: Calculate the expected number of observations

The number of observations within this range is: $6000 \times 0.805 = 4830$

Conclusion:

Approximately 4830 observations are expected to lie between the values of 147 and 185.

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Do you want more details or have any questions about this solution?

Here are some related questions:

1. What is the probability of a stock price being greater than 185 in this dataset?
2. How would the results change if the standard deviation were 15 instead of 13?
3. How can we calculate the exact percentage using z-scores and a normal distribution table?
4. What if we wanted the percentage of stock prices between 147 and 172 instead?
5. How would the results differ if the number of observations were 10,000 instead of 6,000?

Tip: In a normal distribution, using the z-score method can give more precise results when the 68-95-99.7 rule is not detailed enough for your needs.