Suppose a normally distributed set of stock prices with 6000 observations has a mean of 13 use the 68 95 99.7 rule to determine the number of observations in the data set expected to be between the values of 146 and 185

The 68-95-99.7 rule, also known as the empirical rule, applies to normal distributions and gives approximate percentages of data that fall within certain standard deviations from the mean:

- **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.

Let's break down the problem:

1. **Mean** of the stock prices (\(\mu\)) = 13.
2. We need to determine the number of observations expected between 146 and 185.
   
Since 146 and 185 are much larger than the mean, the range given (146 to 185) is far outside the normal range described by the empirical rule (which focuses on data near the mean). It implies that these values are extreme outliers. For stock prices in a normally distributed dataset, values like 146 and 185 are much more than 3 standard deviations from the mean.

To determine whether any observations would fall between 146 and 185, we can estimate:

- **At 3 standard deviations** from the mean, almost all the data (99.7%) should fall. Therefore, values like 146 and 185 are likely beyond this range.

Given that these values are likely beyond 3 standard deviations, the number of observations between 146 and 185 would be negligible—essentially 0 in a normally distributed set of data.

### Conclusion:
The number of observations in the data set expected to be between 146 and 185 is **approximately 0**, given the context of a normal distribution and the values provided.

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### Related questions:
1. How do you calculate probabilities for specific ranges in a normal distribution?
2. What is the relationship between standard deviation and the spread of data in a normal distribution?
3. How can extreme outliers affect the mean in a normally distributed data set?
4. How do z-scores relate to the empirical rule in a normal distribution?
5. How would the results change if the standard deviation was given for this dataset?

### Tip:
When working with normally distributed data, always consider the distance of your values from the mean in terms of standard deviations to estimate how unusual or common certain observations are.

Suppose a normally distributed set of stock prices with 6000 observations has a mean of 13. Use the 68-95-99.7 rule to determine the number of observations in the data set expected to be between the values of 146 and 185.

Determine the expected number of observations between 146 and 185 in a normally distributed dataset of stock prices with a mean of 13, applying the 68-95-99.7 rule. Learn the key concepts of the empirical rule and normal distribution. Suitable for Grades 10-12.

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