Find the bone density scores that are normally distributed with a mean of 0 and a standard deviation of 1, find the percentage of scores that are : a ) significantly high ( or at least 2 standard deviations above the mean. )

To solve this, we'll use the properties of the standard normal distribution, which has a mean (\(\mu\)) of 0 and a standard deviation (\(\sigma\)) of 1.

### Step-by-Step Explanation:

1. **Definition**: A score that is at least 2 standard deviations above the mean is defined as being significantly high.
   
   - This means we are interested in the probability of scores being greater than or equal to 2 standard deviations from the mean.
   
   Mathematically, we need to find:
   \[
   P(Z \geq 2)
   \]
   where \(Z\) is the standard normal variable.

2. **Using the Z-table or Standard Normal Distribution Table**: We can find the probability that a score is less than 2 standard deviations above the mean.

   The cumulative probability for \(Z = 2\) is:
   \[
   P(Z \leq 2) = 0.9772
   \]
   This means 97.72% of the data lies below 2 standard deviations above the mean.

3. **Complement Rule**: To find the percentage of scores that are at least 2 standard deviations above the mean, we subtract this value from 1:
   \[
   P(Z \geq 2) = 1 - 0.9772 = 0.0228
   \]

4. **Convert to Percentage**: Multiply by 100 to get the percentage:
   \[
   0.0228 \times 100 = 2.28\%
   \]

### Conclusion:
- The percentage of bone density scores that are significantly high (at least 2 standard deviations above the mean) is **2.28%**.

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Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. What percentage of scores fall between 1 and 2 standard deviations above the mean?
2. How would the percentages change if the standard deviation were 2 instead of 1?
3. What percentage of scores are more than 3 standard deviations above the mean?
4. How would you calculate the percentage of scores below 2 standard deviations below the mean?
5. What is the probability of scores falling between the mean and 2 standard deviations above it?

**Tip**: The total area under a normal distribution curve is always 1 (or 100%), which helps when calculating probabilities based on standard deviations.

Learn how to calculate the percentage of bone density scores that are significantly high (at least 2 standard deviations above the mean) in a normal distribution. This detailed step-by-step solution explores Z-scores, probability, and the standard normal distribution, suitable for Grades 9-12.

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