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.