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Part 1
Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. Draw a graph and find the probability of a bone density test score greater than negative 1.63.

### Part 1: Probability of a Bone Density Test Score Greater than -1.63

Given:
- The test scores are normally distributed.
- Mean (\(\mu\)) = 0
- Standard deviation (\(\sigma\)) = 1
- We need to find the probability that the test score is greater than -1.63.

#### Step 1: Visualize the Problem
To understand this, let's plot the standard normal distribution curve. The curve is bell-shaped, centered at 0, and spans the range of possible bone density scores. We want to calculate the area under this curve to the right of \(x = -1.63\).

#### Step 2: Calculate the Probability
The probability corresponds to the area under the normal distribution curve. To find this, we look for the cumulative distribution function (CDF) value for \(x = -1.63\), then subtract it from 1 since we are interested in the area greater than -1.63.

We calculate the CDF value for \(x = -1.63\) using standard normal distribution tables or a calculator:

\[
P(X > -1.63) = 1 - P(X \leq -1.63)
\]

Using a standard normal table or calculator:
\[
P(X \leq -1.63) \approx 0.0516
\]

Thus:
\[
P(X > -1.63) = 1 - 0.0516 = 0.9484
\]

#### Step 3: Graph
The graph of the standard normal distribution will have a shaded region to the right of \(x = -1.63\), covering approximately 94.84% of the area under the curve.

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Would you like a detailed graph of the normal distribution curve showing this probability?

### Relative Questions:
1. What is the z-score in a normal distribution?
2. How do you interpret the area under a normal distribution curve?
3. What does a negative z-score indicate in the context of bone density?
4. How would the result change if the mean were not 0?
5. Can you explain why we subtract from 1 when finding probabilities greater than a certain value?

#### Tip:
To find probabilities for any normal distribution, you can standardize using z-scores, which simplifies comparisons to the standard normal table.

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. Draw a graph and find the probability of a bone density test score greater than negative 1.63.

Learn how to calculate the probability that a bone density test score is greater than -1.63 using the normal distribution. This problem involves standard normal distribution, the cumulative distribution function, and graphing techniques. Suitable for students in Grades 10-12.

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