Question

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. Find the probability that a given score is less than and draw a sketch of the region.

Answer 1

In order to find the probability of a test score of less than 0, we will first calculate the z-score, and then we'll use the standard normal distribution to find the probability.

Since we are considering distribution with a mean of 0 and standard deviation of 1, this is a standard normal distribution.

1. **Step 1 - Calculate Z-Score:**

The Z-Score formula is as follows:

Z = (X - μ) / σ

Here,

X = Given test score = 0

μ = Mean = 0

σ = Standard Deviation = 1

Substituting the values into the formula, we get Z = (0 - 0) / 1 = 0

So, the Z-Score is 0.

2. **Step 2 - Find Probability:**

For any given standard normal distributed variable, the probability of the variable being less than the mean (which is 0 in this case since it's a standard normal distribution) is 0.5 or 50%. This is because, in a normal distribution, the graph is symmetric about the mean. Therefore, the area less than the mean will be equal to the area more than the mean which will be 50% of the total.

3. **Step 3 - Draw a sketch of the region:**

- Draw a normal distribution curve (It should look like a bell shape).
- Label the mean (0) at the center.
- Since the Z score is 0 which corresponds to the mean, simply shade the region under the curve that lies to the left of the mean.
- This shaded region represents all test scores less than 0.

Finally, the probability of a given score less than 0 is 0.5 or 50%. And the sketch will show half of the distribution curve shaded which represents the required region.

Answer 2

0.5000

Explanation:

According To The Question,

The Area under the Standard Normal Curve, We Can Use the Statistical Tables , That Reported Area As P(z < a) .

Thus, We need to use the following Relationship.

• P(z > a) = 1 - P(z < a)

Now solve, P(z > 0) = 1 - P(z < 0) ⇔ 1 - 0.5000 ⇔ 0.5000

(Diagram, Please Find in Attachment)