https://file.aiquickdraw.com/mathbot/file/172947565266454bd1x2u.jpg

The problem involves a normally distributed variable with a mean of 0 and a standard deviation of 1, and it asks to find the probability of a bone density test score being greater than -1.76.

The correct region to sketch on the normal curve for a probability of a score *greater than* -1.76 would be the right tail of the distribution, starting at -1.76 and extending to infinity. Option A shows this correctly, where the shaded region represents the area to the right of -1.76, corresponding to the probability of a score greater than -1.76.

### Now, let's calculate the probability:

1. We need to find the **z-score** corresponding to -1.76. This is given directly as the z-score, since the distribution has a mean of 0 and a standard deviation of 1.
2. Using a standard normal table or a calculator, the cumulative probability up to -1.76 is approximately 0.0392.
3. Since we are interested in the probability **greater than** -1.76, we subtract this from 1:
   \[
   P(X > -1.76) = 1 - P(X \leq -1.76) = 1 - 0.0392 = 0.9608
   \]

Thus, the probability is **0.9608**.

Would you like further clarification, or do you have any additional questions?

Here are five related questions to explore:
1. What is the probability of a score greater than 1.76 in a normal distribution?
2. How do you calculate probabilities for values between two z-scores, for example, between -1.76 and 1.76?
3. How does the area under the normal curve change with different standard deviations?
4. What is the total area under the normal distribution curve, and why is it significant?
5. Can you explain the symmetry of the normal distribution and how it affects probability calculations?

**Tip**: When solving normal distribution problems, it is always useful to check if you're dealing with a "greater than" or "less than" probability to apply the correct cumulative probability subtraction.

Assume that a randomly selected subject is given a bone density test. The test scores are normally distributed with a mean of 0 and a standard deviation of 1. Find the probability of a bone density test score greater than -1.76.

This solution demonstrates how to calculate the probability that a bone density test score is greater than -1.76 using a standard normal distribution with a mean of 0 and standard deviation of 1. The problem covers key concepts such as Z-scores and the use of the cumulative probability function.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation