Math Problem Statement

find the area of the shaded region. the graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1. round the answer to four decimal places as needed.

Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1. Round the answer to four decimal places as needed.

Solution

The graph shows a standard normal distribution, where the mean is 0 and the standard deviation is 1. The shaded region is bounded by two z-scores: z=0.86z = -0.86 and z=1.26z = 1.26.

To find the area of the shaded region, we calculate the cumulative area under the standard normal curve between these two z-scores. This is done by using the cumulative distribution function (CDF) for the standard normal distribution.

Steps:

  1. Find the cumulative probability corresponding to z=1.26z = 1.26.
  2. Find the cumulative probability corresponding to z=0.86z = -0.86.
  3. Subtract the cumulative probability of z=0.86z = -0.86 from that of z=1.26z = 1.26 to get the area of the shaded region.

Let's calculate this now.The area of the shaded region between the z-scores z=0.86z = -0.86 and z=1.26z = 1.26 is approximately 0.7013 (rounded to four decimal places).

Would you like further details on this calculation or have any other questions?

Here are some related questions for further exploration:

  1. How is the cumulative distribution function (CDF) used in normal distribution calculations?
  2. What is the significance of z-scores in statistics?
  3. How do you calculate probabilities for standard normal distributions without a table?
  4. What are the properties of a standard normal distribution?
  5. How can confidence intervals be visualized using normal distribution graphs?

Tip: You can use standard normal tables or statistical software to quickly find areas under the normal curve based on z-scores.

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Math Problem Analysis

Mathematical Concepts

Normal Distribution
Z-Scores
Cumulative Distribution Function (CDF)

Formulas

P(a < Z < b) = CDF(b) - CDF(a)

Theorems

Standard Normal Distribution

Suitable Grade Level

Grades 11-12