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. The shaded region corresponds to z = -0.94.

Solution

The problem involves finding the area of the shaded region in a standard normal distribution (with a mean of 0 and standard deviation of 1). The graph shows a shaded region to the left of the z-score z=0.94z = -0.94.

To find the area of this shaded region, we need to find the cumulative probability corresponding to a z-score of z=0.94z = -0.94. This is typically done using the cumulative distribution function (CDF) of the standard normal distribution.

Using a z-table or a calculator, the area (probability) to the left of z=0.94z = -0.94 is approximately:

P(Z0.94)0.1736P(Z \leq -0.94) \approx 0.1736

Thus, the area of the shaded region is approximately 0.1736, which represents the probability that a standard normal variable is less than or equal to -0.94.

Would you like further details or have any questions?

Here are five related questions you might find useful:

  1. How is the area under the normal curve related to probabilities?
  2. What is the significance of the z-score in normal distribution problems?
  3. How do you calculate z-scores for non-standard normal distributions?
  4. How can the area to the right of a z-score be found?
  5. How does changing the mean and standard deviation affect the normal curve?

Tip: When using z-tables, remember that most tables give the area to the left of the z-score, so for right-tail areas, you must subtract from 1.

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Math Problem Analysis

Mathematical Concepts

Probability
Standard Normal Distribution
Z-Scores

Formulas

Cumulative Distribution Function (CDF) for Standard Normal Distribution
P(Z ≤ z)

Theorems

Empirical Rule for Normal Distributions
Properties of Z-Score Tables

Suitable Grade Level

Grades 11-12, College Level