Math Problem Statement
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 .
To find the area of this shaded region, we need to find the cumulative probability corresponding to a z-score of . 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 is approximately:
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:
- How is the area under the normal curve related to probabilities?
- What is the significance of the z-score in normal distribution problems?
- How do you calculate z-scores for non-standard normal distributions?
- How can the area to the right of a z-score be found?
- 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
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