Math Problem Statement
P(-2 < Z < 1)
Solution
To calculate the probability , where is a standard normal random variable, you need to use the standard normal distribution (Z-distribution).
Steps:
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Find the cumulative probability for : This represents the probability that is less than or equal to 1.
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Find the cumulative probability for : This represents the probability that is less than or equal to -2.
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Subtract the two probabilities: To find the probability that is between -2 and 1, subtract the cumulative probability at from the cumulative probability at .
Using a Z-table or a calculator:
- The cumulative probability at is approximately .
- The cumulative probability at is approximately .
Now, calculate:
Final Answer:
Would you like more details or have any questions?
Here are some related questions:
- How do you calculate probabilities for values greater than 1 in the standard normal distribution?
- What is the meaning of the Z-score in a normal distribution?
- How do you find the inverse of a Z-score for a given probability?
- What is the difference between a normal distribution and a standard normal distribution?
- Can you calculate probabilities for non-standard normal distributions?
Tip: Always remember that the total area under the curve of a normal distribution is 1, and probabilities correspond to areas under the curve.
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Math Problem Analysis
Mathematical Concepts
Standard Normal Distribution
Z-Score
Probability Calculation
Formulas
P(Z < a) = cumulative probability for Z = a
Theorems
Z-distribution
Properties of Normal Distribution
Suitable Grade Level
Grades 10-12
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