Math Problem Statement
Solution
Let’s solve these problems step by step. Since is a standard normal variable, it has a mean of 0 and a standard deviation of 1. We will use the standard normal distribution table (Z-table) to find the probabilities.
1.
To find this probability, we first find the cumulative probability up to and subtract the cumulative probability up to .
- From the Z-table, .
- since the standard normal distribution is symmetric about zero.
Thus,
2.
To find this probability, we calculate .
- From the Z-table, .
Therefore,
3.
Since the Z-table gives values for positive Z, we can use the symmetry of the standard normal distribution: .
- From the Z-table, .
Thus,
4.
We find the cumulative probability up to and subtract the cumulative probability up to .
- .
- From the Z-table, .
Therefore,
Would you like further details on these calculations or have any questions?
Related Questions:
- How do you calculate probabilities for non-standard normal distributions?
- What is the significance of the area under the standard normal curve?
- How does the symmetry of the standard normal distribution simplify calculations?
- How can you calculate probabilities for ranges not centered around zero?
- What are the properties of a cumulative distribution function (CDF)?
Tip:
When using Z-tables, always ensure that you understand whether the table provides cumulative probabilities from the left or if it gives tail probabilities.
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Math Problem Analysis
Mathematical Concepts
Standard Normal Distribution
Probability
Cumulative Distribution Function
Symmetry of Normal Distribution
Formulas
P(a ≤ Z ≤ b) = P(Z ≤ b) - P(Z ≤ a)
P(Z ≥ c) = 1 - P(Z ≤ c)
Symmetry of normal distribution: P(Z ≤ -x) = 1 - P(Z ≤ x)
Theorems
Standard Normal Distribution Properties
Cumulative Distribution Function (CDF)
Suitable Grade Level
College/University Level (or AP Statistics)