Math Problem Statement

P(-2 < Z < 1)

Solution

To calculate the probability P(2<Z<1)P(-2 < Z < 1), where ZZ is a standard normal random variable, you need to use the standard normal distribution (Z-distribution).

Steps:

  1. Find the cumulative probability for Z=1Z = 1: This represents the probability that ZZ is less than or equal to 1.

  2. Find the cumulative probability for Z=2Z = -2: This represents the probability that ZZ is less than or equal to -2.

  3. Subtract the two probabilities: To find the probability that ZZ is between -2 and 1, subtract the cumulative probability at Z=2Z = -2 from the cumulative probability at Z=1Z = 1.

Using a Z-table or a calculator:

  • The cumulative probability at Z=1Z = 1 is approximately 0.84130.8413.
  • The cumulative probability at Z=2Z = -2 is approximately 0.02280.0228.

Now, calculate: P(2<Z<1)=P(Z<1)P(Z<2)=0.84130.0228=0.8185P(-2 < Z < 1) = P(Z < 1) - P(Z < -2) = 0.8413 - 0.0228 = 0.8185

Final Answer:

P(2<Z<1)0.8185P(-2 < Z < 1) \approx 0.8185

Would you like more details or have any questions?

Here are some related questions:

  1. How do you calculate probabilities for values greater than 1 in the standard normal distribution?
  2. What is the meaning of the Z-score in a normal distribution?
  3. How do you find the inverse of a Z-score for a given probability?
  4. What is the difference between a normal distribution and a standard normal distribution?
  5. 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