Math Problem Statement

Evaluate the following probabilities based on the standard normal distribution: a. P(z < 0) b. P(z < -2.06) c. P(-0.96 < z < 1.31) d. P(z > 1.59) Round all answers to at least 3 decimal places.

Solution

From the image, you are working on a question that involves calculating probabilities using the standard normal distribution. Let's walk through each part of the question.

You already got parts a, b, and c correct:

  • a. P(z<0)=0.500P(z < 0) = 0.500 is correct.
  • b. P(z<2.06)=0.980P(z < -2.06) = 0.980 is correct.
  • c. P(0.96<z<1.31)=0.736P(-0.96 < z < 1.31) = 0.736 is correct.

However, part d appears to be incorrect. Let's correct it:

d. P(z>1.59)P(z > 1.59)

To calculate P(z>1.59)P(z > 1.59), we need to first find the cumulative probability for z=1.59z = 1.59 using the standard normal distribution table or a calculator.

  • The cumulative probability for z=1.59z = 1.59 is approximately 0.9441.
  • Since we want P(z>1.59)P(z > 1.59), we need to subtract this from 1:

P(z>1.59)=10.9441=0.0559P(z > 1.59) = 1 - 0.9441 = 0.0559

So, the correct answer for part d is:

P(z>1.59)0.056P(z > 1.59) \approx 0.056

Would you like any further details on how this was calculated or clarification on the standard normal distribution?

Here are five related questions you might find useful:

  1. How do you find probabilities for negative zz-scores in a normal distribution?
  2. What is the total area under the standard normal curve?
  3. How do you calculate P(a<z<b)P(a < z < b) for different zz-values?
  4. How can you use a calculator or statistical software to compute normal probabilities?
  5. What are some applications of the standard normal distribution in real-world problems?

Tip: Always remember that the total area under the normal distribution curve equals 1, so for P(z>x)P(z > x), you subtract the cumulative probability from 1.

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Math Problem Analysis

Mathematical Concepts

Probability
Standard Normal Distribution
Z-scores

Formulas

Cumulative probability formula for the standard normal distribution: P(z ≤ x)
Complement rule: P(z > x) = 1 - P(z ≤ x)

Theorems

Standard Normal Distribution Properties
Symmetry of the Normal Distribution

Suitable Grade Level

Grades 9-12