Math Problem Statement

Find the probabilities for each of the following values and convert them into percentage form: 1) P(Z < 1.04) 2) P(Z <= -0.5) 3) P(Z > -1.65) 4) P(-0.87 < Z <= 1.31) 5) P(-0.52 < Z < 1.33)

Solution

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Problem Statement:

Find the probabilities for each of the following given and convert into percentage form.

GIVEN:

  1. P(Z<1.04)P(Z < 1.04)
  2. P(Z0.5)P(Z \leq -0.5)
  3. P(Z>1.65)P(Z > -1.65)
  4. P(0.87<Z1.31>)P(-0.87 < Z \leq 1.31 >)
  5. P(0.52<Z<1.33)P(-0.52 < Z < 1.33)

I will now solve these problems using the Z-table and then convert them to percentages.

Let me calculate them for you.Here are the probabilities converted into percentage form:

  1. P(Z<1.04)P(Z < 1.04) = 85.08%
  2. P(Z0.5)P(Z \leq -0.5) = 30.85%
  3. P(Z>1.65)P(Z > -1.65) = 95.05%
  4. P(0.87<Z1.31)P(-0.87 < Z \leq 1.31) = 71.28%
  5. P(0.52<Z<1.33)P(-0.52 < Z < 1.33) = 60.67%

Would you like more details on how these were calculated or do you have any other questions?

Here are 5 relative questions to expand on this topic:

  1. How are Z-scores calculated from raw data in a normal distribution?
  2. What is the importance of the standard normal distribution in statistics?
  3. How would you calculate the probability for a Z-value not found in a Z-table?
  4. What is the difference between P(Zz)P(Z \leq z) and P(Z<z)P(Z < z)?
  5. Can you explain why the total area under the standard normal curve equals 1?

Tip: When working with Z-scores, remember that they represent the number of standard deviations away from the mean, making them useful for comparing different data sets.

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Math Problem Analysis

Mathematical Concepts

Probability
Standard Normal Distribution
Z-Scores

Formulas

Z = (X - μ) / σ

Theorems

Standard Normal Distribution Table

Suitable Grade Level

Grades 10-12