Math Problem Statement

For a standard normal distribution, find:

P(-1.99 < z < -0.23)

Solution

To find the probability for the standard normal distribution between z=1.99z = -1.99 and z=0.23z = -0.23, you can use a standard normal table (Z-table) or a calculator. The Z-table gives the cumulative probability from the left of the curve up to a given z-value.

  1. Find the cumulative probability for z=1.99z = -1.99: From the Z-table, P(Z<1.99)0.0233P(Z < -1.99) \approx 0.0233.

  2. Find the cumulative probability for z=0.23z = -0.23: From the Z-table, P(Z<0.23)0.4090P(Z < -0.23) \approx 0.4090.

  3. Calculate the probability between the two values: P(1.99<Z<0.23)=P(Z<0.23)P(Z<1.99)P(-1.99 < Z < -0.23) = P(Z < -0.23) - P(Z < -1.99) P(1.99<Z<0.23)=0.40900.0233=0.3857P(-1.99 < Z < -0.23) = 0.4090 - 0.0233 = 0.3857

So, P(1.99<Z<0.23)0.3857P(-1.99 < Z < -0.23) \approx 0.3857.

Would you like more details or have any questions?

Relative questions:

  1. What is the cumulative probability for z=1.99z = 1.99?
  2. How do you calculate the probability P(Z>z)P(Z > z) for a given z-value?
  3. What is the probability between two positive z-values (e.g., P(0.5<Z<1.5)P(0.5 < Z < 1.5))?
  4. How can you calculate probabilities using a calculator instead of a Z-table?
  5. How do you interpret z-values in a real-world context (e.g., in height or test scores)?

Tip:

Always remember that the Z-table typically provides the area to the left of a z-value, so subtracting cumulative probabilities is key for finding the area between two z-values.

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Math Problem Analysis

Mathematical Concepts

Probability
Standard Normal Distribution
Z-Scores

Formulas

P(a < Z < b) = P(Z < b) - P(Z < a)
Z-score formula

Theorems

Standard Normal Distribution
Cumulative Distribution Function

Suitable Grade Level

Grades 11-12 (High School) or College Level