Math Problem Statement

For a standard normal distribution, find: P(z < 2.23). Express the probability as a decimal rounded to 4 decimal places.

Solution

The question asks for the probability P(Z<2.23)P(Z < 2.23) from the standard normal distribution, where the mean μ=0\mu = 0 and standard deviation σ=1\sigma = 1.

Step-by-Step Solution:

  1. Locate Z = 2.23 on the Z-table, which gives the cumulative probability from -\infty to 2.23.

  2. From the Z-table, the cumulative probability for Z=2.23Z = 2.23 is:

    P(Z<2.23)=0.9871P(Z < 2.23) = 0.9871

  3. Answer:
    Expressing the probability rounded to 4 decimal places:

    P(Z<2.23)=0.9871P(Z < 2.23) = 0.9871

This means the area to the left of Z=2.23Z = 2.23 under the standard normal curve is 0.9871.


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Related Questions:

  1. What is the cumulative probability for Z<1.96Z < 1.96?
  2. How do you interpret probabilities in a standard normal distribution?
  3. What is the probability P(Z>2.23)P(Z > 2.23) using the complement rule?
  4. How do we calculate probabilities when Z-scores are negative?
  5. What is the total area under the standard normal curve?

Tip: When calculating probabilities for normal distributions, if Z-scores aren't integers, a Z-table or software tools (like Excel or calculators) can be very handy.

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Math Problem Analysis

Mathematical Concepts

Probability
Standard Normal Distribution
Z-scores

Formulas

Cumulative distribution function for Z-score

Theorems

Empirical Rule
Properties of the Normal Distribution

Suitable Grade Level

Grades 10-12