Math Problem Statement

A certain distribution has a mean of 32 and a standard deviation of 4. Find the z-score for a value of 20. Z-Score = [?]

Solution

To find the z-score for a value of 20, use the z-score formula:

z=Xμσz = \frac{X - \mu}{\sigma}

where:

  • X=20X = 20 (the value for which we're finding the z-score),
  • μ=32\mu = 32 (the mean of the distribution),
  • σ=4\sigma = 4 (the standard deviation).

Substitute the values into the formula:

z=20324=124=3z = \frac{20 - 32}{4} = \frac{-12}{4} = -3

So, the z-score for a value of 20 is:

z=3z = -3

Would you like further explanation or have any questions?

Here are some related questions you might find useful:

  1. How do you interpret a z-score of -3 in terms of standard deviations from the mean?
  2. What is the probability of finding a value less than 20 in this distribution?
  3. How would the z-score change if the value were closer to the mean, like 28?
  4. How does the z-score formula apply to non-normal distributions?
  5. What does a positive z-score indicate compared to a negative one?

Tip: Remember, a z-score tells you how many standard deviations a value is from the mean, allowing comparisons across different distributions.

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Math Problem Analysis

Mathematical Concepts

Statistics
Normal Distribution
Z-Scores

Formulas

Z-score formula: z = (X - μ) / σ

Theorems

Z-score interpretation in normal distribution

Suitable Grade Level

Grades 9-11