Math Problem Statement

A set of data items is normally distributed with a mean of 40 and a standard deviation of 4. Convert 48 to a z-score.

Solution

To convert the given value (48) to a z-score, we can use the z-score formula:

z=xμσz = \frac{x - \mu}{\sigma}

Where:

  • x=48x = 48 (the value we are converting),
  • μ=40\mu = 40 (the mean),
  • σ=4\sigma = 4 (the standard deviation).

Now, let's substitute the values into the formula:

z=48404=84=2z = \frac{48 - 40}{4} = \frac{8}{4} = 2

Thus, the z-score for 48 is 2.

Would you like further details or have any questions?

Here are 5 related questions:

  1. What does a z-score of 2 signify in a normal distribution?
  2. How would you interpret a negative z-score?
  3. How can z-scores be used to compare data from different distributions?
  4. What percentage of data falls above a z-score of 2 in a standard normal distribution?
  5. How would the z-score change if the standard deviation increased to 5?

Tip: Z-scores help determine how far a value is from the mean in terms of standard deviations. A z-score of 2 means the value is 2 standard deviations above the mean.

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Math Problem Analysis

Mathematical Concepts

Statistics
Normal Distribution
Z-score

Formulas

z = (x - μ) / σ

Theorems

Z-Score Theorem

Suitable Grade Level

Grades 10-12