Math Problem Statement

A set of test scores is skewed to the left, has a mean of 74%, and a standard deviation of 12%.  All test scores were between 50 and 100.    If Tamia's test score has a Z-score of 1.2, how many standard deviations is Tamia's test score above the mean?

Round your answer to one decimal place. 

Solution

The Z-score formula is given by:

Z=XμσZ = \frac{X - \mu}{\sigma}

where:

  • ZZ is the Z-score,
  • XX is Tamia's test score,
  • μ\mu is the mean of the test scores,
  • σ\sigma is the standard deviation.

In this problem, you are given:

  • Z=1.2Z = 1.2 (Tamia's Z-score),
  • μ=74%\mu = 74\% (mean),
  • σ=12%\sigma = 12\% (standard deviation).

The Z-score indicates how many standard deviations a data point is above or below the mean. Since Tamia's Z-score is 1.2, her score is 1.2 standard deviations above the mean.

Answer: Tamia's test score is 1.2 standard deviations above the mean.

Would you like further details or have any questions?

Follow-up Questions:

  1. How do you interpret a negative Z-score?
  2. What does it mean for a data set to be skewed to the left?
  3. How would Tamia's Z-score change if her score were lower?
  4. Can a Z-score be used to compare scores from different tests?
  5. What does a Z-score of 0 represent in this context?

Tip:

A Z-score of 1.2 indicates that Tamia's score is better than approximately 88% of all test scores, assuming a normal distribution, even though the data is skewed.