Question

2. The midterm grades of a kinesiology test are normally distributed with µ = 72 and σ = 5.0. Mary earned a 72 on the exam, while Ashley earned a 67. What are the Z-scores of each? Describe what the z-scores mean and how that relates to standard deviation. Show all of your work mathematically.

Answer 1

Mary's z-score is 0, meaning her score is the same as the mean, and Ashley's z-score is -1, indicating her score is one standard deviation below the mean. Z-scores represent the distance of a score from the mean in terms of standard deviations.

Step-by-step explanation:

Z-Scores Calculation

To calculate the z-scores for Mary and Ashley's exam grades, we use the formula:
z = (x - μ) / σ, where x is the score, μ is the mean, and σ is the standard deviation.

Mary's score is 72. So her z-score is calculated as follows:

z = (72 - 72) / 5.0 = 0

Ashley's score is 67. So her z-score is calculated as follows:

z = (67 - 72) / 5.0 = -1

Interpretation of Z-Scores

A z-score tells us how many standard deviations a value is from the mean. Mary's z-score of 0 means her grade is exactly at the mean. Ashley's z-score of -1 indicates her grade is one standard deviation below the mean.

In terms of standard deviation, this means that about 68% of scores lie within one standard deviation of the mean (between a z-score of -1 and 1) in a normal distribution.