On her first test in algebra, Mary earned a score of 75, and she made an 85 on her first psychology test. If the average score on the algebra test was a 68, with s=7, and the average on the psychology test was 91, with s=6, is Mary doing better in psychology or algebra. Why?
Z = (score-mean)/SD
Compute both Z scores.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores. Compare proportions.
To determine whether Mary is doing better in psychology or algebra, we need to compare her scores relative to the average scores and the standard deviations of each test.
Let's first calculate the z-scores for her scores in each subject.
The z-score formula is:
z = (x - μ) / σ
where:
- x is the individual score,
- μ is the population mean, and
- σ is the population standard deviation.
For the algebra test:
- x = 75 (Mary's score)
- μ = 68 (average score)
- σ = 7 (standard deviation)
Using the formula, we can calculate the z-score for Mary's algebra test:
z_algebra = (75 - 68) / 7 = 1
For the psychology test:
- x = 85 (Mary's score)
- μ = 91 (average score)
- σ = 6 (standard deviation)
Using the formula, we can calculate the z-score for Mary's psychology test:
z_psychology = (85 - 91) / 6 = -1
Now, let's interpret the z-scores:
- A positive z-score means Mary's score in algebra is above the average, relative to the standard deviation.
- A negative z-score means Mary's score in psychology is below the average, relative to the standard deviation.
Since Mary's z-score for the algebra test is 1 (positive), it indicates that her score is above the average for algebra relative to the standard deviation.
On the other hand, her z-score for the psychology test is -1 (negative), which suggests that her score is below the average for psychology relative to the standard deviation.
Therefore, based on the z-scores, Mary is doing better in algebra compared to psychology because her score in algebra is above the average relative to the standard deviation, while her score in psychology is below the average relative to the standard deviation.