Relatively speaking, which statistics exam did you do better on? On your first statistics exam you scored a 72 and on the second exam you scored an 89. Hey, you’re improving! Or are you? On the first exam, the class mean was 71 with a standard deviation of 15. On the second exam the class mean was 95 with a standard deviation of 5. Convert your scores into z scores and determine whether your grade improved over the course of the semester relative to the other students in the class.

Z = (score-mean)/SD

Which Z score is highest?

To determine whether your grade improved over the course of the semester relative to the other students in the class, we can convert your scores into z-scores and compare them.

To convert your first exam score of 72 into a z-score, we use the formula:

z = (x - μ) / σ

where x is the value of your score, μ is the mean, and σ is the standard deviation.

For the first exam:
x = 72
μ = 71
σ = 15

Plugging in the values, we get:

z1 = (72 - 71) / 15 = 0.07

For the second exam:
x = 89
μ = 95
σ = 5

Plugging in the values, we get:

z2 = (89 - 95) / 5 = -1.2

Comparing the two z-scores, we can see that your z-score for the first exam is 0.07, indicating that your score is just above the mean. On the other hand, your z-score for the second exam is -1.2, indicating that your score is below the mean.

Therefore, based on the z-scores, it appears that your grade did not improve over the course of the semester relative to the other students in the class, as your z-score for the second exam is lower than that of the first exam.

To determine whether your grade improved relative to the other students in the class, we will calculate the z-scores for your scores on both exams.

The formula for calculating z-score is:
z = (x - μ) / σ

Where:
- x is the value you want to convert to a z-score (your score)
- μ is the mean of the data set (class mean)
- σ is the standard deviation of the data set

Let's calculate the z-scores for both of your exam scores:

For the first exam:
x1 = 72
μ1 = 71
σ1 = 15

z1 = (x1 - μ1) / σ1
z1 = (72 - 71) / 15
z1 = 1 / 15
z1 ≈ 0.07

For the second exam:
x2 = 89
μ2 = 95
σ2 = 5

z2 = (x2 - μ2) / σ2
z2 = (89 - 95) / 5
z2 = -6 / 5
z2 = -1.2

Now, let's interpret the z-scores:

A positive z-score means that your score is above the mean, while a negative z-score means that your score is below the mean.

For the first exam, you obtained a z-score of approximately 0.07, meaning that your score was slightly above the mean.

For the second exam, you obtained a z-score of -1.2, indicating that your score was below the mean.

Comparing the z-scores, we find that your score improved relative to the class mean from the first exam to the second exam. Although your score on the second exam was below the class mean, it is still higher in relation to the class distribution than your score on the first exam.