Two high school students took equivalent language tests, one in German and one in French. The student taking the German test, for which the mean was 66 and the standard deviation was 8, scored an 82, while the student taking the French test, for which the mean was 27 and the standard deviation was 5, scored a 35. Compare the scores.
To compare the scores, we can use the concept of z-scores. The z-score measures how many standard deviations a particular score is from the mean.
1. For the German test:
- Mean = 66
- Standard deviation = 8
- Given score = 82
To calculate the z-score for the German test, use the following formula:
z = (x - mean) / standard deviation
Substitute the given values into the formula:
z = (82 - 66) / 8
= 16 / 8
= 2
The z-score for the German test is 2.
2. For the French test:
- Mean = 27
- Standard deviation = 5
- Given score = 35
To calculate the z-score for the French test, use the formula again:
z = (x - mean) / standard deviation
Substitute the given values into the formula:
z = (35 - 27) / 5
= 8 / 5
= 1.6
The z-score for the French test is 1.6.
Now, we can compare the scores by comparing the z-scores. Since the z-score for the German test is 2 and the z-score for the French test is 1.6, we can conclude that the student's score on the German test is comparatively higher than their score on the French test.
To compare the scores of the two students, we need to standardize the scores using z-scores. The formula to calculate the z-score is:
z = (x - μ) / σ
where:
- x is the individual score
- μ is the mean
- σ is the standard deviation
Let's calculate the z-scores for both students:
For the German test:
x_german = 82
μ_german = 66
σ_german = 8
z_german = (82 - 66) / 8
z_german = 16 / 8
z_german = 2
For the French test:
x_french = 35
μ_french = 27
σ_french = 5
z_french = (35 - 27) / 5
z_french = 8 / 5
z_french = 1.6
Now let's compare the z-scores. A higher z-score indicates a higher score relative to the mean.
z_german = 2
z_french = 1.6
Since 2 is greater than 1.6, we can conclude that the student who took the German test scored higher compared to the student who took the French test.