The same 100 students sat for a Chemistry test and a Biology test. The Chemistry test yielded the same mean as the biology test but a different standard

Question

deviation of 2.5. The Biology test had a standard deviation of 7.95.
I. Which test was done well and why?
II. If a student scored 19 in both tests, which of the test did he do better relative to the other
students?

Answer 1

To determine which test was conducted better and evaluate the performance of a student relative to their peers, we can consider the standard deviations and how they relate to the scores.

I. Which test was done well and why?

The Chemistry test had a standard deviation of 2.5 while the Biology test had a standard deviation of 7.95. A lower standard deviation indicates that the scores are more closely clustered around the mean (i.e., there is less variability in scores), while a higher standard deviation indicates more variability in scores.

In this case, the Chemistry test, with a standard deviation of 2.5, shows that the scores were more consistent and less spread out than the Biology test, which had a standard deviation of 7.95. Therefore, we can conclude that the Chemistry test was done better in terms of consistency of student performance. A lower standard deviation suggests that the test may have been more effective in assessing students' knowledge uniformly.

II. If a student scored 19 in both tests, which of the tests did he do better relative to the other students?

To determine which test the student did better in relative to peers, we need to calculate the z-scores for the score of 19 in each test. The z-score will show how many standard deviations a score is from the mean.

The formula for the z-score is: \[ z = \frac{(X - \mu)}{\sigma} \] where \(X\) is the score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Since we know both tests have the same mean, we can denote the mean as \(\mu\).

Assuming the same mean for both tests, we can calculate the z-scores for both tests:

1. Chemistry Test: \[ z_C = \frac{(19 - \mu)}{2.5} \]

2. Biology Test: \[ z_B = \frac{(19 - \mu)}{7.95} \]

Since the scores are equal (19) and the mean is the same, we cannot calculate the numerical z-scores without knowing the exact value of the mean. However, we can analyze the relationship between the z-scores:

• The Chemistry z-score will have a smaller denominator (2.5) compared to the Biology z-score (7.95). Thus: \[ z_C > z_B \] resulting in a higher z-score for Chemistry than for Biology.

Conclusion:

The student performed better relative to their peers in the Chemistry test since the z-score for that test will be higher, indicating that their score is further above the mean in comparison to the distribution of scores in the Biology test.