The question asks us to determine which student had the best score on either the Physics or the Chemistry test. When comparing scores from different tests, especially when the average scores (means) and spread of scores (standard deviations) are different, simply comparing the raw scores can be misleading. To compare performance across different tests, we use standardized scores, like Z-scores.

A Z-score tells us how many standard deviations an individual score is away from the mean of the group. A positive Z-score means the score is above the average, and a negative Z-score means it's below the average. A higher Z-score indicates a relatively better performance compared to the group.

The formula to calculate a Z-score is:

$$Z = \frac{X - \mu}{\sigma}$$

Where:

• \(X\) is the individual score
• \(\mu\) is the mean of the group
• \(\sigma\) is the standard deviation of the group

Let's look at the given scores and statistics from the paragraph:

Calculating Z-scores for Student Achievement

Now, let's calculate the Z-score for each student in both Physics and Chemistry.

Rekha's Z-scores:

• Physics: Rekha scored 30. The mean is 50, and the SD is 20.

  $$Z_{Physics, Rekha} = \frac{30 - 50}{20} = \frac{-20}{20} = -1.0$$

• Chemistry: Rekha scored 26. The mean is 20, and the SD is 5.

  $$Z_{Chemistry, Rekha} = \frac{26 - 20}{5} = \frac{6}{5} = 1.2$$

Rekha's Z-scores are -1.0 in Physics and 1.2 in Chemistry. Her best relative score is 1.2 in Chemistry.

Reshma's Z-scores:

• Physics: Reshma scored 56. The mean is 50, and the SD is 20.

  $$Z_{Physics, Reshma} = \frac{56 - 50}{20} = \frac{6}{20} = 0.3$$

• Chemistry: Reshma scored 22.5. The mean is 20, and the SD is 5.

  $$Z_{Chemistry, Reshma} = \frac{22.5 - 20}{5} = \frac{2.5}{5} = 0.5$$

Reshma's Z-scores are 0.3 in Physics and 0.5 in Chemistry. Her best relative score is 0.5 in Chemistry.

Sameer's Z-scores:

• Physics: Sameer scored 46. The mean is 50, and the SD is 20.

  $$Z_{Physics, Sameer} = \frac{46 - 50}{20} = \frac{-4}{20} = -0.2$$

• Chemistry: Sameer scored 19. The mean is 20, and the SD is 5.

  $$Z_{Chemistry, Sameer} = \frac{19 - 20}{5} = \frac{-1}{5} = -0.2$$

Sameer's Z-scores are -0.2 in Physics and -0.2 in Chemistry. His best relative score is -0.2.

Comparing Best Relative Scores

Let's compare the highest Z-score achieved by each student on either test:

• Rekha: 1.2 (Chemistry)
• Reshma: 0.5 (Chemistry)
• Sameer: -0.2 (Physics or Chemistry)

Comparing these highest Z-scores, Rekha has the largest Z-score (1.2). This means Rekha's score in Chemistry was the furthest above the class average (in terms of standard deviations) compared to the best relative scores of Reshma and Sameer.

Therefore, Rekha had the best score on either test when considering relative performance using Z-scores.

Revision Table: Z-Scores for Physics and Chemistry Tests

Additional Information on Standard Scores

Z-scores are a type of standard score. Standard scores are useful for comparing scores from different distributions. Besides Z-scores, other types of standard scores exist, such as T-scores. A T-score transforms a Z-score to a scale with a mean of 50 and a standard deviation of 10 using the formula \(T = 50 + 10Z\). This transformation is often used to avoid negative scores and decimals.

In this problem, using Z-scores clearly showed Rekha had the strongest performance relative to her classmates in one of the tests compared to Reshma and Sameer.