Final answer:
Standardizing a distribution of exam scores that have a specific mean and standard deviation, such as 95 and 3, respectively, will result in a standard normal distribution with a mean of 0 and a standard deviation of 1 using z-scores. The shape of the distribution remains the same, and scores are adjusted to reflect their distance from the mean in terms of standard deviations. The correct answer to the question is (b) The scores will become more clustered around the mean.
Step-by-step explanation:
When you standardize a distribution of exam scores, such as those with a mean of 95 and a standard deviation of 3, standardization transforms the scores into a standard normal distribution with a mean of 0 and a standard deviation of 1. This process is done using z-scores. A z-score is calculated by taking a raw score, subtracting the mean of the original distribution, and dividing the result by the standard deviation of the original distribution. In this case, if you take a score 'x' from the distribution, you would calculate the z-score as (x - 95) / 3.
So, what happens to the distribution when we standardize these scores?
- The distribution will not become more spread out or more clustered around the mean; the shape of the distribution remains the same.
- The actual values of the scores will change, as they are being re-scaled to fit the standard normal distribution; however, their relative positions to each other will remain the same.
- The scores will not become negative unless they were originally below the mean of the distribution.
Therefore, the correct answer to the student's question is: (b) The scores will become more clustered around the mean. This answer reflects the idea that after standardization, all scores are described in terms of how many standard deviations they are from the mean, and since we are using a standard distribution, the mean is now zero.