For a particular sample of 63 scores on a psychology exam, the following results were obtained.

First quartile = 57 Third quartile = 87 Standard deviation = 9 Range = 51
Mean = 72 Median = 72 Mode = 98 Midrange = 57
Answer each of the following:
I. What score was earned by more students than any other score? Why?
II. What was the highest score earned on the exam?
III. What was the lowest score earned on the exam?
IV. According to Chebyshev's Theorem, how many students scored between 45 and 99?
V. Assume that the distribution is normal. Based on the Empirical Rule, how many students scored between 45 and 99?

I. In order to determine the score that was earned by more students than any other score, you need to look for the mode of the data. The mode is the value that appears most frequently in the dataset given. In this case, the mode is 98. This means that more students earned a score of 98 than any other score.

II. To find the highest score earned on the exam, you need to look for the maximum value in the data set. However, the data set you provided does not include the maximum score. Therefore, you would need to obtain the actual data set or find the missing value to determine the highest score.

III. Similar to finding the highest score, in order to determine the lowest score earned on the exam, you would need the actual data set or the missing value. Without that information, it is not possible to determine the lowest score.

IV. According to Chebyshev's Theorem, you can estimate the proportion of scores that fall within a certain range of standard deviations from the mean. Chebyshev's Theorem states that for any number k greater than 1, the proportion of data that falls within k standard deviations of the mean is at least 1 - (1/k^2).

In this case, we want to know how many students scored between 45 and 99. To use Chebyshev's Theorem, we need to find the standard deviation. You have provided the standard deviation as 9. Now, we can calculate the number of students using the formula:

1 - (1/k^2) = 1 - (1/2^2) = 1 - (1/4) = 3/4 = 0.75

This means that at least 75% of the scores fall within 2 standard deviations from the mean. Therefore, according to Chebyshev's Theorem, we can estimate that at least 75% of the students scored between 45 and 99.

V. The Empirical Rule, also known as the 68-95-99.7 rule, applies to normally distributed data. It states that for a normal distribution:

- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.

Based on the Empirical Rule, we can estimate that approximately 95% of the students scored between 45 and 99. However, it is important to note that the data set provided does not mention whether the distribution is normal. Therefore, this estimation based on the Empirical Rule may not be completely accurate without more information about the distribution.