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?
Please show all of your work.
It seems that you might have a typo. With the mode deviating so much from the mean and median, the distribution definitely is not normal. Do you have more than one typo?
I. The mode is the score that occurs most frequently in a data set. In this case, the mode is 98. Hence, more students earned a score of 98 than any other score.
II. The highest score earned on the exam is given by the range, which is the difference between the highest and lowest scores. In this case, the range is 51, and since the lowest score is not given explicitly, we can calculate the highest score as follows:
Highest score = Lowest score + Range
= Lowest score + 51
III. The lowest score earned on the exam is not explicitly given and needs to be calculated using the highest score and the range. Using the equation mentioned in the previous part, we can isolate the lowest score:
Lowest score = Highest score - Range
= (Lowest score + 51) - 51
= Lowest score
Therefore, the lowest score is equal to itself, and we don't have enough information to determine its value.
IV. Chebyshev's Theorem provides a lower bound on the proportion of values within a certain range, given the standard deviation. According to Chebyshev's Theorem, at least (1 - 1/k^2) of the values lie within k standard deviations from the mean, where k is any positive constant greater than 1.
In this case, the range of scores we are interested in is from 45 to 99, which is a distance of 54 from the mean (72), or 6 standard deviations (since the standard deviation is 9).
So, using Chebyshev's Theorem, we can calculate the lower bound for the proportion of students scoring between 45 and 99:
Lower bound = 1 - 1/k^2
= 1 - 1/6^2
= 1 - 1/36
= 35/36
Hence, at least 35/36 (or approximately 97.22%) of the students scored between 45 and 99, according to Chebyshev's Theorem.
V. The Empirical Rule, also known as the 68-95-99.7 rule, applies to approximately normal distributions. It states that about 68% of the values are within one standard deviation of the mean, about 95% are within two standard deviations, and about 99.7% are within three standard deviations.
In this case, the range of scores we are interested in is from 45 to 99, which is a distance of 54 from the mean (72), or 6 standard deviations (since the standard deviation is 9). Applying the Empirical Rule, we can calculate the proportion of students scoring between 45 and 99:
68% + 95% + 99.7% = 262.7%
However, the sum goes beyond 100%, indicating that the distribution is not a perfect normal distribution. Therefore, we cannot directly apply the Empirical Rule to determine the proportion of students scoring between 45 and 99.
Note: The Empirical Rule is only applicable to approximately normal distributions, and in this case, the proportion of scores falling within the given range cannot be accurately determined using this rule.