For a particular sample of 80 scores on a psychology exam, the following results were obtained.
First quartile = 65 Third quartile = 98 Standard deviation = 8 Range = 56
Mean = 83 Median = 81 Mode = 59 Midrange = 64
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 67 and 99?
V. Assume that the distribution is normal. Based on the Empirical Rule, how many students scored between 67 and 99?
Please show all of your work
I. To find the score that was earned by more students than any other score, we need to identify the mode of the data. The mode is the score that occurs most frequently in the dataset. In this case, the mode is given as 59. This means that more students earned a score of 59 than any other score.
II. The highest score earned on the exam can be determined by finding the maximum value in the dataset. However, the dataset provided does not explicitly mention the highest score. Instead, we are given the range, which is the difference between the highest and lowest scores. The range is mentioned as 56, suggesting that the highest score can be obtained by adding the range to the lowest score. So, the highest score can be calculated as: Lowest Score + Range = Lowest Score + 56.
III. The lowest score earned on the exam can be determined by finding the minimum value in the dataset. However, the dataset provided does not explicitly mention the lowest score. Instead, we are given the range, which is the difference between the highest and lowest scores. The range is mentioned as 56, suggesting that the lowest score can be obtained by subtracting the range from the highest score. So, the lowest score can be calculated as: Highest Score - Range = Highest Score - 56.
IV. According to Chebyshev's Theorem, we can determine the proportion of data within a certain number of standard deviations from the mean. Specifically, for any given value k, at least (1 - 1/k^2) of the data will lie within k standard deviations from the mean. In this case, we want to calculate the number of students who scored between 67 and 99.
First, let's calculate the mean and standard deviation for the dataset:
Mean (μ) = 83
Standard Deviation (σ) = 8
Now, let's calculate the number of standard deviations away from the mean for the lower and upper bounds:
Lower Bound = (67 - Mean) / Standard Deviation
Upper Bound = (99 - Mean) / Standard Deviation
Substituting the values into the equations:
Lower Bound = (67 - 83) / 8
Upper Bound = (99 - 83) / 8
Calculate the number of standard deviations:
Lower Bound = -2
Upper Bound = 2
According to Chebyshev's Theorem, at least (1 - 1/k^2) of the data will lie within k standard deviations from the mean. So, for k = 2, at least (1 - 1/2^2) = 1 - (1/4) = 3/4 = 0.75 (or 75%) of the data will lie within 2 standard deviations from the mean.
There were 80 students in total, so the number of students who scored between 67 and 99 is:
Number of students = 0.75 * 80 = 60
V. The Empirical Rule, also known as the 68-95-99.7 rule, applies to normal distributions. According to this rule, approximately 68% of the data will lie within 1 standard deviation of the mean, 95% will lie within 2 standard deviations, and 99.7% will lie within 3 standard deviations.
Since the lower bound (67) and upper bound (99) are within 2 standard deviations from the mean, we can conclude that approximately 95% of the students scored between 67 and 99. Therefore, the number of students who scored between 67 and 99 is:
Number of students = 0.95 * 80 = 76
Please note that while the Empirical Rule assumes a normal distribution, the given dataset does not explicitly state that the scores follow a normal distribution.