statistics
posted by Mandy on .
For a particular sample of 53 scores on a psychology exam, the following results were obtained.
First quartile = 44 Third quartile = 68 Standard deviation = 8 Range = 55
Mean = 54 Median = 54 Mode = 71 Midrange = 64
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 30 and 78?
V. Assume that the distribution is normal. Based on the Empirical Rule, how many students scored between 22 and 86?
Please show all of your work.

I. 71. The mode, by definition is the answer that appears most frequently.
II. The midrange is 64, which is by definition the arithmetic mean of the largest and the smallest values in a sample or other group. The range is 55. Let H = highest score, and L = lowest score. Then,
(H+L)/2 = 64
HL = 55
Use algebra to solve for H and L
III. See part II
IV
In general terms, the chebyshevâ€™s theorem states that at least (1 1/k2) of the elements of any distribution lie
Within k standard deviations of the mean (where k = a number greater than 1). 30 is 24 less than the mean of 54; 78 is 24 more than 54; 24 is 3 times the standard deviation of 8; So k = 3, so at least 1  1/9, or at least 8/9 of the total 53 scores lie between 30 and 78.
V. 22 is 4 standard deviations below the mean of 54, and 86 is 4 standard deviations above 54; According to the empirical rule, 68% of the values lie within 1 standard deviation of the mean, 95% of the values lie within 2 standard deviations of the mean, and 99.73% lie within 3 standard deviations of the mean. So even more than 99.73% would lie within 4 standard deviations of the mean; this number has to be a whole number, so the entire set of 53 scores lies within this range.