SOLUTION: For a particular sample of 80 scores on a psychology exam, the following results were obtained. First quartile = 52 Third quartile = 78 Standard deviation = 7 Range = 45 Mean =

For a particular sample of 80 scores on a psychology exam, the following results were obtained.
First quartile = 52  Third quartile = 78 Standard deviation = 7  Range = 45
Mean = 65   Median = 65   Mode = 76  Midrange = 62
Answer each of the following:
I.       What score was earned by more students than any other score? 

76

Why? 

Because that what "MOde" means -- Just remember that "MOst" and "MOde"
both starts with "MO".

II.     What was the highest score earned on the exam? 
III.    What was the lowest score earned on the exam?

We'll answer those together:

Let H = the highest score
Let L = the lowest score

RANGE = H-L = 45

MIDRANGE =  = 62

So we have this system of equations:



Multiply the second one through by 2



That's easy to solve H=84.5, L=39.5

IV.     According to Chebyshev's Theorem, how many students scored 
between 37 and 93?

Let M = the mean
Let S = the standard deviation.

Chebyshev's theorem says that at least this fraction 1-1/K² of 
the data lies between M-KS and M+KS 

Standard deviation = S = 7, Mean = M = 65

  M-KS=37        and      M+KS=93
65-K·7=37               65+K·7=93 
   -7K=-28                  7K=28
     K=4                     K=4
So you see, we didn't have to solve both, since either one gave K=4

Now we must calculate the fraction 1-1/K² = 1-1/4² = 1-1/16 = 15/16

So at least 15/16th of the 80 scores were between 37 and 93.
15/16ths of 80 is 75.  So at least 75 of the 80 students scored
between 37 and 93


V.       Assume that the distribution is normal.  Based on the Empirical
Rule, how many students scored between 51 and 79?

Let M = the mean = 65
Let S = the standard deviation = 7

The Empirical rule is sometimes called the "68-95-99.7" rule.

Approximately 68% of the data lies between M-S and M+S
Approximately 95% of the data lies between M-2S and M+2S
Approximately 99.7% of the data lies between M-3S and M+3S

So we calculate all those:
M-S = 65-7 = 58                  M+S = 65+7 = 72
M-2S = 65-2·7 = 65-14 = 51       M+2S = 65+2·14 = 65+14 = 79
M-3S = 65-3·7 = 65-21 = 44       M+3S = 65+3·21 = 65+21 = 86

So:
Approximately 68% of the data lies between 58 and 72
Approximately 95% of the data lies between 51 and 79
Approximately 99.7% of the data lies between 44 and 86

You only wanted the middle answer, 95%

Edwin