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Stella and the Statistical Six (Posted on 2008-04-18) Difficulty: 3 of 5
Exactly seven statistics students were enrolled in a statistics class. One of them, Stella, had been out sick and missed the most recent exam. The other six had taken the exam and each of them knew all six scores. When Stella asked about how the test went--she was worried about her own makeup exam--the other students made the following (true) statements about their scores.

Shawn: This was a pretty hard test, but it was structured just like the last one. There were 100 questions each worth one point. As usual, the professor didn't give out any partial credit. The good news is that nobody missed every single question. At least I scored above the mean, but sadly by less than two points.

Simon: That's the format for every test. But for THIS test, no two of us got the same score. My score, for example, was exactly twice the lowest score.

Sally: Yes, yes. But don't you find it more interesting that our mean score was a whole number, yet none of us actually scored exactly the mean. As for my score, I'm just glad I did better than Seth.

Seth: Bah! Whole number means happen all the time. What's REALLY intriguing is that the standard deviation of our scores (remember, we're the entire population taking the exam) was also a whole number. That's much less likely!

Steven: You want rare? What if I told you that if you were to discard the highest and lowest scores and calculate the mean and standard deviation for that population of four, then not only would both STILL be whole numbers but the mean wouldn't change AT ALL and the standard deviation would be reduced by exactly a factor of three. Can you believe it? It's true!

Sylvia: Steven's just jealous because I got a perfect score and he got the lowest score.

Stella thought about this information and then went to the professor (who heard the entire conversation) and told her what each student's score was. The professor was so impressed that Stella got an A without having to take a makeup exam.

Can you duplicate Stella's feat? What score did each of the six test takers receive?

NOTE: This problem can be solved without spreadsheets or computers, although solutions that use those devices are also welcome.

No Solution Yet Submitted by Paul    
Rating: 4.0000 (2 votes)

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Solution computer solution Comment 1 of 1

The variables in the program are the second letter of each of the students' names. Sylvia doesn't need a variable as her score is 100. The inner four scores are figured first, and only Simon's, Seth's and Sally's scores need to go through all possibilities. Shawn's is the total of those other three, plus 4, all divided by 3, as his/her score is one above the mean:

h = (i+e+a+h)/4 + 1
4h = i+e+a+h + 4
3h = i+e+a + 4
h = (i+e+a + 4) / 3

As a result, we're assured that the mean is one less than Shawn's score.

Then, once Steven's score (at half of Simon's) is found to be lower than the four others and a perfect square variance is obtained for the middle four scores, the mean of the best and worst scores has to match the mean of the middle four and the variance of all six is tested to see if it's a perfect square.

FOR i = 2 TO 98 STEP 2 ' sImon
 FOR e = 2 TO 98 ' sEth
  IF e <> i THEN
   FOR a = e + 1 TO 98 ' sAlly
    IF a <> i THEN
     h = (i + e + a + 4) / 3  ' sHawn
     IF h = INT(h) THEN
      t = i / 2   ' sTeven
      IF t < e AND t < h THEN
       mean = h - 1
       var = ((i - mean) * (i - mean) + (a - mean) * (a - mean) + (e - mean) * (e - mean) + (h - mean) * (h - mean)) / 4
       sd = INT(SQR(var) + .5)
       IF sd * sd = var THEN
        IF (100 + t) / 2 = mean THEN
         varBig = ((i - mean) * (i - mean) + (a - mean) * (a - mean) + (e - mean) * (e - mean) + (h - mean) * (h - mean) + (t - mean) * (t - mean) + (100 - mean) * (100 - mean)) / 6
         sdBig = INT(SQR(varBig) + .5)
         IF sdBig * sdBig = varBig THEN
          PRINT "Sylvia Simon  Seth  Sally Shawn Steven mean   sd1  sd2"
          PRINT USING "######"; 100; i; e; a; h; t; mean; sd; sdBig
         END IF
        END IF
       END IF
      END IF
     END IF
    END IF

Sylvia Simon  Seth  Sally Shawn Steven mean   sd1  sd2
   100    60    58    76    66    30    65     7    21

  Posted by Charlie on 2008-04-18 15:39:16
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