## Question 819:

Asked on October 15, 2009

Tags:
Empirical Rule ,
Chebyshev's Theorem ,
Midrange

1

## Answer:

No answer provided yet.To answer this question we need to know the definition for many of the terms.

- The score which occurs the most is the mode, which is 81. By definition the mode is the score which occurs the most if a set of data. If the data are normal, the mean and mode are the same.

- Since we were not given the maximum score or minimum score we need to deduce them from what were are give. To find it we can use the mid-range and the range. The mid-range is a statistic that you'll probably never encounter (I haven't in 10 years) but it is just the (max-min)/2. It is a measure of central tendency but is worse than the mean and median in being susceptible to outliers. To find the max and the min values we need to setup two simulatenous equations (remember this from Algebra I?).
- Since the range is 66 and it is the difference between the max and min equation 1 is. Max-Min = 66
- The mid-range is 65 so using the equation (Min+Max) /2 = 65 simplifies to Min+Max = 130, which is our second equation.
- Solving for Max in the 1st equation we get Max = 66 + Min.
- Substituting Max in the second equation we get Min + 66 + Min = 130.
- Simplifying we get 2Min +66 = 130 = 2Min = 64 = Min = 32.
- Now that we know the minimum value is 32, we put that back in the 1st equation for the range = Max -32 = 66 = Max = 98.

- So the maximum score was a 98.
- The minimum score was a 32.
- Chebyshev's Theorem states that the proportion of data that lie within K standard deviations of the mean is at least 1 - 1/K
^{2} where K is the number of standard deviations above 1. So with a mean of 79 and SD of 10, that is 2 standard deviations above and below the mean. Plugging it into the equation we get 1 - 1/2^{2} = 3/4 = at least 75% will score between 59 and 99. 75% of 50 is .75*50 = 37.5 or approximately 38 of the 50 students. - The empirical rule states that approximately 68,95 and 99.7% of data will fall within 1,2 and 3 standard deviations of the mean. Since the mean is 79 the scores of 69 and 89 are 1 standard deviation of the mean, so roughly 68% of students will score between these two scores, or .68*50 = 34 students.