12 10 As you plan your next by checking out you come across

12. 10 As you plan your next by checking out you come across the following historie information about Dr. grade distribution: 78 and m 6. (a) Ira 90 or above is an A, use Tchebysheff inequality to determine the maximum percent of students a who could get an A. b) If a To or above is passing, use Tchebysheff i ality to determine the minimum percent of students who will pass

Solution

A.

Getting the distance from the mean in terms of standard deviations,

k = (X - u)/s = (90 - 78)/6 = 2

Thus, k = 2 standard deviations from the mean.

As 1 - 1/k^2 is within k standard deviations from the mean, then

1 - 1/2^2 = 3/4

are within 2 standard deviations.

The remaining 1/4 is cut into 2, the lower ones and the higher ones.

Thus, (1/4)/2 = 1/8 = 12.5% are higher than 90. [ANSWER, 12.5%]

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B.

Getting the distance from the mean in terms of standard deviations,

k = (X - u)/s = (70 - 78)/6 = -1.3333

Thus, k = 1.3333 standard deviations from the mean.

As 1 - 1/k^2 is within k standard deviations from the mean, then

1 - 1/1.3333^2 = 0.4375

are within 1.3333 standard deviations.

Thus, those outside this is 1 - 0.4375 = 0.5625.

Half of those are those who will fail, or less than 70.

Thus, those who fail is

0.5625/2 = 0.28125

Therefore, those who pass are

1 - 0.28125 = 0.78175 or 78.175% [ANSWER, 78.175%]