Question

The distribution of scores on the test written by a group of students is skewed with the mean of 80 points and standard deviation of 5 points.

According to Tchebysheff’s Theorem, at least 75 % of scores will fall into the interval between _____ points and ______ points.

At least______% of the scores fall between 65 and 95 hours.

At most_____ % of the scores are smaller than 65 points.

At most_____% of the scores are smaller than 65 points or larger than 95 points.

Answer 1

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Concept:

Tchebesheff' Theorm states that within k standard deviations from mean at least 1-1/k^2 %age of area under curve is covered.

So,

1.

75% = 1-1/2^2 * 100% or k =2. So, 80+/- 2*5 = 70 to 90

Answer: At least 75 % of scores will fall into the interval between 70 points and 90 points.

2.

65 and 95 are 80 +/- 3*5. So, 1-1/3^2 * 100% = 88.89%

Answer: At least 88.89% of the scores fall between 65 and 95 hours.

3. 65 is 3 deviation less than 80.

So, < (1-1/k^2) / 2 = (1-1/3^2)*100% / 2 44.445%

Answer: At most 44.445% of the scores are smaller than 65 points.

4.

Again 65 and 95 are 3 deviations from mean.

That would be 1/k^2 = 1/3^2 = 1/9 * 100% = 11.11%

Answer: At most 11.11% of the scores are smaller than 65 points or larger than 95 points.