The test scores for a large statistics class have an unknown

Question

3. The test scores for a large statistics class have an unknown distribution with mean of 70 and standard deviation of 10. Use the Chebyshev's theorem to nd k so that (a) at least 50% of the scores are within k standard deviations of the mean. (b) at most 10% of the scores are more than k standard deviations away from the mean

Answer 1

Chebyshev's inequality is:

P(|X-μ)≥kσ)≤1/k², where μ=mean and σ=standard deviation. σ² is the variance. This means the probability that the data lie at least (or more than) k standard deviations from the mean does not exceed 1/k².

This can also be stated:

P(|X-μ)≤kσ)≥1-1/k². This means the probability that the data are within k standard deviations of the mean is at least 1-1/k².

(a) Given that P≥50%, P≥0.5 for at least 50% of the scores to be within k standard deviations (10k) from the mean 70, then:

0.5=1-1/k², 1/k²=0.5, 1=0.5k², 2=k², k=√2 (about 1.41).

(b) Given that P≤10%, P≤0.1 for at most 10% of the scores to be within k standard deviations of the mean, then:

0.1=1/k², 0.1k²=1, k²=10, k=√10 (about 3.16).