3. Chebyshev’s theorem states that given any random variable X with mean µ and standard deviation σ, the probability that X lies within k standard deviations from the mean µ is at least 1−1/k 2 . In...

3. Chebyshev’s theorem states that given any random variable X with mean µ and standard deviation σ, the probability that X lies within k standard deviations from the mean µ is at least 1−1/k 2 . In symbols, we write P(µ−kσ ≤ X ≤ µ+kσ) ≥ 1− 1 k 2 .

(a) Suppose X is normal. Compare the results of Chebyshev’s theorem with the 68−95− 99.7 rule.

(b) Suppose Xn is the sampling distribution of the sample mean x for a population with population mean µ and population variance σ. Use Chebyshev’s theorem to construct a confidence interval for µ with confidence level of at least 90%. Use the fact that the expected value of Xn is µ and the standard deviation is σ/ √ n, where n is the size of the sample. Compare the formula found with the one for a 90% confidence interval, where Xn is assumed to be normal.