Suppose we obtain a sample mean of 80 from a sample of size 100 from an infinite population with $\sigma=50$. Then the probability is 0.75 that the error in estimating the population mean using Chebyshev's theorem is:
A) between 30 and 130 .
B) less than 6.
C) between 74 and 86 .
D) less than 10 .

Question

Quizplus · Accepted Answer

Chebyshev's theorem states that for any distribution, the probability that a value lies within k standard deviations of the mean is at least $1 - \frac{1}{k^2}$. Here, the standard error is $\frac{\sigma}{\sqrt{n}} = \frac{50}{10} = 5$. For a probability of 0.75, $1 - \frac{1}{k^2} = 0.75$, solving gives $k \approx 2$. Thus, the error is less than $2 \times 5 = 10$.

Suppose We Obtain a Sample Mean of 80 from a Sample