Disclaimer: This is part of a larger homework question, but I'm getting stuck on one point.
Assume that $n/3$ people voted for a candidate in an election, and in an exit poll $k'/k$ people voted for the candidate. (sample of original population - chosen with replacement).
What is $P[1/6 < k′/k < 1/2]$. I understand that Chebyshev’s inequality needs to be applied, so the answer will be:
$P[1/6 < k′/k < 1/2] = P[|k'/k - \mu| \leq 1/6] \geq 1 - \frac{\sigma^2}{(\frac{1}{6})^2}$
The problem I have relates to the mean and variance of the Bernoulli variables. From the information of the problem, I'm assuming that the mean is $1/3$ since n/3 people voted for the candidate in the election, and then the variance is $1/3*2/3 = 2/9$. When I try to use these values, then the $P[1/6 < k′/k < 1/2] \geq -7$ which makes no sense. What am I misunderstanding?