Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
Your $\sigma$ is for a single person, not a $k$ person sample, I guess. Also, $1-\ldots$ cannot be $>1$. –  Hagen von Eitzen Nov 10 '12 at 15:35
    
Corrected the issue with the prob being greater than 1, but I'm not sure how to calculate the mean and variance in this case. –  Slruh Nov 10 '12 at 15:42
add comment

1 Answer

up vote 0 down vote accepted

The expected value for $k'$ is $\frac13 k$ and the variance is $\frac29 k$. Hence the $\mu$ for $\frac{k'}{k}$ is $\frac13$ and standard deviation $\sigma$ for $\frac{k'}k$ is $$\frac{\sqrt{\frac29k}}{k}=\frac13\sqrt{\frac{2}{k}}. $$
Plug this into Chebyshev.

share|improve this answer
    
Thank you! I knew I must have been messing up the mean/variance calculation. –  Slruh Nov 10 '12 at 15:55
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.