1
$\begingroup$

Let p be the (unknown) true fraction of voters who support a particular candidate A for office. To estimate p, we poll a random sample of n voters. Let Fn be the fraction of voters who support A among n randomly selected voters.

(a) Using Chebyshev’s inequality, calculate an upper bound on the probability that if we poll 100 voters, our estimate Fn differs from p by more than 0.1. (Hint: you may need to use the fact that x(1 − x) ≤ 1/4 for any 0 ≤ x ≤ 1).

What I have tried :

Chebyshev's Inequality : $$ P(|X-\mu| \geq k\sigma) \leq 1/k^2 $$

So since p is our true fraction of voters : $$ \mu = p $$

Now I know I have to use the 100 voters and the fact p differs from 0.1 but I'm not sure how to fit it into my inequality. Also, I have thought of find the standard deviation but since I can't calculate the $$ Var(X) = E[X^2] - \mu $$since I can't find E[X^2]

$\endgroup$
1
  • 1
    $\begingroup$ Here your $X$ is $F_n$. What is the distribution of $F_n$? Binomial. The moments are known for such distribution. The hint uses $x(1-x)$ should have helped you think about the variance of the Bernoulli distribution $p(1-p)$. $\endgroup$
    – Frostic
    Mar 27, 2018 at 19:12

1 Answer 1

1
$\begingroup$

To use Chebyshev's inequality, you need to know the mean and variance of the random variable (in this case Fn). You are correct that the mean is p since it is the sample proportion which is k/n where k is the number of successes of a binomial (n,p). The variance of Fn is p(1-p)/n. Trouble is we don't know p. So we can estimate it OR take the worst case. This is where the hint comes into play. Estimate the variance by 1/(400) if n=100. So you need to find the k so that k/20=.1 and use it for Chebyshev's inequality.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .