Question

In the Survey Accuracy problem, 49% of the respondents said they planned to vote for Barak Obama (V=0.49). If respondents answer truthfully, what is P(V < 0.49)? (Write your answer to the closet hundredth)

Answer 1

Let X_i = 1 , answer truthfully

= 0 , otherwise

Let $\hat{p}$ be the estimator to estimate p where p is proportion of respondents planned to vote for Obama.

Here p = 0.49

$\hat{p}$ is the estimator of sample proportion of those answer truthfully . $\hat{p}$ =  $\sum X_{i}/n$ ; n is the sample size.

E($\hat{p}$) = p ;  $\hat{p}$ is unbiased.

Assumption that sample size (n) is very large then

Z = ($\hat{p}$ - p ) $\div$ $\sqrt{Var(\hat{p})}$ asymptotically follows N(0,1) .

Here V is $\hat{p}$ . Then P [ V $\leq$ 0.49 ] = P [ $\hat{p}$ $\leq$ 0.49 ]

= P [ $\hat{p}$ $\leq$ p ]

= P [ ( $\hat{p}$ - p ) $\leq$ 0 ]

= P [ Z $\leq$ 0 ]

= 1 / 2 = 0.5

Ans : 50 %