Question

To estimate the percentage of a state's voters who support the current

Answer 1

Answer:

The Herald's estimate is likely closest to the actual  percentage of voters who support the governor for reelection.

Step-by-step explanation:

The law of large numbers, in probability concept, states that as we increase the sample size, the mean of the sample ($\bar x$) approaches the whole population mean ($\mu$).

That is, as n → ∞, $\bar x$ → $\mu$.

In this case we are estimating the  percentage of a state's voters who support the current governor for reelection.

A Normal approximation to binomial can be applied to approximate the distribution of p if the following conditions are satisfied:

• $n\hat p \geq 10$
• $n (1-\hat p) \geq 10$

The sample selected by all the three newspaper are quite large.

Check if Normal approximation to binomial can be applied to approximate the distribution of p for all three.

• For n = 1000 and $\hat p$ = 0.61 check the conditions as follows:

       $n\hat p=1000\times 0.61=610>10\\n(1-\hat p)=1000\times (1-0.61)=390>10$

       Thus, $p\sim N(n\hat p,n\hat p(1-\hat p))=N(610, 237.9)$.

• For n = 800 and $\hat p$ = 0.57 check the conditions as follows:

       $n\hat p=800\times 0.57=456>10\\n(1-\hat p)=800\times (1-0.57)=344>10$

       Thus, $p\sim N(n\hat p,n\hat p(1-\hat p))=N(456, 196.08)$

• For n = 600 and $\hat p$ = 0.71 check the conditions as follows:

       $n\hat p=600\times 0.71=426>10\\n(1-\hat p)=600\times (1-0.71)=174>10$

       Thus, $p\sim N(n\hat p,n\hat p(1-\hat p))=N(426, 123.54)$

So, for all the three sample, the proportions follows a Normal distribution.

Now, according to the law of large numbers, as the sample size increases the sample mean approaches the population mean. In case of proportions, the sample mean is the sample proportion itself.

So, as the sample size increases the sample proportions approaches the true proportion value.

The Herald has the largest sample size, i.e. n = 1000, so their estimate of the percentage of voters in their  sample who support the governor would closest to the actual percentage.

Thus, The Herald's estimate is likely closest to the actual  percentage of voters who support the governor for reelection.