Question

There are 2,000 eligible voters in a precinct. A total of 500 voters are randomly selected and asked whether they plan to vote for the Democratic incumbent or the Republican challenger. Of the 500 surveyed, 350 said they would vote for the Democratic incumbent. Using the 0.99 confidence coefficient, what are the confidence limits for the proportion that plan to vote for the Democratic incumbent?a. 0.647 and 0.753 b. 0.612 and 0.712 c. 0.397 and 0.797 d. 0.826 and 0.926

Answer 1

`0.7 - 2.58 \sqrt{(0.7(1-0.7))/(500)}=0.647`

`0.7 + 2.58 \sqrt{(0.7(1-0.7))/(500)}=0.753`

And the 99% confidence interval would be given (0.647;0.753).

So the correct answer would be:

a. 0.647 and 0.753

Explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The population proportion have the following distribution

`p \sim N(p,\sqrt{(p(1-p))/(n)})`

Solution to the problem

The estimated population proportion for this case is:

`\hat p = (350)/(500)=0.7`

The confidence interval would be given by this formula

`\hat p \pm z_(\alpha/2) \sqrt{(\hat p(1-\hat p))/(n)}`

For the 99% confidence interval the value of
`\alpha=1-0.99=0.01` and
`\alpha/2=0.005`, with that value we can find the quantile required for the interval in the normal standard distribution.

`z_(\alpha/2)=2.58`

And replacing into the confidence interval formula we got:

`0.7 - 2.58 \sqrt{(0.7(1-0.7))/(500)}=0.647`

`0.7 + 2.58 \sqrt{(0.7(1-0.7))/(500)}=0.753`

And the 99% confidence interval would be given (0.647;0.753).

So the correct answer would be:

a. 0.647 and 0.753