A total of 500 voters are randomly selected in a certain precinct 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 level of confidence, what are the confidence limits for the proportion that plan to vote for the Democratic incumbent

In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which

z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].

Of the 500 surveyed, 350 said they would vote for the Democratic incumbent.

This means that [tex]n = 500, \pi = \frac{350}{500} = 0.7[/tex]

99% confidence level

So [tex]\alpha = 0.01[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.01}{2} = 0.995[/tex], so [tex]Z = 2.575[/tex].

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.7 - 2.575\sqrt{\frac{0.7*0.3}{500}} = 0.6472[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.7 + 2.575\sqrt{\frac{0.7*0.3}{500}} = 0.7528[/tex]

The limits are 0.6472 and 0.7528

ahirohit963 ahirohit963 · Answer 2 · 2021-11-24T13:56:27+01:00

The 99% confidence interval is (0.6472,0.7528) and this can be determined by using the confidence interval formula.

Given :

A total of 500 voters are randomly selected in a certain precinct 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.
0.99 level of confidence.

The confidence interval is given by:

[tex]\rm CI =\hat{p} \pm z\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex] --- (1)

99% confidence level means [tex]\alpha = 0.01[/tex] and the means the value of:

[tex]1-\dfrac{\alpha }{2}=1-\dfrac{0.01}{2} = 0.995[/tex]

So, the p-value is 0.995 and the z value is 2.575.

Given that n = 500 and the value of [tex]\rm \hat{p}[/tex] is :

[tex]\rm \hat {p} = \dfrac{350}{500} = 0.7[/tex]

Now, put the values of known terms in the equation (1).

[tex]\rm CI =0.7 \pm 2.575\sqrt{\dfrac{0.7(1-0.7)}{500}}[/tex]

[tex]\rm CI =0.7 \pm 2.575\sqrt{\dfrac{0.7\times 03}{500}}[/tex]

So, the lower limit is given by:

[tex]\rm 0.7 - 2.575\sqrt{\dfrac{0.7\times 0.3}{500}} = 0.6472[/tex]

And the upper limit is given by:

[tex]\rm 0.7 + 2.575\sqrt{\dfrac{0.7\times 0.3}{500}} = 0.7528[/tex]

So, the 99% confidence interval is (0.6472,0.7528).

For more information, refer to the link given below:

https://brainly.com/question/23017717