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Sagot :
Answer:
The limits are 0.6472 and 0.7528
Step-by-step explanation:
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
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
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