Answer :
To construct a 90% confidence interval for the support of the incumbent candidate, we calculate the margin of error using the z-score for 90% confidence and apply it to the sample proportion of 54%, resulting in a confidence interval of 49.94% to 58.06%.
To construct a 90% confidence interval for the percentage of votes that the incumbent has received in an election, based upon exit polls of 400 voters indicating that 54% of the voters supported the incumbent candidate, we use the following formula for a confidence interval for a population proportion:
Confidence interval = p ± z*(√[p(1-p)/n])
Where:
p is the sample proportion (0.54)
n is the sample size (400)
z is the z-score corresponding to the desired confidence level (1.645 for 90% confidence)
First, we calculate the standard error (SE):
SE = √[p(1-p)/n] = √[0.54(1-0.54)/400] ≈ 0.02469
Next, find the margin of error (ME):
ME = z*SE = 1.645*0.02469 ≈ 0.0406 (or 4.06%)
Finally, apply the margin of error to the sample proportion:
Lower bound = p - ME = 0.54 - 0.0406 = 0.4994 (or 49.94%)
Upper bound = p + ME = 0.54 + 0.0406 = 0.5806 (or 58.06%)
Thus, we estimate with 90 percent confidence that between 49.94 percent and 58.06 percent of all voters supported the incumbent in this election.
Answer:
The 90% confidence interval for the percentage of votes that the incumbent has received in this election is (0.499, 0.581).
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].
For this problem, we have that:
[tex]n = 400, p = 0.54[/tex]
90% confidence level
So [tex]\alpha = 0.1[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.1}{2} = 0.95[/tex], so [tex]Z = 1.645[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.54 - 1.645\sqrt{\frac{0.54*0.46}{400}} = 0.499[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.54 + 1.645\sqrt{\frac{0.54*0.46}{400}} = 0.581[/tex]
The 90% confidence interval for the percentage of votes that the incumbent has received in this election is (0.499, 0.581).