Respuesta :
Answer:
(a) Hence, the margin of error reported by The Marist Poll was correct.
(b) Based on a 95% confidence interval the poll does not provide convincing evidence that more than 70% of the population think that licensed drivers should be required to retake their road test once they turn 65.
Step-by-step explanation:
We are given that the Marist Poll published a report stating that 66% of adults nationally think licensed drivers should be required to retake their road test once they reach 65 years of age.
It was also reported that interviews were conducted on 1,018 American adults, and that the margin of error was 3% using a 95% confidence level.
(a) Margin of error formula is given by;
Margin of Error = [tex]Z_(_\frac{\alpha}{2}_) \times \sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex]
where, [tex]\alpha[/tex] = level of significance = 1 - 0.95 = 0.05 or 5%
Standard of error = [tex]\sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex]
Also, [tex]\hat p[/tex] = sample proportion of adults nationally think licensed drivers should be required to retake their road test once they reach 65 years of age = 66%
n = sample of American adults = 1.018
The critical value of z for level of significance of 2.5% is 1.96.
So, Margin of Error = [tex]Z_(_\frac{\alpha}{2}_) \times \sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex]
= [tex]1.96 \times \sqrt{\frac{0.66(1-0.66)}{1,018} }[/tex] = 0.03 or 3%
Hence, the margin of error reported by The Marist Poll was correct.
(b) Now, the pivotal quantity for 95% confidence interval for the population proportion who think that licensed drivers should be required to retake their road test once they turn 65 is given by;
P.Q. = [tex]\frac{\hat p-p}{ \sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex] ~ N(0,1)
So, 95% confidence interval for p = [tex]\hat p \pm \text{Margin of error}[/tex]
= [tex]0.66 \pm 0.03[/tex]
= [0.66 - 0.03 , 0.66 + 0.03]
= [0.63 , 0.69]
Hence, based on a 95% confidence interval the poll does not provide convincing evidence that more than 70% of the population think that licensed drivers should be required to retake their road test once they turn 65 because the interval does not include the value of 70% or more.
Using the information given, we have that
a) Applying the formula, it is found that the margin of error is of 3%.
b) The confidence interval is (0.63, 0.69), which is entirely below 70%, which means that is does not provide convincing evidence that more than 70% of the population think that licensed drivers should be required to retake their road test once they turn 65.
Item a:
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]\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 z-score that has a p-value of [tex]\frac{1+\alpha}{2}[/tex].
The margin of error is of:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
95% confidence level
So [tex]\alpha = 0.95[/tex], z is the value of Z that has a p-value of [tex]\frac{1+0.95}{2} = 0.975[/tex], so [tex]z = 1.96[/tex].
For this problem, we have that:
- 66% of adults think licensed drivers should be required to retake their road test once they reach 65 years of age, hence [tex]p = 0.66[/tex].
- Sample of 1018 adults, hence [tex]n = 1018[/tex].
The margin of error is of:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]M = 1.96\sqrt{\frac{0.66(0.34)}{1018}}[/tex]
[tex]M = 0.03[/tex]
Item b:
The interval is the sample proportion plus/minus the margin of error, hence:
0.66 - 0.3 = 0.63
0.66 + 0.3 = 0.69.
The confidence interval is (0.63, 0.69), which is entirely below 70%, which means that is does not provide convincing evidence that more than 70% of the population think that licensed drivers should be required to retake their road test once they turn 65.
A similar problem is given at https://brainly.com/question/16807970
