Respuesta :
Answer:
The 80% confidence interval for the population proportion of new car buyers who prefer foreign cars over domestic cars is (0.23, 0.27).
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].
Suppose a sample of 768 new car buyers is drawn. Of those sampled, 192 preferred foreign over domestic cars.
This means that [tex]n = 768, \pi = \frac{192}{768} = 0.25[/tex]
80% confidence level
So [tex]\alpha = 0.2[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.2}{2} = 0.95[/tex], so [tex]Z = 1.28[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.25 - 1.28\sqrt{\frac{0.25*0.75}{768}} = 0.23[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.25 + 1.28\sqrt{\frac{0.25*0.75}{768}} = 0.27[/tex]
The 80% confidence interval for the population proportion of new car buyers who prefer foreign cars over domestic cars is (0.23, 0.27).
