Respuesta :
Answer:
a) The 95% confidence interval for the true proportion of respondents who think current environmental regulations are too strict or not too strict is (0.1678, 0.2122). This means that we are 95% sure that the true proportion of respondents who think current environmental regulations are too strict or not too strict is between 0.1678 and 0.2122.
b) The 99% confidence interval for the true proportion of respondents who think current environmental regulations are too strict or not too strict is (0.1608, 0.2192). This means that we are 99% sure that the true proportion of respondents who think current environmental regulations are too strict or not too strict is between 0.1608 and 0.2192.
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 interval [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 = 1200, \pi = \frac{229}{1200} = 0.19[/tex]
(a) 95% confidence interval.
So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.19 - 1.96\sqrt{\frac{0.19*0.81}{1200}} = 0.1678[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.19 + 1.96\sqrt{\frac{0.19*0.81}{1200}} = 0.2122[/tex]
The 95% confidence interval for the true proportion of respondents who think current environmental regulations are too strict or not too strict is (0.1678, 0.2122). This means that we are 95% sure that the true proportion of respondents who think current environmental regulations are too strict or not too strict is between 0.1678 and 0.2122.
(b) 99% confidence interval.
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}} = 0.19 - 2.575\sqrt{\frac{0.19*0.81}{1200}} = 0.1608[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.19 + 2.575\sqrt{\frac{0.19*0.81}{1200}} = 0.2192[/tex]
The 99% confidence interval for the true proportion of respondents who think current environmental regulations are too strict or not too strict is (0.1608, 0.2192). This means that we are 99% sure that the true proportion of respondents who think current environmental regulations are too strict or not too strict is between 0.1608 and 0.2192.
