Respuesta :
Answer:
The 90​% confidence interval for the proportion of parents who had children in grades​ K-12 were satisfied with the quality of education the students receive is (0.405, 0.455). 0.42 = 42% is part of the confidence interval, so we are 90% sure that there is no evidence that​ parents' attitudes toward the quality of education have changed.
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]\pi = 462, n = \frac{462}{1075} = 0.4298[/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}} = 0.4298 - 1.645\sqrt{\frac{0.4298*0.5702}{1075}} = 0.405[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.4298 + 1.645\sqrt{\frac{0.4298*0.5702}{1075}} = 0.455[/tex]
The 90​% confidence interval for the proportion of parents who had children in grades​ K-12 were satisfied with the quality of education the students receive is (0.405, 0.455). 0.42 = 42% is part of the confidence interval, so we are 90% sure that there is no evidence that​ parents' attitudes toward the quality of education have changed.
