Respuesta :
Using the z-distribution, we have that:
a) The 90% confidence interval for the proportion of all U.S. adults who could name all three branches of government is (0.339, 0.381). It means that we are 90% sure that the true population proportion is within these values.
b) The entire confidence interval is below 50%, which means that it provides convincing evidence that less than half of all U.S. adults could name all three branches of government.
What is a confidence interval of proportions?
A confidence interval of proportions is given by:
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which:
- [tex]\pi[/tex] is the sample proportion.
- z is the critical value.
- n is the sample size.
Item a:
For the parameters, we have that:
- 36% of adults in the United States could name all three branches of government, hence [tex]\pi = 0.36[/tex].
- Sample of 1416 adults, hence [tex]n = 1416[/tex].
- 90% confidence level, hence[tex]\alpha = 0.9[/tex], z is the value of Z that has a p-value of [tex]\frac{1+0.9}{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}} = 0.36 - 1.645\sqrt{\frac{0.36(0.64)}{1416}} = 0.339[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.36 + 1.645\sqrt{\frac{0.36(0.64)}{1416}} = 0.381[/tex]
The 90% confidence interval for the proportion of all U.S. adults who could name all three branches of government is (0.339, 0.381). It means that we are 90% sure that the true population proportion is within these values.
Item b:
The entire confidence interval is below 50%, which means that it provides convincing evidence that less than half of all U.S. adults could name all three branches of government.
More can be learned about the z-distribution at https://brainly.com/question/15850972
