Question

According to a recent study by the Annenberg Foundation, only 36% of adults in the United States could name all three branches of government. This was based on a survey given to a random sample of 1416 U.S. adults.
a. Construct and interpret a 90% confidence interval for the proportion of all U.S. adults who could name all three branches of government.
b. Does the interval from part (a) provide convincing evidence that less than half of all U.S. adults could name all three branches of government? Explain your answer.

Answer 1

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:

$\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}$

In which:

• $\pi$ 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 $\pi = 0.36$.

• Sample of 1416 adults, hence $n = 1416$.

• 90% confidence level, hence$\alpha = 0.9$, z is the value of Z that has a p-value of $\frac{1+0.9}{2} = 0.95$, so $z = 1.645$.

The lower limit of this interval is:

$\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.36 - 1.645\sqrt{\frac{0.36(0.64)}{1416}} = 0.339$

The upper limit of this interval is:

$\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.36 + 1.645\sqrt{\frac{0.36(0.64)}{1416}} = 0.381$

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 brainly.com/question/15850972