Question

Three branches 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.

Short answer

a. The confidence interval lie between $(0.3350,0.3850)$

b. Yes, there is convincing evidence that less than half of adults can name all branches of government.

Step-by-step solution

Given Information

It is given that $c=90\%=0.90$

$\hat{p}=0.36,n=1416$

For confidence interval$1-\alpha =0.90$,$z_{a/2}=1.96$using probability table.

Construct and Interpret 90% Confidence Interval

The conditions are:

Random: US adults are selected randomly, it is satisfied.

Normal: Success are $1416(0.36)=509.76$and failure are $1416(1-0.36)=906.24$which are larger than ten.

Independent: $1416$adults is surely less than $10\%$of total population.

All conditions are satisfied.

Now,

Margin of Error: $E=z_{\alpha /2}\times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

$=1.96\times \sqrt{\frac{0.36(1-0.36)}{1416}}=0.0250$

Confidence Interval: $\hat{p}-E=0.6-0.0250=0.3350$

and $\hat{p}+E=0.6+0.0250=0.3850$

hence, interval is$(0.3350,0.3850)$

To check if the interval from part (a) provide convincing evidence that less than half of all U.S. adults could name all three branches of government 

The interval does not have $0.5$. All values are below $0.5$. It shows that population proportion is less than $0.5$. Hence, there is convincing evidence that less than half of adults can name all branches of government.