To find the confidence interval for the percentage of Americans that own a dog, Paulo needs to use the formula for the margin of error (E) in a confidence interval. The margin of error can be computed as follows:
[tex]\[
E = z \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
\][/tex]
where:
- [tex]\(\hat{p}\)[/tex] is the sample proportion,
- [tex]\(n\)[/tex] is the sample size, and
- [tex]\(z\)[/tex] is the z-score corresponding to the desired confidence level.
In Paulo's case:
- [tex]\(\hat{p} = 0.452\)[/tex]
- [tex]\(n = 950\)[/tex]
- [tex]\(z = 1.645\)[/tex] (for a 90% confidence level)
Firstly, calculate the standard error (SE):
[tex]\[
\text{SE} = \sqrt{\frac{0.452 \cdot (1 - 0.452)}{950}}
\][/tex]
Upon calculation, the standard error is found to be approximately:
[tex]\[
\text{SE} \approx 0.01614721745623522
\][/tex]
Next, calculate the margin of error (E):
[tex]\[
E = 1.645 \cdot 0.01614721745623522 \approx 0.02656217271550694
\][/tex]
Now, the confidence interval can be determined using [tex]\(\hat{p} \pm E\)[/tex]:
Lower bound:
[tex]\[
\hat{p} - E = 0.452 - 0.02656217271550694 \approx 0.4254378272844931
\][/tex]
Upper bound:
[tex]\[
\hat{p} + E = 0.452 + 0.02656217271550694 \approx 0.47856217271550694
\][/tex]
Therefore, the 90% confidence interval for the percentage of Americans who own a dog is approximately between [tex]\(42.5\%\)[/tex] and [tex]\(47.9\%\)[/tex].
Among the given options, the correct conclusion is:
[tex]\[
\text{"90\% confidence that between 42.5\% and 47.9\% Americans own a dog."}
\][/tex]
