To solve this problem, let's break it down step-by-step and understand the process Paulo followed to create a confidence interval.
1. Sample Proportion and Size:
- The sample proportion ([tex]\( \hat{p} \)[/tex]) is [tex]\(0.452\)[/tex].
- The sample size ([tex]\( n \)[/tex]) is 950.
2. Z-Score for 90% Confidence Level:
- The Z-score corresponding to a 90% confidence level is [tex]\(1.645\)[/tex].
3. Computing the Standard Error:
- The standard error of the sample proportion (SE) is calculated using the formula:
[tex]\[
SE = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}
\][/tex]
Substituting the values:
[tex]\[
SE = \sqrt{\frac{0.452 \cdot (1 - 0.452)}{950}}
\][/tex]
4. Calculating the Margin of Error (E):
- The margin of error is obtained by multiplying the Z-score by the standard error:
[tex]\[
E = 1.645 \cdot SE
\][/tex]
Substituting the standard error:
[tex]\[
E = 1.645 \cdot \sqrt{\frac{0.452 \cdot (1 - 0.452)}{950}}
\][/tex]
After computing, the margin of error ([tex]\( E \)[/tex]) is approximately [tex]\(0.0266\)[/tex].
5. Determining Confidence Interval:
- The lower bound of the confidence interval is given by [tex]\( \hat{p} - E \)[/tex]:
[tex]\[
\text{Lower Bound} = 0.452 - 0.0266 \approx 0.425
\][/tex]
- The upper bound of the confidence interval is given by [tex]\( \hat{p} + E \)[/tex]:
[tex]\[
\text{Upper Bound} = 0.452 + 0.0266 \approx 0.479
\][/tex]
Converting to percentages:
[tex]\[
\text{Lower Bound Percentage} \approx 42.5\%
\][/tex]
[tex]\[
\text{Upper Bound Percentage} \approx 47.9\%
\][/tex]
Based on this analysis, the conclusion we can draw from Paulo's sample data is that it can be said with [tex]\(90\%\)[/tex] confidence that between [tex]\(42.5\%\)[/tex] and [tex]\(47.9\%\)[/tex] of Americans own a dog. Therefore, the correct complete statement is:
It can be said with [tex]$90 \%$[/tex] confidence that between [tex]$42.5 \%$[/tex] and [tex]$47.9 \%$[/tex] of Americans own a dog.
