To calculate the margin of error for a proportion and to interpret it correctly, we'll start by identifying the relevant variables and formulas.
- The sample size \( n = 650 \).
- The sample proportion \( \hat{p} = 0.32 \) (32% favoring reelection).
To find the margin of error for a 90% confidence interval, we can use the formula:
\[ \text{Margin of Error} = z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]
For a 90% confidence interval, the z-score is approximately 1.645.
Calculating the standard deviation (standard error) of the proportion:
\[ \sqrt{\frac{0.32 \times (1 - 0.32)}{650}} = \sqrt{\frac{0.32 \times 0.68}{650}} = \sqrt{\frac{0.2176}{650}} \approx \sqrt{0.000334} \approx 0.0183 \]
Now calculating the margin of error:
\[ \text{Margin of Error} = 1.645 \times 0.0183 \approx 0.0301 \]
Now, we can create the confidence interval:
\[ \hat{p} \pm \text{Margin of Error} \Rightarrow 0.32 \pm 0.0301 \]
Thus, the confidence interval is:
\[ (0.32 - 0.0301, 0.32 + 0.0301) \approx (0.2899, 0.3501) \]
Rounding the interval, we have:
\[ (0.29, 0.35) \]
Therefore, the correct interpretation of the margin of error for the 90% confidence interval is:
With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.29, 0.35).