To calculate the margin of error for a poll, you can use the formula:
\[ \text{Margin of Error} = z \times \sqrt{\frac{p(1-p)}{n}} \]
where:
- \( z \) is the z-score corresponding to the desired confidence level (for a 95% confidence level, \( z \approx 1.96 \)),
- \( p \) is the proportion of respondents who support candidate A (in decimal form),
- \( n \) is the total number of respondents.
Here, we have:
- \( p = 0.12 \) (12% support for candidate A),
- \( n = 450 \).
First, we need to calculate \( p(1-p) \):
\[ p(1 - p) = 0.12(1 - 0.12) = 0.12 \times 0.88 = 0.1056 \]
Next, we calculate the standard error (SE):
\[ SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.1056}{450}} \approx \sqrt{0.00023467} \approx 0.0153 \]
Now, we can calculate the margin of error:
\[ \text{Margin of Error} = z \times SE = 1.96 \times 0.0153 \approx 0.0300 \]
To express the margin of error as a percentage, we multiply by 100:
\[ \text{Margin of Error} \approx 0.0300 \times 100 \approx 3.0% \]
Thus, the margin of error for this poll, rounded to the nearest tenth of a percent, is:
\[ \text{Margin of Error} = 3.0% \]