Sure, let's find the Margin of Error for the poll at the 90% confidence level. Here’s a detailed, step-by-step breakdown:
1. Determine the sample size and sample proportion:
- The sample size, [tex]\( n \)[/tex], is 480 people.
- The sample proportion, [tex]\( \hat{p} \)[/tex], is 0.87 (since 87% of the respondents said they liked dogs).
2. Find the Z-score corresponding to the desired confidence level:
- At a 90% confidence level, the Z-score (from the Z-table or standard normal distribution table) is approximately 1.6449.
3. Calculate the standard error of the proportion:
The standard error (SE) of the sample proportion can be calculated using the formula:
[tex]\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\][/tex]
Plugging in the values:
[tex]\[
SE = \sqrt{\frac{0.87 \times (1 - 0.87)}{480}}
\][/tex]
4. Compute the standard error:
[tex]\[
SE = \sqrt{\frac{0.87 \times 0.13}{480}}
\][/tex]
[tex]\[
SE = \sqrt{\frac{0.1131}{480}}
\][/tex]
[tex]\[
SE \approx 0.015350
\][/tex]
5. Calculate the margin of error (MOE):
The margin of error is obtained by multiplying the standard error by the Z-score:
[tex]\[
MOE = Z \times SE
\][/tex]
Substituting the values:
[tex]\[
MOE = 1.6449 \times 0.015350
\][/tex]
6. Compute the margin of error:
[tex]\[
MOE \approx 1.6449 \times 0.015350 = 0.025249
\][/tex]
7. Rounding the margin of error to six decimal places:
The margin of error, rounded to six decimal places, is:
[tex]\[
MOE \approx 0.025249
\][/tex]
Thus, the Margin of Error for this poll at the 90% confidence level is 0.025249 when rounded to six decimal places.
