Question

In a recent poll, 500 people were asked if they liked dogs, and 56% said they did. Find the margin of error of this poll, at the 99% confidence level.

Give your answer to three decimals

Answer 1

To calculate the margin of error for a poll at a given confidence level, we can use the formula:

\[
ME = z \times \sqrt{\frac{p(1-p)}{n}}
\]

where:
- \(ME\) is the margin of error,
- \(z\) is the z-score corresponding to the confidence level,
- \(p\) is the sample proportion (as a decimal), and
- \(n\) is the sample size.

### Step 1: Find the values

- The sample size \(n\) is 500.
- The proportion \(p = 56\%\) means \(p = 0.56\).
- The complement of the proportion \(1 - p = 1 - 0.56 = 0.44\).

### Step 2: Find the z-score for 99% confidence level

For a 99% confidence interval, the z-score (z) is approximately 2.576 (you can find this value in a z-table or from standard normal distribution values for a two-tailed test).

### Step 3: Plug the values into the formula

Now we can substitute our values into the margin of error formula:

\[
ME = 2.576 \times \sqrt{\frac{0.56 \times 0.44}{500}}
\]

Calculating the inside of the square root first:

\[
0.56 \times 0.44 = 0.2464
\]

Now, divide by the sample size \(n\):

\[
\frac{0.2464}{500} = 0.0004928
\]

Now, take the square root:

\[
\sqrt{0.0004928} \approx 0.022187
\]

Finally, multiply by the z-score to find the margin of error:

\[
ME = 2.576 \times 0.022187 \approx 0.057228
\]

### Step 4: Rounding

Rounding to three decimal places gives us a margin of error of:

\[
\text{Margin of Error} \approx 0.057
\]

Thus, the final answer for the margin of error at the 99% confidence level is:

\[
\boxed{0.057}
\]