The Correct Answer and Explanation is:
The correct answer is 0.097.
The margin of error is a statistic that expresses the amount of random sampling error in a survey’s results. It quantifies the uncertainty of using a sample proportion to estimate the proportion for the entire population. To calculate the margin of error for a proportion, we use the following formula:
Margin of Error (ME) = z* * √[p̂(1-p̂) / n]
In this formula:
- p̂ (p-hat) is the sample proportion.
- n is the sample size.
- z* is the critical value, which is determined by the confidence level.
First, we identify the values from the problem. The sample size n is 100 people. The sample proportion p̂ is 17%, which we write as the decimal 0.17.
Next, we need to find the critical value z* for a 99% confidence level. A 99% confidence level means that 99% of the distribution is centered around the mean, leaving 1% of the probability in the tails of the standard normal distribution. This 1% is split equally between the two tails, so there is 0.5% (or 0.005) in each tail. The z-score that corresponds to a cumulative probability of 0.995 (1 minus 0.005) is approximately 2.576. This is our z* value.
Now, we can plug all the values into the formula:
ME = 2.576 * √[(0.17 * (1 – 0.17)) / 100]
We calculate the part inside the square root first:
ME = 2.576 * √[(0.17 * 0.83) / 100]
ME = 2.576 * √[0.1411 / 100]
ME = 2.576 * √[0.001411]
Taking the square root gives:
ME ≈ 2.576 * 0.03756
Finally, we perform the multiplication:
ME ≈ 0.09676
The question asks for the answer to be rounded to three decimal places. Rounding 0.09676 gives us 0.097.
