PLEASE HELP A survey was given to a random sample of 25 voters in the United States to ask about their preference for a presidential candidate.
Of those surveyed, 18 respondents said that they preferred Candidate A. At the 95% confidence level, what is the margin of error for this survey expressed as a proportion to the nearest thousandth?
To calculate the margin of error for a survey expressed as a proportion, you need to understand the formula for the margin of error. In this case, you are given the number of respondents who preferred Candidate A, which is 18 out of the total sample size of 25.
The margin of error formula for proportions is:
Margin of Error = critical value * square root of [(p * (1 - p)) / n]
In this formula:
- The critical value corresponds to the desired confidence level. For a 95% confidence level, the critical value is approximately 1.96.
- "p" represents the proportion of respondents who preferred Candidate A. It is calculated by dividing the number of respondents who preferred Candidate A (18) by the total sample size (25). Therefore, p = 18/25 = 0.72.
- "n" represents the sample size, which is given as 25.
Now, let's substitute the values into the formula:
Margin of Error = 1.96 * √[(0.72 * (1 - 0.72)) / 25]
Calculating this expression will give you the margin of error as a proportion. Let's do the calculation:
Margin of Error = 1.96 * √[(0.72 * 0.28) / 25]
= 1.96 * √(0.2016 / 25)
= 1.96 * √0.008064
≈ 1.96 * 0.090
Rounding to the nearest thousandth, the margin of error for this survey expressed as a proportion is approximately 0.176.