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, we first need to determine the sample proportion and the critical value.
Step 1: Calculate the sample proportion
The sample proportion (p̂) is calculated by dividing the number of individuals who preferred Candidate A by the total number of individuals surveyed.
p̂ = number of individuals who preferred Candidate A / total number of individuals surveyed
In this case, p̂ = 18/25 = 0.72
Step 2: Calculate the critical value
The critical value is based on the desired confidence level. For a 95% confidence level, we need to find the critical value corresponding to a 2.5% area in the tail of the standard normal distribution.
For a two-tailed test, the critical value is typically found using a standard normal distribution table or a calculator. The critical value at a 95% confidence level is approximately 1.96.
Step 3: Calculate the margin of error
The margin of error can be calculated using the following formula:
Margin of error = Critical value × Standard error
The standard error is calculated by taking the square root of the sample proportion multiplied by the complement of the sample proportion and dividing it by the square root of the sample size.
Standard error = sqrt((p̂ × (1 - p̂)) / n)
In this case, the sample size (n) is 25.
Calculating the standard error:
Standard error = sqrt((0.72 × (1 - 0.72)) / 25) ≈ 0.092
Calculating the margin of error:
Margin of error = 1.96 × 0.092 ≈ 0.18
Therefore, the margin of error for this survey, expressed as a proportion to the nearest thousandth, is approximately 0.180.