Final answer:
A 90% confidence interval for a population proportion is found using the sample proportion, the Z-value for the desired confidence level (1.645 for 90% confidence), and the standard error of the proportion. The margin of error is calculated using this Z-value and the standard error, which then defines the range of the confidence interval.
Step-by-step explanation:
Finding a 90% Confidence Interval for a Population Proportion
To find a 90% confidence interval for a population proportion (p), we use the sample proportion (p') as our point estimate. The sample proportion is calculated by dividing the number of individuals with the characteristic of interest by the total sample size. In this case, the student found that out of 525 randomly selected voters, 347 indicated preference for a certain candidate, so p' = 347/525.
Next, we use the standard error (SE) for a proportion, calculated using the formula SE = sqrt[(p'(1 - p')/n)], and the Z-value for the desired confidence level. For a 90% confidence interval, the Z-value is approximately 1.645 (representing the 90% of the central area under a standard normal distribution, plus 5% in one tail).
The formula for the confidence interval is:
p' ± Z*(SE)
Plugging in our values,
SE = sqrt[(347/525)*(1 - 347/525)/525]
Margin of Error (ME) = Z * SE
The 90% confidence interval is then calculated by subtracting and adding the margin of error to the sample proportion.
Margin of Error (ME)
The margin of error is half the width of the confidence interval and gives us an idea of how precise our estimate is. It depends on both the size of the sample and the variance within the sample.