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The margin of error is a measure of how confident you can be that the true population parameter lies within a certain range of the sample statistic. Confidence intervals are a range of values that is likely to contain the true population parameter with a certain degree of confidence. In other words, the margin of error is a measure of precision, while the confidence interval is a measure of accuracy.
Often in statistics we use to estimate the value of a with a certain level of confidence.
Every confidence interval takes on the following form:
Confidence Interval = [lower bound, upper bound]
The margin of error is equal to half the width of the entire confidence interval.
For example, suppose we have the following confidence interval for a population mean:
95% confidence interval = [12.5, 18.5]
The width of the confidence interval is 18.5 – 12.5 = 6. The margin of error is equal to half the width, which would be 6/2 = 3.
The following examples show how to calculate a confidence interval along with the margin of error for several different scenarios.
Example 1: Confidence Interval & Margin of Error for Population Mean
We use the following formula to calculate a confidence interval for a population mean:
Confidence Interval = x +/- z*(s/√n)
where:
- x: sample mean
- z: the z-critical value
- s: sample standard deviation
- n: sample size
Example: Suppose we collect a random sample of dolphins with the following information:
- Sample size n = 40
- Sample mean weight x = 300
- Sample standard deviation s = 18.5
We can plug these numbers into the to find the 95% confidence interval:
The margin of error would be equal to half the width of the confidence interval, which is equal to:
Margin of Error: (305.733 – 294.267) / 2 = 5.733.
Example 2: Confidence Interval & Margin of Error for Population Proportion
We use the following formula to calculate a confidence interval for a population proportion:
Confidence Interval = p +/- z*(√p(1-p) / n)
where:
- p: sample proportion
- z: the chosen z-value
- n: sample size
Example: Suppose we want to estimate the proportion of residents in a county that are in favor of a certain law. We select a random sample of 100 residents and ask them about their stance on the law. Here are the results:
- Sample size n = 100
- Proportion in favor of law p = 0.56
We can plug these numbers into the to find the 95% confidence interval:
The 95% confidence interval for the true population proportion is [.4627, .6573].
The margin of error would be equal to half the width of the confidence interval, which is equal to:
Margin of Error: (.6573 – .4627) / 2 = .0973.