Confidence Intervals
Estimating with Margin of Error
Interval estimates
A confidence interval gives a plausible range of values for a population parameter based on sample data. It combines a point estimate with a margin of error that reflects sampling variability.
Common confidence interval form
| Parameter | General form | Interpretation |
|---|---|---|
| Population mean | Estimate of the true mean | |
| Population proportion | Estimate of the true proportion |
Confidence level
A 95% confidence interval method will capture the true parameter in about 95% of repeated samples, under the method's assumptions.
Building and interpreting a confidence interval
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Step 1: Identify the parameter of interest.
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Step 2: Compute the point estimate and standard error.
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Step 3: Choose the critical value for the desired confidence level.
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Step 4: Form the interval estimate.
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Step 5: Interpret the interval in context, not as a probability about the fixed parameter.
Wider or narrower intervals?
Wider interval
- Higher confidence level
- Smaller sample size
- Greater variability
Narrower interval
- Lower confidence level
- Larger sample size
- Less variability
What happens to a confidence interval when sample size increases, assuming everything else stays the same?
Larger samples reduce standard error, which narrows the interval.
Correct answer: It usually becomes narrower
What is the main practical meaning of a 95% confidence interval?
The interpretation is about the long-run performance of the method, not a probability that the parameter lies in one particular interval.
Correct answer: It is a method that would capture the true parameter in about 95% of repeated samples.