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Confidence Intervals
Estimating with Margin of Error
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A confidence interval gives a range of values that are consistent with sample data and likely to contain a population parameter. Instead of reporting only one estimate, it shows both the estimate and the uncertainty around it. This makes statistical conclusions more informative and helps students judge how precise a result is. Confidence intervals are widely used in science, medicine, polling, and engineering.
A typical confidence interval is built from a sample statistic plus or minus a margin of error. The margin of error depends on the variability in the data, the sample size, and the chosen confidence level such as 95%. Larger samples usually produce narrower intervals because they reduce sampling variability. A confidence interval does not say the parameter moves around, but rather that the method used will capture the true parameter in a stated proportion of repeated samples.
Key Facts
- General form: estimate ± margin of error
- For a mean with known population standard deviation: x̄ ± z*σ/√n
- Standard error of the mean: SE = σ/√n or approximately s/√n
- Margin of error: ME = critical value × standard error
- Common 95% critical value for a normal model: z* ≈ 1.96
- Increasing n decreases SE because SE is proportional to 1/√n
Vocabulary
- Confidence interval
- A range of values calculated from sample data that is used to estimate a population parameter.
- Sample statistic
- A numerical summary from a sample, such as a sample mean or sample proportion.
- Population parameter
- A fixed but usually unknown value that describes a population, such as the true mean.
- Margin of error
- The amount added to and subtracted from an estimate to create the interval.
- Standard error
- The typical amount a sample statistic varies from sample to sample.
Common Mistakes to Avoid
- Saying a 95% confidence interval means there is a 95% chance the true parameter is in this specific interval, which is wrong because the parameter is fixed and the 95% refers to the long run success rate of the method.
- Using the sample standard deviation s as if it were the population standard deviation σ without checking the formula, which is wrong because this changes whether a z interval or another method is appropriate.
- Thinking a higher confidence level makes the interval narrower, which is wrong because higher confidence requires a larger critical value and therefore a wider interval.
- Ignoring sample size when comparing intervals, which is wrong because larger samples usually give smaller standard errors and more precise intervals.
Practice Questions
- 1 A sample mean is x̄ = 50, the population standard deviation is σ = 12, and the sample size is n = 36. Find the 95% confidence interval for the population mean using z* = 1.96.
- 2 A poll estimates that 42% of voters support a candidate with a margin of error of 3 percentage points. Write the confidence interval and state its lower and upper bounds.
- 3 Two studies estimate the same population mean at 80. Study A reports a 90% confidence interval of 76 to 84, and Study B reports a 95% confidence interval of 74 to 86. Explain which study is more precise and why.