Overview
This lecture explains confidence intervals: what they are, how to construct them (especially 95% and 99% intervals), how to interpret them, and how to adjust confidence levels.
What is a Confidence Interval?
- A confidence interval gives a range of plausible values for a population parameter, not just a single point estimate.
- Confidence intervals account for sample variability by providing a range instead of a single value.
- Using a confidence interval increases the likelihood of capturing the true population value.
Constructing a 95% Confidence Interval
- For a normally distributed point estimate, use: point estimate ± 1.96 × standard error.
- In the example, the sample proportion supporting solar energy was 0.887 with a standard error of 0.01.
- The 95% confidence interval: 0.887 ± 1.96 × 0.01 = 0.8674 to 0.9066.
- Interpretation: "We are 95% confident that the actual proportion is between 86.74% and 90.66%."
Changing Confidence Levels
- Confidence level is the percentage of intervals expected to capture the parameter in repeated sampling (e.g., 95%, 99%).
- A 90% confidence interval is narrower than a 95% interval; a 99% interval is wider.
- 99% confidence interval formula: point estimate ± 2.58 × standard error.
Example: 99% Confidence Interval for Wind Power Support
- Sample proportion supporting wind turbines: 0.848; standard error: 0.0114.
- 99% confidence interval: 0.848 ± 2.58 × 0.0114 = 0.8186 to 0.8774.
- Interpretation: "We are 99% confident that the true proportion is between 81.9% and 87.7%."
Interpreting Confidence Intervals
- Confidence intervals only relate to the population parameter, not individual data points or future samples.
- The confidence level refers to the expected proportion of intervals capturing the parameter across many samples.
- Do not describe the confidence level as the probability that the interval contains the parameter.
Key Terms & Definitions
- Confidence Interval — a range of values likely to contain the population parameter.
- Point Estimate — a single value (e.g., sample mean or proportion) used to estimate a population parameter.
- Standard Error — the estimated standard deviation of a sampling distribution.
- Confidence Level — the proportion of confidence intervals that would contain the parameter if repeated samples were taken.
- Z-star (Z*) — critical value from the normal distribution associated with the desired confidence level.
Action Items / Next Steps
- Practice constructing confidence intervals for different confidence levels.
- Avoid using the word "probability" when interpreting confidence intervals in assignments or exams.