Overview
This lecture covers how confidence intervals change with confidence levels and their relationship with margin of error, using examples and analogies for clarity.
Confidence Intervals & Width
- Confidence interval is the range likely to contain the population parameter, given a confidence level.
- The width of a confidence interval equals the upper value minus the lower value.
- For the Arnold Schwarzenegger example at 95% confidence, the interval was 0.546 to 0.584, resulting in a width of 0.038.
- The sample proportion lies at the center of the confidence interval.
- Margin of error is half the width of the confidence interval.
Margin of Error Calculation
- Margin of error = (Upper value – Lower value) ÷ 2.
- For the 95% confidence interval, margin of error is 0.019 or 1.9%.
- Changing confidence level to 80% (with other factors constant) gives a narrower interval: 0.553 to 0.577.
- Students are tasked to compute width and margin of error for this new (80%) interval.
Confidence Level vs. Error
- Increasing confidence level widens the interval, thus increasing the margin of error.
- Higher confidence means a greater chance of capturing the parameter, but allows more error.
- Lower error requires a narrower interval, which reduces confidence.
- There's a trade-off: higher confidence raises error and vice versa.
Key Terms & Definitions
- Confidence Interval — Range where the true population parameter is expected to lie, given a confidence level.
- Confidence Level — The probability (%) that a confidence interval contains the true parameter.
- Margin of Error — Half the width of the confidence interval, representing the maximum expected error.
- Width — The distance between the upper and lower bounds of the confidence interval.
Action Items / Next Steps
- Calculate the width and margin of error for the 80% confidence interval (0.553 to 0.577).
- Prepare for the next section on achieving both high confidence and low error in statistical studies.