Confidence Intervals
A confidence interval (CI) is a range of values, derived from sample data, that is likely to contain the true population parameter. It provides an estimate of the parameter along with a measure of uncertainty.
Why Use Confidence Intervals?
- Precision: Provides a range rather than a single point estimate, offering more information about the parameter.
- Uncertainty: Quantifies the level of confidence in the estimate.
- Interpretability: Helps in decision-making by considering the possible range of values for a parameter.
Key Components of a Confidence Interval
- Point Estimate: The central value (e.g., sample mean) used as the best estimate of the population parameter.
- Margin of Error: The maximum expected difference between the point estimate and the true parameter.
- Confidence Level: The probability that the interval contains the true parameter, typically expressed as a percentage (e.g., 95%).
Common Confidence Levels:
- 90% Confidence Level: Less precise, smaller margin of error.
- 95% Confidence Level: Most commonly used.
- 99% Confidence Level: More precise, larger margin of error.
Formula for Confidence Interval
For a Population Mean ($\mu$):
\[\text{CI} = \bar{x} \pm Z \cdot \frac{\sigma}{\sqrt{n}}\]Where:
- $\bar{x}$: Sample mean
- $Z$: Z-score corresponding to the confidence level
- $\sigma$: Population standard deviation (or sample standard deviation $s$ if $\sigma$ is unknown)
- $n$: Sample size
For a Population Proportion ($p$):
\[\text{CI} = \hat{p} \pm Z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\]Where:
- $\hat{p}$: Sample proportion
- $Z$: Z-score corresponding to the confidence level
- $n$: Sample size
Z-Scores for Common Confidence Levels
| Confidence Level | Z-Score ($Z$) |
|---|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
Example 1: Confidence Interval for a Mean
A sample of 50 students has a mean test score of 75 with a standard deviation of 10. Calculate the 95% confidence interval for the population mean.
Solution:
\[\text{CI} = \bar{x} \pm Z \cdot \frac{s}{\sqrt{n}}\] \[\text{CI} = 75 \pm 1.96 \cdot \frac{10}{\sqrt{50}}\] \[\text{CI} = 75 \pm 1.96 \cdot 1.414 \approx 75 \pm 2.77\] \[\text{CI} = [72.23, 77.77]\]The 95% confidence interval for the population mean is $[72.23, 77.77]$.
Example 2: Confidence Interval for a Proportion
In a survey of 200 people, 60% said they prefer coffee over tea. Calculate the 95% confidence interval for the population proportion.
Solution:
\[\text{CI} = \hat{p} \pm Z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\] \[\text{CI} = 0.60 \pm 1.96 \cdot \sqrt{\frac{0.60 \cdot 0.40}{200}}\] \[\text{CI} = 0.60 \pm 1.96 \cdot \sqrt{0.0012} \approx 0.60 \pm 1.96 \cdot 0.0346\] \[\text{CI} = 0.60 \pm 0.0679\] \[\text{CI} = [0.532, 0.668]\]The 95% confidence interval for the population proportion is $[0.532, 0.668]$.
Interpretation of Confidence Intervals
A 95% confidence interval means that if we were to take 100 random samples and calculate the CI for each, approximately 95 of them would contain the true population parameter.
Important Notes:
- The CI does not imply that the true parameter has a 95% probability of lying within the interval. The interval either contains the parameter or it doesn’t.
- Larger sample sizes result in narrower confidence intervals (more precision).
- Higher confidence levels result in wider confidence intervals (more certainty).
Applications of Confidence Intervals
- Business: Estimating average customer satisfaction scores.
- Healthcare: Estimating the effectiveness of a new treatment.
- Polling: Predicting election outcomes based on survey results.
Conclusion
Confidence intervals provide a robust method for estimating population parameters with a quantifiable level of uncertainty. Mastering CIs enables better decision-making and clearer communication of statistical findings.
Next Steps: Hypothesis Testing