Confidence Intervals and Statistical Inference

Overview of Statistical Inference

• Statistical inference has two main components:

  • Hypothesis Testing: Formulating null and alternative hypotheses based on the research question, leading to decisions to either reject or not reject the null hypothesis.

  • Estimation: Providing an estimate of the population mean and a likely range, often communicated through confidence intervals.

Confidence Intervals

• Confidence intervals specify boundaries within which we believe the population values will fall.

• They complement p-values, offering a range of values instead of merely a binary decision (reject/don’t reject).

• Derived from the same data, confidence intervals can provide richer information about population means.

Normal Formula for Confidence Intervals

• A standard formula for a 95% confidence interval (CI) is:

  • CI = Mean ± (1.96 × Standard Error)

• The standard error (SE) is calculated as:

  • SE = Standard Deviation / √Sample Size

• Interval Interpretation: A CI indicates there is a 95% probability that it contains the true population mean.

Example: Air Pollution in Suburbs

• Suburb 1:

  • Sample Mean = 230 parts per million (ppm)

  • Standard Deviation = 35 ppm

  • Sample Size = 15

Calculating CI for Suburb 1

1. Calculate Standard Error:

   • SE = 35 / √15

2. Determine lower limit:

   • Lower limit = 230 - (1.96 × SE)

3. Determine upper limit:

   • Upper limit = 230 + (1.96 × SE)

4. Result:

   • 95% CI = [212, 248] ppm

• Interpretation: 95% probability that the mean particulate level lies between 212 and 248 ppm.

Important Note on Interpretation

• Misinterpretation to avoid: "There is a 95% probability that the population mean is within the confidence interval."

• Correct interpretations:

  • "There is a 95% probability that the confidence interval contains the true population mean."

  • "With 95% confidence, the population mean is between the lower and upper limits of the confidence interval."

Level of Confidence and Interval Widths

• Confidence intervals can be adjusted:

  • 99% CI is calculated with:

    • CI = Mean ± (2.58 × SE)

  • A 99% confidence interval will be wider than a 95% CI, reflecting greater certainty.

  • E.g., 99% CI for Suburb 1 could range from [207, 253] ppm.

Impact on Hypothesis Testing

• Constructing confidence intervals for mean differences:

  • Using example data from Suburb 2 and calculating a 95% CI for the mean difference helps indicate significant differences in pollution levels.

• If intervals do not overlap, it suggests differences in means.

• Each CI derived provides specific bounds for true population mean differences.

Relationship Between P-Values and Confidence Intervals

• Both p-values and confidence intervals are derived from the same data and should provide consistent conclusions.

• A p-value informs likelihood based on hypothesis testing while CIs give probable ranges of parameter estimates.

• Limitations of hypothesis testing: It primarily focuses on the null hypothesis and lacks insights on the alternative hypothesis.

Conclusion

• Confidence intervals offer a valuable means of estimating population parameters and, when interpreted correctly, furnish essential insights for decision-making.

• Important not to confuse the interpretations; ensure awareness of how confidence levels impact intervals as well.

• Note that further readings on Bayesian statistics may provide additional perspectives, though not covered in this module.