Confidence Intervals Lecture

Introduction

• Statistical tests and p-values are not sufficient to conclude analysis.
• We need an estimate of the population parameter based on sample data.
• Point Estimate: A single statistic as the best guess of an unknown quantity (e.g., sample average for population mean).

Variability and Uncertainty

• There's variability, randomness, and uncertainty in the estimate of a population parameter due to sample variability.
• Confidence Interval (CI): A range of plausible values for the population parameter.

Example of a Confidence Interval

• Experiment: Mean pulse rate of students taking a midterm (PM 510).
• Sample: 50 students, two TAs measure pulse rates of 10 students each at random.
• Results:
  • TA1 average pulse rate: 75.2 beats per minute
  • TA2 average pulse rate: 76.1 beats per minute

Form of a Confidence Interval

• CI = Point Estimate ± (Constant * Standard Error)
  • Standard Error: Depends on sample size (n) and variability (standard deviation).
  • Known population variance: Use Z-distribution.
  • Unknown population variance: Use T-distribution.

Confidence Interval Calculation

• Known standard deviation: Use Z value (e.g., 1.95996 for 95% CI).
• Unknown standard deviation: Use T value based on degrees of freedom (n-1).

Example Calculation

• IQ of LA residents: Sample of 7, average IQ = 99.6, population SD = 15.
• CI = 99.6 ± 1.95996 * (15/√7)
• CI Result: [88.5, 110.7]

Interpretation of Confidence Intervals

• We are X% confident that the true population mean falls within the interval.
• Wrong Statement: "There's a X% chance that the mean is between those values."

Hypothesis Testing and Confidence Intervals

• Two-sided hypothesis test is significant if CI does not contain the hypothesized parameter.
• Example: LA IQ vs. National average (100), 95% CI includes 100.

Confidence Intervals with Unknown Population SD

• Use T-distribution with n-1 degrees of freedom.
• TA1 Example: 10 students, mean pulse = 75.2, SD = 2.7
• CI = 75.2 ± 2.26216 * (2.7/√10)
• Result: [73.3, 77.1]

Common Misunderstandings

• CI does not imply a probability that the true mean is within the interval.
• Simulation shows 93% of CIs contain the mean, not exactly 90%.

Conclusion

• Confidence intervals provide a range of values for population parameters with a specified confidence level.
• They aid in understanding variability and uncertainty in statistical estimates.