Basics of Confidence Intervals

8.1: Basics of Confidence Intervals

Point Estimators

  • Definition: A point estimator is a statistic calculated to estimate a population parameter.
    • Example:
    • To estimate the population mean, the sample mean is used as the point estimator: \bar{x}.
    • To estimate the population proportion, the sample proportion is used as the point estimator: \hat{p}.
    • In general, if a parameter is denoted as \theta, the point estimator is denoted as \hat{\theta}.
  • Characteristics:
    • Point estimators are easy to find, but they have drawbacks:
    • With larger sample sizes, estimates are typically better.
    • Point estimators do not provide information on estimate accuracy.

Confidence Intervals

  • Definition: A confidence interval provides an interval estimate around a parameter, which has a probability of being accurate.
  • Structure:
    • General form of a confidence interval:
    • CI = [\hat{\theta} - E, \hat{\theta} + E]
      • Where:
      • \hat{\theta} is the point estimator.
      • E is the margin of error.
  • Interpretation of Confidence Intervals:
    • Statistical Interpretation:
    • States the probability that the confidence interval contains the population parameter.
    • For instance, if a 95% confidence interval is given as 0.65 < p < 0.73:
      • Interpretation: “There is a 95% chance that the interval 0.65 to 0.73 contains the true population proportion.”
      • Relevant fact: Out of 100 confidence intervals calculated, 95 would capture the true population proportion, while 5 would not.
    • Incorrect Interpretation:
      • Saying “there is a 95% chance that the true value of p will fall between 0.65 and 0.73” is inaccurate. The true value is a fixed parameter, and the interval's chance pertains to capturing it.
    • Real World Interpretation:
    • This interpretation is contextual and does not convey probability. It simply informs people of the estimated range for the parameter based on the data.
      • Example: If you find that the mean age of women marrying in 2013 is between 26 and 28, you state that the mean age of women marrying in 2013 is between 26 and 28 years.
  • Confidence Levels:
    • Common confidence levels are 90\%, 95\%, and 99\%.
    • Definitions:
    • Confidence level (C) is related to alpha level (( \alpha )).
      • For two-tailed tests: C = 1 - \alpha.
      • Example: If \alpha = 0.10, then C = 1 - 0.10 = 0.90 or 90\%.
    • For one-tailed tests: The total area considered is 2 * \alpha; thus, for the confidence level,
      • C = 1 - 2\alpha.

Examples and Their Solutions

  • Example Confidence Intervals:
    • For the mean age of women marrying in 2013, the interval is defined but unspecified in the provided text.
    • Statistical Interpretation: There is a 95% chance that the interval contains the mean age women got married in 2013.
    • Real World Interpretation: The mean age that a woman married in 2013 is between 26 and 28 years of age.
    • For the proportion of Americans who have tried marijuana in 2013, the interval is also defined but unspecified in the text.
    • Statistical Interpretation: There is a 99% chance that the interval contains the proportion of Americans who have tried marijuana as of 2013.
    • Real World Interpretation: The proportion of Americans who have tried marijuana as of 2013 is between 0.35 and 0.41.

Impact of Sample Size and Confidence Level on Confidence Intervals

  • Sample Size Influence:
    • As sample size increases, the interval narrows. Larger samples yield closer point estimates to the true population value.
  • Confidence Level Influence:
    • Higher confidence levels result in wider intervals. There is a trade-off between width and precision:
    • Increasing confidence may lead to an imprecise estimation due to a larger margin of error.
    • Conversely, pursuing a more precise measurement results in less confidence in the estimate.

Homework Questions

  1. Compute the effect on a confidence interval when sample size increases from 25 to 50.
  2. Analyze the effect on a 95% confidence interval upon increasing the confidence level to 99%.
  3. Determine the impact on a 95% confidence interval upon decreasing the confidence level to 90%.
  4. Evaluate the outcome on a confidence interval when sample size decreases from 100 to 80.
  5. Given a 95% confidence interval of 6353 \text{ km} < \mu < 6384 \text{ km} (where \mu is the mean diameter of the Earth), state both interpretations.
    • Statistical Interpretation: The mean diameter of the Earth is contained within this interval with 95% confidence.
    • Real World Interpretation: The mean diameter of the Earth lies between 6353 km and 6384 km.
  6. For a Gallup poll in 2013 yielding a 95% confidence interval of 0.52 < p < 0.60 (where p is the proportion of Americans who believe health care is the government’s responsibility):
    • Real World Interpretation: The proportion of Americans who believe it is the government’s responsibility for health care lies between 52\% and 60\%.
    • Statistical Interpretation: There is a 95% chance that the interval contains the true proportion of Americans believing in health care responsibility.

Miscellaneous

  • Visual Representation:
    • Figures accompany text to illustrate the relationship between confidence levels and interval widths as well as sample size effect on confidence intervals.
  • Licensing Note: This page titled 8.1: Basics of Confidence Intervals is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Kathryn Kozak via source content that was edited to the style and standards of the LibreTexts platform.