Day 19
Confidence Interval for a Single Proportion



EPSY 5261 : Introductory Statistical Methods

More EPSY Statistics Courses 📈

  • EPSY 5261: Tell your friends! 😎
  • EPSY 5262: Continuation of this course 😃
    • Focus on multiple linear regression & ANOVA
  • EPSY 5221: Measurement (Test design/Validity)
  • EPSY 5244: Survey Design
  • EPSY 8251/52: Advanced Statistical Methods
    • More R Studio work
    • Multiple Regression and Beyond

Learning Goals

At the end of this lesson, you should be able to …

  • Identify when to answer a research question with a confidence interval.
  • Explain the need for creating a confidence interval to do statistical inference.
  • Know how to calculate a confidence interval by hand and using R Studio.
  • Interpret a confidence interval.
  • Explain how the confidence level we choose affects our interval.

Confidence Intervals

  • We have uncertainty in our sample estimates because of sampling variability (i.e., samples vary)
  • We need something to quantify the uncertainty in our estimates.

…

→ Confidence Intervals

Methodology for a Confidence Interval

\[ 95\%~\text{CI} = \text{Sample Statistic} \pm \underbrace{(2 \times SE)}_{\text{Margin of Error}} \] - Adding and subtracting 2 standard errors gives an estimate for the margin of error. - In practice, we determine the exact multiplier used in the margin of error by using the z-distribution (for CI for proportions)

Use z-Distribution

Assumptions

  • The values in the population can only take on 2 categories (e.g., “yes”, “no”).
  • The values in the population are independent from each other.
  • Sample sizes for both groups is greater than 10

\[ n(\hat{p}) > 10 \qquad n(1-\hat{p}) > 10 \]

Formula

\[ \text{CI} = \hat{p} \pm (z^* \times \text{SE}) \]

Table 19.1 in Textbook

Formula (Update)

\[ \text{CI} = \hat{p} \pm \bigg(z^* \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\bigg) \]

What is \(z^*\)?

  • Recall the z-distribution (same one as used for z-test).
    • We need to know our degrees of freedom (df)
  • We will use this to find the \(z^*\) value based on the desired confidence level.

Example: \(z^*\) for a 95% Confidence Interval

Interpretation

  • When interpreting a CI you need to include:
    • Confidence level
    • Population parameter
    • Interval Estimate

Example: We are 95% confident that the proportion of people that are left handed is between 0.08 and 0.12.

Confidence Interval for a Single Proportion Activity

Summary

  • For a research question asking for an estimate, the best way to answer is with a confidence interval.
  • The confidence interval allows us to account for uncertainty by including sampling variability in our estimate of the parameter.
  • With a higher confidence level we expect a larger confidence interval (more uncertainty in the estimate).