Day 18
Confidence Interval for a Single Mean



EPSY 5261 : Introductory Statistical Methods

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.

Inferential Methods

  • Hypothesis Testing
    • Answers a yes/no type question
    • Example: Is the average movie length longer than 110 minutes?
  • Confidence Intervals
    • Provides us an estimate taking into account uncertainty
    • Example: How long is the average movie?

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

\[ 95\%~\text{CI} = \text{Sample Statistic} \pm \underbrace{(2 \times SE)}_{\text{Margin of Error}} \] - To get a confidence interval, we add and subtract a specified number of standard errors from the sample statistic.

Determining the SE Multiplier

  • 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 t-distribution (for CI for means)

Use t-Distribution

Assumptions

  • Data comes from a population with a normal distribution.
  • We can proceed if the distribution of the sample looks reasonably normal…OR…
    • If the sample size is large enough (\(>30\); CLT)
    • In practice, better to use a simulation method to get the standard error (then we don’t need to worry about sample size)
  • Independence: must have independent observations

Formula

\[ \text{CI} = \bar{x} \pm (t^* \times \text{SE}) \]

Table 19.1 in Textbook

Formula (Update)

\[ \text{CI} = \bar{x} \pm (t^* \times \frac{\text{SD}}{\sqrt{n}}) \]

What is \(t^*\)?

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

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

Confidence Interval for a Single Mean 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).