Objectives and Concepts

  • Use the seeing-theory tools to develop an intuitive understanding of the meaning of the Frequentist confidence interval.

Background

Repeated Samplilng

At the heart of frequentist inference is the idea of infinite repeated sampling which means that as we repeat an experiment many times, our estimates of population paramers converge upon the true values (which are unknowable).

You’ll examine the repeated sampling idea in the contexts of the sampling distribution and confidence intervals.

Instructions

Confidence Intervals

Navigate to the confidence intervals portion of the seeing theory webpage.

From the dropdown menu, select either the Normal or T distribution.

Keep with the default settings for \(n\) and \(1 - \alpha\) and hit the Start Sampling button.

The simulation generates a set of random numbers from your chosen distribution, calculates their mean, and then creates a confidence interval.

Each confidence interval either contains, or does not contain, the true population mean \(\mu\).

Confidence Level: \(1 - \alpha\)

After you’ve pondered the confidence intervals for a while, take a note of how often the confidence intervals contain \(mu\). Also note the widths of the confidence intervals.

Next, adjust the slider for \(1 - \alpha\) so that you create 50% confidence intervals.

  • Take note of:
    • The widths of the confidence intervals
    • The proportion of times that the confidence intervals contain the true population mean \(\mu\).

Finally, adjust the slider so that you create 99% confidence intervals. Again note the widths of the intervals and the proportion of times that the interval contains the true value.

Sample Sizes \(n\)

Now set the confidence level to 95%.

Let the simulation generate more confidence intervals for a while and take note of the widths.

Next, change n from 5 to 30 and watch what happens to the width.

The width of a confidence interval is a measure of the precision of our estimate of the mean value.

  • Do you recall the difference between accuracy and precision?
  • Can we assess accuracy in using confidence intervals?
    • Note that with a real sample of data the true value of the population mean \(\mu\) is unknown.

Questions

  • Q1 (4 pts.): As a group, list four questions you have about anything related to the course. These can be about R, specific concepts, assignment questions, or anything else you’d like to know!

Get as far as you can on the following, they are not graded. We’ll talk about them in class after you experiment on Seeing Theory.

  • Q2 (0 pts.): What happens to the widths of the confidence intervals as you change the confidence level (\(1 - \alpha\))? Your answer should describe the relative widths of a 90% interval and a 99% interval.
  • Q3 (0 pts.): What happens to the proportion of times the interval contains the true population mean?
  • Q4 (0 pts.): Why do you think the width of the interval changes as you change the confidence level?
  • Q5 (0 pts.): What happens to the widths of the confidence intervals as you increase the sample size from 5 to 30?
  • Q6 (0 pts.): Consider the fact that when we calculate a confidence interval from data, we have no way to know if it contains the true population mean. Given that we don’t know whether our interval contains the value we want to know about, try to describe the intuition we can draw from a confidence interval.