The Concept of Confidence Intervals

The concept of confidence in a confidence interval (CI) is subtle because it depends on thinking about taking all possible samples from a known population distribution, computing the CI for each sample, and then determining if the CI captures the parameter or not. This thought process is complicated by the fact that the parameter is unknown, and taking all possible samples is impossible. The ciSim() function (with no arguments) can simulate this process and provide a graphical result of the CIs relative to the parameter from many (but not all possible) samples.

The plot below is an example from ciSim(). The known population mean is depicted by the solid vertical line (μ=100 in this example). Each horizontal line is one CI (the endpoints of the horizontal line is the CI and the point in the middle is the sample mean). The CIs that do not capture μ are in red. The percentage of CIs that capture μ is shown at the top along with the margin-of-error (m.e.), or half the width of the CI.

In RStudio, this plot will have a gear icon in the upper-left corner that will open a dialog box that allows you to modify n, the level of confidence, and the type of confidence region. Pressing the “rerandomize” button will construct another set of CIs. The plot updates automatically when any item is changed.

The table below can be used to record your results from ciSim() for different sample sizes and levels of confidence (press “rerandomize” twenty times and mentally average the percentages to form a general conclusion.). Then answer the questions below the table.

90% Confidence 95% Confidence 99% Confidence
n= % contain μ m.e. % contain μ m.e. % contain μ m.e.
10            
25            
50            
  1. How did the percentage of CIs that captured μ change with increasing n.
  2. How did the percentage of CIs that captured μ change with increasing level of confidence.
  3. How did the margin-of-error change with increasing n.
  4. How did the margin-of-error change with increasing level of confidence.
  5. What two things can you, as a statistician, do to make a confidence interval narrower (i.e., reduce the margin-of-error)? Are there any negative consequences to either of decision?


Confidence Region Calculations

follow these steps when answering all confidence region questions:
  1. Identify the level of confidence (i.e., 100(1-α)%).
  2. Identify whether a lower confidence bound (greater than HA), upper confidence bound (less than HA), or confidence interval (not equals HA) should be constructed.
  3. Find Z* (Include a drawing to illustrate your calculation and your distrib() code).
  4. Calculate the confidence region values (i.e., x̄+Z*SE).
  5. Specifically interpret the confidence region with a complete sentence.

See the last section of the module reading and here for a demonstration of these steps.


For each situation below, calculate and interpret the confidence region.

  1. α=0.10, HA: μ>75, n=50, σ=12, x̄=79.5.
  2. α=0.05, HA: μ≠14, n=25, σ=6, x̄=11.2.
  3. α=0.01, HA: μ>880, n=80, σ=90, x̄=918.
  4. α=0.10, HA: μ<15000, n=50, σ=8000, x̄=13700.