Confidence Intervals

Estimating the Value of a Parameter

The chapter begins with a discussion of estimating the population proportion and then presents a discussion of estimating a population mean. The most important concept for students to understand is the meaning of the level of confidence in a confidence interval. In addition, we recommend that you require students to construct a couple of confidence intervals by hand, but then move away from by hand computation in favor of technology.

Section 9.1 Estimating a Population Proportion

The chapter begins with estimating a population proportion. The reason for this is that estimating a population proportion utilizes the normal model, which is familiar to students. In addition, most of the estimation that students experience in the media is based on proportions.

  • Begin with the discussion of point estimates. It is very important to remind students that point estimates will vary from sample to sample. For this reason, there is a sampling distribution associated with point estimates such as the sample proportion or sample mean. Remind students of the sampling distribution of the sample proportion (Section 8.2).
  • Because statistics such as vary from sample to sample, we prefer to report a range of values along with a level of confidence the range includes the unknown population parameter. It is not a bad idea to allow students to associate higher levels of confidence with wider intervals. For example, I might be 80% confident the proportion of Americans who favor a certain policy is between 0.4 and 0.48. Do you think the interval will be wider or narrower with a higher level of confidence?
  • With the concept of level of confidence introduced, it is now important that students truly understand its meaning. To accomplish this, we strongly encourage you to either use a simulation or applet.
    • The confidence intervals for a proportion applet in StatCrunch is a great tool for students to see that level of confidence refers to the proportion of intervals that capture the unknown parameter.  Change the "Sample size" to 200.  Click "1000 intervals" in this applet.  Any interval that is green is an interval that includes the population proportion; any red interval is an interval that does not include the population proportion.  The population proportion is 0.2, so we would expect 950 of the intervals to capture the population proportion.  In addition, the applet may be used to help students see that intervals that do not capture the population proportion are the result of a sample statistic that is in the tails of the sampling distribution.  To help students see this, consider using the left column titled "Intervals".  Select an interval that is red. The raw data for this sample pops up.  Compute the sample proportion for the sample data.  Show that the sample proportion is more than 2 standard errors from the population proportion of 0.2 (and this is why the interval does not include 0.2).  You might consider drawing a theoretical sampling distribution above the StatCrunch applet output so students can see this result.
    • Now change the confidence level to 0.9.  Ask students to determine the proportion of applets they expect include 0.2.  Now click "1000 intervals" to determine the proportion for 1000 independent random samples that include 0.2.
  • Next, show where the formula for constructing confidence intervals comes from as shown in the text. Use this formula to construct a confidence interval by hand. Emphasize the correct interpretation of the confidence interval and emphasize that that level of confidence refers to the proportion of random samples that will result in a confidence interval that captures the parameter, not the probability the interval captures the parameter.
  • Once the formula is understood, I recommend using technology to demonstrate the relation between level of confidence and width of the confidence interval. I also recommend using technology to demonstrate the role sample size plays in the margin of error.
  • Extra Example Education Relative to other nations, how do fourth graders in the United States rank in terms of reading and math ability? Are they in the bottom 50% or in the top 50%?   In a survey of 700 registered voters in the United States conducted by Conquest Communications Group, 258 correctly answered in the top 50%. Note: By the age of 15, U.S. students drop to the 50th percentile in reading and below the 25th percentile in mathematics.(a) What is the variable of interest in this study? Is it qualitative or quantitative?
    (b) Determine the appropriate 95% confidence interval based on the variable of interest. Interpret the interval.
Section 9.2 Estimating a Population Mean

Now we are ready to estimate a population mean. I do not to cover estimating a population mean under the assumption the population standard deviation is known. While presenting the estimation of a population mean with sigma known has pedagogical merits if one begins the discussion of estimation with the mean because this approach does not have the added difficulty of introducing a new distribution (Student's t-distribution). However, by introducing estimation with proportions first, this pedagogical benefit is not necessary. It is unreasonable to expect to know the population standard deviation without knowing the population mean. For these reasons, I forego introducing estimation of the mean with sigma known.

  • I strongly recommend the use of a simulation to help students see the properties of Student's t-distribution. The following YouTube video shows this simulation.
  • Show how the critical values in Student's t-distribution are found, and show how the critical values of Student's t-distribution approach the critical values of the normal distribution. The larger critical values for smaller degrees of freedom are needed to account for the increased variability associated with using the sample standard deviation as an estimate of the population standard deviation in determining the margin of error.
  • Continue to emphasize the meaning of level of confidence and the interpretation of a confidence interval.
Estimating a Population Standard Deviation

This section is optional and if you are squeezed for time, feel free to skip it without loss of continuity.  Personally, I do not cover this topic.  To estimate a population standard deviation requires that we introduce a new probability distribution – the chi-square distribution.

  • As we suggested for introducing Student’s t distribution, we recommend the use of a simulation to help students see the properties of the chi-square distribution. Guidelines for the simulation may be found in Objective 1.   Emphasize that the chi-square distribution is a skewed right distribution. You may also want to mention that as the degrees of freedom increases, the chi-square distribution becomes symmetric.
  • Illustrate how to find critical values while continuing to emphasize the lack of symmetry in the distribution.
  • The methods of this section are more sensitive to departures from the normality requirement, so be sure to emphasize the verification of model requirements.
  • Continue to emphasize the meaning of level of confidence and the interpretation of a confidence interval.
Putting It Together: Which Procedure Do I Use?

It is my experience that students will have difficulty reading problems and ascertaining which parameter is to be estimated. For this reason, I wrote the Putting It Together section to provide a mix of estimating proportions and means. Students must figure out which parameter is being estimated first, and then employ the correct procedure.

Estimating with Bootstrapping

This section introduces one of the more recent breakthroughs in statistical inference – the ability to estimate a parameter without having access to an underlying probability distribution – such as the normal model. Bootstrapping was devised in 1979 by Bradley Efron of Stanford University and allows one to estimate virtually any parameter through In traditional introductory statistics courses, we discuss how to estimate a proportion, a mean, and sometimes a standard deviation. We do not discuss estimation of a median or quartile, however, because we don’t discuss the statistic’s probability distribution. The bootstrap method can be used to estimate any parameter you want, and the approach is always the same.  

  • Bootstrapping requires the use of statistical software, such as StatCrunch, so that you may illustrate the method manually. Watch the video below to learn about bootstrapping. 
  • We recommend first doing a bootstrap manually by resampling with replacement from sample data. Once the mechanics of the bootstrap method are understood, rely on software’s algorithms or applets to perform bootstraps.
  • If pressed for time, this is still worthwhile to cover. Consider assigning this section’s video for homework and require students to submit their own bootstrap interval.
  • I highly recommend the article by Tim Hesterberg to learn more about Bootstrapping.