VI. Additional Applications

This section applies the reasoning and methods developed up to this point to specific applications. ActivStats tries to direct students' attention away from memorizing different methods and toward the similarities in reasoning and form of these methods. As much as possible, each method uses the Density tool in the same way with only differences in the formula for the standard error.

21. Testing Differences in Means

We consider tests and intervals for differences between means. The natural null pattern here is that there is no difference.

A video shows a laboratory that measures nutrition components of hot dogs and students begin to analyze the sodium levels of two kinds of hot dogs by making boxplots in Data Desk.

Students test whether there is a difference in sodium levels using a two-sample t test and the Density tool. An exposition discusses the assumptions required for this test and finds them to be few.

Finally, students return to Data Desk to perform the test there.

We recall the video on whether a restricted diet prolongs life and use a confidence interval from a two-sample t procedure to estimate the change in lifetimes. Another visit to Data Desk shows how to compute these confidence intervals for any data.

We then introduce pooled-variance methods for both testing and confidence intervals.

When we are willing to assume that the variances of two populations are equal, we can use a pooled t procedure to test whether the means of the two populations are equal. However, when we have a computer to do the calculations, two-sample t procedures are probably a better choice.

The density tool lets us compare the two methods of assessing differences between means, and we can conclude that in most situations the two-sample method gives up very little in return for requiring fewer assumptions.

Teacher's Notes: This lesson should be a straightforward application of methods and reasoning developed in the previous section. Using the same density tool with only a change in standard error formula reinforces the insight that the reasoning of inference is the same for different tests.

The bottom line for this lesson is that in testing differences between means of independent populations using computers, students should use the two-sample methods by default. Computers calculate the degree of freedom approximation, which makes this method preferable. Students should opt for pooled-t methods only when sample sizes are markedly different and variances are plausibly identical. Students need to be able to recognize paired data because even though they may look similar, two-sample tests are inappropriate for paired data.

22. Inference for Proportions

We recall the investigation of the sampling distribution of the mean, repeat that idea with proportions, and discover that, for large enough samples, proportions also have a Normal sampling distribution.

The sampling distribution of the sample proportion has a mean of p and a standard deviation of (p(1-p)/n).

We then find results that correspond to all that we have done for means, finding confidence intervals for population proportions and testing hypotheses about their value. Finally, we consider differences between proportions.

Teacher's Notes: This lesson extends students' inference abilities to a common kind of data. It is easy to find current examples on the Internet, so this is a good place to have students search for recent surveys and the like and interpret the conclusions, which are usually presented as percentages.

Proportions present one new challenge. Because the standard deviation of the sampling distribution depends on the mean, we find that hypothesizing a value for the proportion also hypothesizes a value for the standard deviation, leading us to work as if we knew the standard deviation. Conversely, when finding confidence intervals, we must rely on the estimated proportion both in estimating the center of the sampling distribution and in estimating the standard deviation of that distribution, which is now a standard error.

Once again, the underlying message is that all of these methods work in essentially the same way.

23. Contingency Tables and Chi Square

A dataset about an "incident" in which some children died introduces the question of whether the boys and girls had an equal chance of dying. We examine the data in a contingency table and test the difference in the two proportions, finding that we cannot reject the null hypothesis.

The dataset returns with data on women added to the table. Now we can't test for a difference in two proportions because there are three. We consider instead a test for equality of the three proportions and settle on chi square. Standardized residuals are particularly interesting because they provide a path to the chi square test and diagnose whether any cell of the table is particularly important.

To test the equality of several proportions in a row or column of a table, use the chi-square statistic. The chi-square statistic sums the squares of the standardized residuals.

Upon turning the page, we learn that the incident is the sinking of the Titanic., and see a video on the subject. An examination of the data in Data Desk reveals marked class differences in survival rates, and we return to examine these differences with the tool.

We compare what the data might have looked like had survival been independent of class (but with the same margins). We consider samples drawn according to the observed association and note that chi-square grows with sample size.

Teacher's Notes: There is a wealth of information about the Titanic on the Worldwide web. ActivStats' page for this lesson notes only a few interesting pages. Among the questions that students might consider is the quality of the data. A Project (under the PROJ icon in the tool bar) raises some specific questions and links to an appropriate web site for further investigation. This subject is rich enough to support a class discussion or group projects.

ActivStats provides a χ² table tool that itself can teach a number of things. As with the t-table tool, the diagram of the density at the top of the table changes with the df of the selected cell in the table. However, unlike the t family, the χ² densities change shape radically at small degrees of freedom and then proceed to drift to the right for larger degrees of freedom, showing that the expected value of χ² grows with the df. As with the t-table, students can insert new columns for specified P-values.

The tables are all available conveniently from the Appendix in the final lesson of the Lesson Book.

24. Regression with Inference

We start in Data Desk and simulate the sampling distribution of the slope of a regression. We return to the least squares tool, comparing what we already know about finding the least squares line with the kind of table that most statistics programs produce.

Regression slopes are just like means when it comes to statistical inference. We make similar assumptions, and construct confidence intervals and significance tests in much the same way as for means.

A video tells the story of Edwin Hubble's discovery that the universe is expanding, and notes the value of knowing the true slope of the relationship between a galaxy's distance from earth and its velocity away from us. We examine Hubble's original data to find that slope.

We examine the formulas for standard errors in regression, and notice where to find the calculated values in Data Desk's regression tables. Finally, we consider several other regression examples.

Teacher's Notes: This lesson is more an application of the ideas and methods of inference than a thorough discussion of regression analysis. Students spend more time working with Data Desk than with any special visualization tools in ActivStats.