Significance Tests for a Mean (Lesson 9.6)
Chapter 9 - Day 7
Use the four-step process to perform a significance test about a population mean.
Use a confidence interval to make a conclusion about a two-sided test for a population mean.
Understand how to wisely use and interpret the results of significance tests.
There has been some recent research which suggests that "normal" body temperature might be changing. We are going to test if this claim is true for high school students by taking the temperature of all the students in class (think of this as a random sample of all high school students). We suggest standing at the door with a fancy forehead thermometer. Inform students of their temperature and have them record it on the whiteboard. Next, enter all the values into the 1 Quantitative Variables applet at www.statsmedic.com/applets. Students will need to know the mean, standard deviation, and sample size for the sample data.
Students should be able to do the full 4-step significance test on the first page without too much support. Note that the activity states that the conditions have been met, so students do not have to check them.
You may choose to debrief page 1 of the activity before moving to page 2, or you can have students do the entire activity at once. On page 2 of the activity, students will for the first time make the strong connection between confidence intervals (Chapter 8) and significance tests (Chapter 9). It turns out that the decision to reject or fail to reject the null hypotheses will be the same whether using a confidence interval or a significance test. The big connection to make is that the confidence level and the alpha level for the significance test are complementary (add up to 1). So a 95% confidence interval that rejects the null hypothesis means that the significance test will also reject the null hypothesis if alpha = 0.05.
Here are some tips for using the four-step process for inference:
Maintain high expectations for what students should be producing. Clearly communicate these expectations and hold them accountable when grading.
Establish patterns of thinking that will help students later. For example, always have students write a general formula first, followed by the specific formula, followed by numbers plugged in, and then a final answer. We will maintain this expectation for all confidence intervals and significance tests in Chapters 8-11.
Don’t reveal calculator commands or applets yet. Of course, all the work of today’s lesson can be done with T-Test on the TI 83/84 calculator or using the One sample t test for a mean applet. It is important that students become familiar with the formulas and process for performing a significance test. At the end of the chapter we will reveal the calculator commands for significance tests (a good way for them to check their final answers).
We suggest that you allow students to use the four-step templates for all homework questions. You will have to then decide whether or not they get templates for quizzes and tests. We allow templates for the quizzes, but not the test.
There is a 3rd learning target that is not addressed in the activity. There are two ideas here that are worth discussing with students:
Just because a sample result is statistically significant, this does not always mean it has real world practical importance. Decreasing mean wait time at Taco Bell from 92.7 seconds to 92.5 seconds doesn't really help you eat your burrito any faster.
When performing many significance tests, some of them will show as statistically significant even when there is no difference (this is a Type I Error). When you perform 100 significance tests (at alpha = 0.05), you can expect that about 5 of them will incorrectly show as statistically significant. See example below (credit XKCD)