Significance Test for a Mean
Chapter 9 - Day 7 - Lesson 9.3
State and check the Random, 10%, and Normal/Large Sample conditions for performing a significance test about a population mean.
Calculate the standardized test statistic and P-value for a test about a population mean.
Perform a significance test about a population mean.
Activity: Are You Getting Enough Sleep?
In this Activity, we will investigate the claim that AP Stats students are not getting enough sleep. As with Lesson 9.2, students have all of the pieces they need in order to do this significance test. They simply need to put all of the pieces together. Remind students that we use a t-distribution to calculate the P-value here because we are using the sample standard deviation (which is a variable) rather than the population standard deviation.
What To Do When Conditions Are Not Satisfied
In this Activity, the small population size means that a sample size of 10 will violate the 10% condition. Many students will say that we shouldn’t move forward with the significance test. They’re not wrong. When the 10% condition is violated, the formula that we use for standard deviation is no longer valid (we should be using the finite population correction).
We think you should give your students this advice: If a condition is not satisfied, go through all of the proper motions for calculating a confidence interval or carrying out a significance test. Then at the very end of your conclusion, write:
We must be cautious of our results, as the __________ condition was not satisfied.
The reason for this is that on the AP Exam, it is very unlikely that students would be asked to do inference in a situation where the conditions are not satisfied. We would not want them to lose credit for not attempting a problem because they incorrectly thought one of the conditions was not satisfied.
Table B or tcdf?
Table B gives students a range of possible P-values, while tcdf gives students an exact P-value. This is one of the rare occasions where we prefer the technology, but only because we don’t think that Table B provides any additional understanding.
Tips for Using the Four-Step Process
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 maintain this expectation for all confidence intervals and significance tests in Chapters 8-12.
While it is important that students know how to check each condition, it is equally important that they understand why we check the condition. We call this the “so what?”.
Random Condition: so we can generalize to the population.
10% Condition: so sampling without replacement is OK.
Normal/Large Sample Condition: so the sampling distribution of the sample proportions will be approximately Normal and we can use z to find a P-value.
Don’t reveal calculator commands yet. Of course, all the work of today’s lesson can be done with T-Test on the TI 83/84 calculator. It is important that students become very 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. Then we instruct students that they are to use this feature only to check their final answer on a Free Response question (or on a MC question if they wish).