Testing a Claim About a Mean (Lesson 9.5)
Chapter 9 - Day 6
Check the Random and Normal/Large Sample conditions for performing a significance test about a population mean.
Calculate the standardized test statistic for a significance test about a population mean.
Find the P-value for a significance test about a population mean using Table B or technology.
Before starting this lesson, there are a few items to have prepared:
Make copies of Table B, or be sure students have access to Table B in the textbook.
Have students find their Chapter 7 Formula Review Sheet. This will be a helpful resource during the activity in this lesson.
In Lessons 9.3 and 9.4 we were in the world of proportions. For Lessons 9.5 and 9.6 we will be in the world of means. Remind students that there are different formulas and conditions in the world of means than in the world of proportions.
There is a hard stop sign at question #7. We fully expect students to calculate a z-score for question #6 and we need to set this straight by making it a t-score before we can find a P-value. For many students (and many teachers!) this logic is fuzzy. Here is the truth:
The sampling distribution is approximately normal here (because of the large sample) with a mean of 15 and a standard deviation of sigma/sqrt(200). We don't know sigma (the population SD), but it does exist.
When we standardize the xbar = 16.5 value, we must now guess for sigma (using the sample SD) which introduces more variation into the distribution. Therefore the standardized version of the sampling distribution is a t-distribution. We need to use a t-distribution to find the P-value.
Students will already be familiar with Table B from when we calculated t* values for confidence intervals for means. When doing a significance test, look in the row for the specified df and find the two critical values that surround your t-test statistic. Then look to the top of the table for the tail probabilities that surround your P-value.
There is absolutely no new content in this activity. Students will be taking content from across the entire course and putting all the puzzle piece together today. It is our job to then formalize all of this learning with vocabulary, notation, and formulas. We are preparing students to do a full 4-step significance test in the next lesson.
The margin notes in red during the debrief are absolutely essential. This lesson is a perfect example of the Experience First, Formalize Later (EFFL) teaching model.
Notice that we continue to practice the interpretation of the P-value. This is the holy grail of an intro stats class! It is what allows students to understand a significance test, rather than memorize a rule for significance tests (if the P is low, the null must go). During the debrief, we bring attention to the fact that the P-value is just the area of the shaded region in the normal distribution.
In the QuickNotes, you will notice that we provide students with two formulas. We call these the GENERAL FORMULA and the specific formula. The GENERAL FORMULA will work for most significance tests we do in this course (chi-square tests are weird), while the specific formula changes from test to test. In the next lesson, students will be doing a full 4-step significance test, in which we will require them to provide both formulas.
We strongly suggest that students should not yet be made aware of calculator functions and applets that will perform entire significance tests yet. We save this reveal for after the Quiz 9.5 to 9.6.
Students should be ready for a full 4-step significance test in the next lesson!