Estimating a Population Mean: STATE/PLAN/DO/CONCLUDE
Day 85 - Lesson 8.3
Construct and interpret a confidence interval for a population mean.
Determine the sample size required to obtain a C% confidence interval for a population mean with a specified margin of error.
Activity: How many states can you name?
How many states can you name in 60 seconds? Teachers should try this before doing the Activity with students. Make note to the students that we are thinking of the students in the class as a random sample of all AP Stats students (actually this is a convenience sample!)
In the Activity, students are asked to identify whether the variable being measured is categorical or quantitative. Because the variable is quantitative (# of states), this confidence interval will be estimating a mean. If the variable was categorical, students would be estimating a proportion. Students can use their calculators or this applet to find the mean and standard deviation of the data.
The Four-Step Process
This structure was developed specifically to develop student inferential thinking, but it’s also no coincidence that it matches closely with the four-point rubrics for the free response questions on the AP Exam. This structure will be used for all inference problems for the remainder of the course, so it is critical that students become familiar with the expectations.
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 will 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: so the sampling distribution of the sample means will be approximately Normal and we can use t* to do calculations.
Don’t reveal calculator commands yet. Of course, all the work of today’s lesson can be done with TInterval on the TI 83/84 calculator. It is important that students become very familiar with the formulas and process for creating an interval. At the end of the chapter we will reveal the calculator commands for confidence intervals. 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).
Four-Step White Board Critique
We used the Check Your Understanding #2 to do our first White Board Critique. Here is how it works:
(1) Assign students a 4-step problem to work on in pairs.
(2) Monitor the room to support student learning. As each pair finishes, send them to the white board to write up 1 of the steps STATE, PLAN, DO, or CONCLUDE. The first two pairs should handle the PLAN and the DO because they take the longest to write up.
(3) Once all 4 steps are on the board, call the class back together as a group. Ask them to critique the solution on the board as if it were a quiz or test question. Make any revisions with a red marker. This is your opportunity to make clear your expectations for a 4-step problem on an assessment.