Normal Distributions: Finding Areas from Values (Lesson 2.5)
Chapter 2 - Day 6
Find the proportion of values to the left of a boundary in a normal distribution.
Find the proportion of values to the right of a boundary in a normal distribution.
Find the proportion of values between two boundaries in a normal distribution.
This activity is a bit longer than usual (2 pages), so it is important that you are prepared for this lesson. Here are some important details:
The STOP sign at the end of page 1 means a teacher debrief is necessary before students will be able to move on. As part of this debrief, you should have copies of Table A ready to hand out to students.
It is recommended that you go through question #5 as a whole class, modeling for students what work is expected for normal distribution calculations.
This activity starts with 3 questions that are a review of the 68-95-99.7 rule from the previous lesson. In question 4, students will realize that this rule is no longer helpful because the rule only works for values that are 1, 2, or 3 standard deviations away from the mean (it doesn’t work for non-integer values). This provides the rationale for needing a new tool to solve normal distribution problems (Table A!). Dan Meyer would say that we created a headache for students, so that we could sell them aspirin.
We suggest that you work through question 5 as a whole class. This is your chance to model the expectations for problems like this. We suggest requiring students to draw and label a picture and to show their work for the z-score calculation. Once students have seen the model solution for question 5, they should be able to work in groups to do the remaining questions.
Encouraging students to establish good habits now will definitely pay them back later in the course. Here are a few suggestions:
(1) Carefully labeled and shaded normal curve. The N(mean,SD) helps students keep track of the correct mean and standard deviation. Also, later in the course we will have to check that our distributions are approximately normal and this notation will remind us to do that check. Labeling the mean at the center of the curve helps students to identify whether the given value is less than or greater than the mean. Students will also start to think about how far away the given value might be from the mean. The shaded region will help students to check their final answers. If the shaded region is clearly less than 50% of the curve and they have an answer of Area = 0.8943 then they know they have made a mistake.
(2) Showing the work for a z-score. We require this step to try and set up students for success later in the course. When we get to calculating test statistics for significance tests (Chapter 9,10), the formulas for calculating a z-score will get much more complex. We want students in the habit of showing their work. This step also reminds students that a z-score tells us how many standard deviations we are away from the mean.
(3) Find a correct z-score and turn it into an area. Later in the course, we will be doing significance testing. The z-score will be our test statistic and the area will be our P-value. The P-value is the holy grail of introductory statistics.
If students need more help understanding the differences in finding area to the left, area to the right, and area in between, consider having them make this cool flip chart.