Finding Areas and Values in a Normal Distribution
Day 14 - Lesson 2.2
Find the proportion of values in a specified interval in a Normal distribution using Table A or technology.
Find the value that corresponds to a given percentile in a Normal distribution using Table A or technology.
Activity: Will Marty Make it Back to the Future?
Quite possibly the coolest car ever made, the DeLorean DMC-12 features gull-wing doors and a stainless steel body. There were a few models produced with an optional flux capacitor, which allowed for time travel. Lucky for us, Doc kept a special notebook that contained data on the speed of the DeLorean after a 20 second acceleration run (turns out that speeds are normally distributed with a mean of 80 mph and a standard deviation of 7.7 mph).
On page 1 of the Activity, students will work in groups (pairs) to use their knowledge of the 68-95-99.7 rule from yesterday. No problem until they get to problem #5. This is where the 68-95-99.7 rule falls apart and creates the need for a new mathematical tool. Use your debrief here to introduce Table A!
On page 2, we suggest going through question #6 as a whole group. This will allow you to model your expectations for student work. Then let students work in groups (pairs) on the last two questions of the Activity. Notice that in #7 the students are given a value and asked to find an area. On #8 they are given an area and asked to find a value. Have the first few groups to finish the problem come up to the white board and write out the solution. Use your debrief here to critique the student solution, highlighting your expectations if this was a quiz or test question.
This might help to reveal the answer to the final question (LANGUAGE WARNING!)
It is very important to establish clear expectations for “showing work” on these normal distribution calculations.
Picture of the normal curve. Must be labeled N(mean, SD), the center labeled with the mean, and then the shaded area of interest.
Why? 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 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.
Formula for a z-score, then numbers plugged in.
Why? We require this step to try and set up students for success later in the course. When we get to calculating z-test statistics for significance tests (Chapter 9 and 10), the formulas for calculating a z-score will get much more complex. We want students in the habit of writing formulas. This step also reminds students that a z-score tells us how many standard deviations away from the mean a value is.
Find correct z-score and then use Table A to turn it into an area.
Why? Later in the course, we will be doing significance testing. The z-score will be our standardized test statistic and the area will be our P-value.
Calculator vs. Table A
For today, we do not show students how to use the calculator to answer these questions. We want them to work the homework problems without this knowledge. Tomorrow, we will start the lesson by showing students normalcdf and InvNorm, but in the end, we will still require them to be old school. For more explanation, here is why we bother with z-scores and Table A.