Tag Archives: AP Statistics

Recentering Normal Curves, revisited

I wrote here about using a CAS to determine a the new mean of a recentered normal curve from an AP Statistics exam question from the last decade.  My initial post shared my ideas on using CAS technology to determine the new center.  After hearing some of my students’ attempts to solve the problem, I believe they took a simpler, more intuitive approach than I had proposed.

REVISITING:

In the first part of the problem, solvers found the mean and standard deviation of the wait time of one train: \mu = 30 and \sigma = \sqrt{500}, respectively.  Then, students computed the probability of waiting to be 0.910144.

normCAS1

The final part of the question asked how long that train would have to be delayed to make that wait time 0.01.  Here’s where my solution diverged from my students’ approach.  Being comfortable with transformations, I thought of the solution as the original time less some unknown delay which was easily solved on our CAS.

normCAS4

STUDENT VARIATION:

Instead of thinking of the delay–the explicit goal of the AP question–my students  sought the new starting time.  Now that I’ve thought more about it, knowing the new time when the train will leave does seem like a more natural question and avoids the more awkward expression I used for the center.

The setup is the same, but now the new unknown variable, the center of the translated normal curve, is newtime.  Using their CAS solve command, they found

newnorm

It was a little different to think about negative time, but they found the difference between the new time difference (-52.0187 minutes) and the original (30 minutes) to be 82.0187 minutes, the same solution I discovered using transformations.

CONCLUSION:

This is nothing revolutionary, but my students’ thought processes were cleaner than mine.  And fresh thinking is always worth celebrating.

Confidence Intervals via graphs and CAS

Confidence intervals (CIs) are a challenging topic for many students, a task made more challenging, in my opinion, because many (most?) statistics texts approach CIs via z-scores.  While repeatedly calculating CI endpoints from standard deviations explains the underlying mathematical structure, it relies on an (admittedly simple) algebraic technique that predates classroom technology currently available for students on the AP Statistics exam.

Many (most?) statistics packages now include automatic CI commands.  Unfortunately for students just learning what a CI means, automatic commands can become computational “black boxes.”  Both CAS and graphing techniques offer a strong middle ground–enough foundation to reinforce what CIs mean with enough automation to avoid unnecessary symbol manipulation time.

In most cases, this is accomplished by understanding a normal cumulative distribution function (cdf) as a function, not just as an electronic substitute for normal probability tables of values.  In this post, I share two alternatives each for three approaches to determining CIs using a TI-Nspire CAS.

SAMPLE PROBLEM:

In 2010, the mean ACT mathematics score for all tests was 21.0 with standard deviation 5.3.  Determine a 90% confidence interval for the math ACT score of an individual chosen at random from all 2010 ACT test takers.

METHOD 1a — THE STANDARD APPROACH:

A 90% CI excludes the extreme 5% on each end of the normal distribution.  Using an inverse normal command gives the z-scores at the corresponding 5% and 95% locations on the normal cdf.

normCAS6

Of course, utilizing symmetry would have required only one command.  To find the actual boundary points of the CI, standardize the endpoints, x, and equate that to the two versions of the z-scores.

\displaystyle \frac{x-21.0}{5.3} = \pm 1.64485

Solving these rational equations for x gives x=12.28 and x=29.72, or CI = \left[ 12.28,29.72 \right] .

Most statistics software lets users avoid this computation with optional parameters for the mean and standard deviation of non-standard normal curves.  One of my students last year used this in the next variation.

METHOD 1b — INTRODUCING LISTS:

After using lists as shortcuts on our TI-Nspires last year for evaluating functions at several points simultaneously, one of my students creatively applied them to the inverse normal command, entering the separate 0.05 and 0.95 cdf probabilities as a single list.  I particularly like how the output for this approach outputs looks exactly like a CI.

CI4

METHOD 2a — CAS:

The endpoints of a CI are just endpoints of an interval on a normal cdf, so why not avoid the algebra and additional inverse normal command and determine the endpoints via CAS commands?  My students know the solve command from previous math classes, so after learning the normal cdf command, there are very few situations for them to even use the inverse.

CI1

This approach keeps my students connected to the normal cdf and solving for the bounds quickly gives the previous CI bounds.

METHOD 2b (Alas, not yet) — CAS and LISTS:

Currently, the numerical techniques the TI-Nspire family uses to solve equations with statistics commands don’t work well with lists in all situations.  Curiously, the Nspire currently can’t handle the solve+lists equivalent of the inverse normal+lists approach in METHOD 1b.

CI5

But, I’ve also learned that problems not easily solved in an Nspire CAS calculator window typically crack pretty easily when translated to their graphical equivalents.

METHOD 3a — GRAPHING:

This approach should work for any CAS or non-CAS graphing calculator or software with statistics commands.

Remember the “f” in cdf.  A cumulative distribution function is a function, and graphing calculators/software treats them as such.  Replacing the normCdf upper bounds with an x for standard graphing syntax lets one graph the normal cdf (below).

Also remember that any algebraic equation can be solved graphically by independently graphing each side of the equation and treating the resulting pair of equations as a system of equations.  In this case, graphing y=0.05 and y=0.95 and finding the points of intersection gives the values of the CI.

CI2

METHOD 3b — GRAPHING and LISTS:

SIDENOTE:  While lists didn’t work with the CAS in the case of METHOD 2b, the next screen shows the syntax to graph both ends of the CI using lists with a single endpoint equation.

CI3

The lists obviously weren’t necessary here, but the ability to use lists is a very convenient feature on the TI-Nspire that I’ve leveraged countless times to represent families of functions.  In my opinion, using them in METHOD 3b again leverages that same idea, that the endpoints you seek are different aspects of the same family–the CI.

CONCLUSION:

There are many ways for students in their first statistics courses to use what they already know to determine the endpoints of a confidence interval.  And keeping students attention focused on new ways to use old information solidifies both old and new content.  Eliminating unnecessary computations that aren’t the point of most of introductory statistics anyway is an added bonus.

Happy learning everyone…

SBG and Statistics

I’ve been following Standards-Based Grading (SBG) for several years now after first being introduced to the concept by colleague John Burk (@occam98).  Thanks, John!

I finally made the dive into SBG with my Summer School Algebra 2 class this past June & July, and I’ve fully committed to an SBG pilot for my AP Statistics classes this year.

I found writing standards for Algebra 2 this summer relatively straightforward.  I’ve taught that content for decades now and know precisely what I want my students to understand.  I needed some practice writing standards and got better as the summer class progressed.  Over time, I’ve read several teachers’ versions of standards for various courses.  But writing standards for my statistics class prove MUCH more challenging.  In the end, I found myself guided by three major philosophies.

  1. The elegance and challenge of well designed Enduring Understandings from the Understanding by Design (UbD) work of Jay McTighe the late Grant Wiggins helped me craft many of my standards as targets for student learning that didn’t necessarily reveal everything all at once.
  2. The power of writing student-centered “I can …” statements that I learned through my colleague Jill Gough (@jgough) has become very important in my classroom design.  I’ve become much more focused on what I want my students (“learners” in Jill’s parlance) to be able to accomplish and less about what I’m trying to deliver.  This recentering of my teaching awareness has been good for my continuing professional development and was a prime motivator in writing these Standards.
  3. I struggled throughout the creation of my first AP Statistics standards document to find a balance between too few very broad high-level conceptual claims and a far-too-granular long list of skill minutiae.  I wanted more than a narrow checklist of tiny skills and less than overloaded individual standards that are difficult for students to satisfy.  I want a challenging, but reachable bar.

So, following is my first attempt at Standards for my AP Statistics class, and I’ll be using them this year.  In sharing this, I have two hopes:

  • Maybe some teacher out there might find some use in my Standards.
  • More importantly, I’d LOVE some feedback from anyone on this work.  It feels much too long to me, but I wonder if it is really too much or too little.  Have I left something out?

At some point, all work needs a public airing to improve.  That time for me is now.  Thank you in advance on behalf of my students for any feedback.