# Tag Archives: normal distribution

## Stats Exploration Yields Deeper Understanding

or “A lesson I wouldn’t have learned without technology”

Last November, some of my AP Statistics students were solving a problem involving a normal distribution with an unknown mean.  Leveraging the TI Nspire CAS calculators we use for all computations, they crafted a logical command that should have worked.  Their unexpected result initially left us scratching heads.  After some conversations with the great folks at TI, we realized that what at first seemed perfectly reasonable for a single answer, in fact had two solutions.  And it took until the end of this week for another student to finally identify and resolve the mysterious results.  This ‘blog post recounts our journey from a questionable normal probability result to a rich approach to confidence intervals.

THE INITIAL PROBLEM

I had assigned an AP Statistics free response question about a manufacturing process that could be manipulated to control the mean distance its golf balls would travel.  We were told that the process created balls with a normally distributed distance of 288 yards and a standard deviation of 2.8 yards.  The first part asked students to find the probability of balls traveling more than an allowable 291.2 yards.  This was straightforward.  Find the area under a normal curve with a mean of 288 and a standard deviation of 2.8 from 291.2 to infinity.  The Nspire (CAS and non-CAS) syntax for this is:

[Post publishing note: See Dennis’ comment below for a small correction for the non-CAS Nspires.  I forgot that those machines don’t accept “infinity” as a bound.]

As 12.7% of the golf balls traveling too far is obviously an unacceptably high percentage, the next part asked for the mean distance needed so only 99% of the balls traveled allowable distances.  That’s when things got interesting.

A “LOGICAL” RESPONSE RESULTS IN A MYSTERY

Their initial thought was that even though they didn’t know the mean, they now knew the output of their normCdf command.  Since the balls couldn’t travel a negative distance and zero was many standard deviations from the unknown mean, the following equation with representing the unknown mean should define the scenario nicely.

Because this was an equation with a single unknown, we could now use our CAS calculators to solve for the missing parameter.

Something was wrong.  How could the mean distance possibly be just 6.5 yards?  The Nspires are great, reliable machines.  What happened?

I had encountered something like this before with unexpected answers when a solve command was applied to a Normal cdf with dual finite bounds .  While it didn’t seem logical to me why this should make a difference, I asked them to try an infinite lower bound and also to try computing the area on the other side of 291.2.  Both of these provided the expected solution.

The caution symbol on the last line should have been a warning, but I honestly didn’t see it at the time.  I was happy to see the expected solution, but quite frustrated that infinite bounds seemed to be required.  Beyond three standard deviations from the mean of any normal distribution, almost no area exists, so how could extending the lower bound from 0 to negative infinity make any difference in the solution when 0 was already $\frac{291.2}{2.8}=104$ standard deviations away from 291.2?  I couldn’t make sense of it.

My initial assumption was that something was wrong with the programming in the Nspire, so I emailed some colleagues I knew within CAS development at TI.

GRAPHS REVEAL A HIDDEN SOLUTION

They reminded me that statistical computations in the Nspire CAS were resolved through numeric algorithms–an understandable approach given the algebraic definition of the normal and other probability distribution functions.  The downside to this is that numeric solvers may not pick up on (or are incapable of finding) difficult to locate or multiple solutions.  Their suggestion was to employ a graph whenever we got stuck.  This, too, made sense because graphing a function forced the machine to evaluate multiple values of the unknown variable over a predefined domain.

It was also a good reminder for my students that a solution to any algebraic equation can be thought of as the first substitution solution step for a system of equations.  Going back to the initially troublesome input, I rewrote normCdf(0,291.2,x,2.8)=0.99 as the system

y=normCdf(0,291.2,x,2.8)
y=0.99

and “the point” of intersection of that system would be the solution we sought.  Notice my emphasis indicating my still lingering assumptions about the problem.  Graphing both equations shone a clear light on what was my persistent misunderstanding.

I was stunned to see two intersection solutions on the screen.  Asking the Nspire for the points of intersection revealed BOTH ANSWERS my students and I had found earlier.

If both solutions were correct, then there really were two different normal pdfs that could solve the finite bounded problem.  Graphing these two pdfs finally explained what was happening.

By equating the normCdf result to 0.99 with FINITE bounds, I never specified on which end the additional 0.01 existed–left or right.  This graph showed the 0.01 could have been at either end, one with a mean near the expected 284 yards and the other with a mean near the unexpected 6.5 yards.  The graph below shows both normal curves with the 6.5 solution having an the additional 0.01 on the left and the 284 solution with the 0.01 on the right.

The CAS wasn’t wrong in the beginning.  I was.  And as has happened several times before, the machine didn’t rely on the same sometimes errant assumptions I did.  My students had made a very reasonable assumption that the area under the normal pdf for the golf balls should start only 0 (no negative distances) and inadvertently stumbled into a much richer problem.

A TEMPORARY FIX

The reason the infinity-bounded solutions didn’t give the unexpected second solution is that it is impossible to have the unspecified extra 0.01 area to the left of an infinite lower or upper bound.

To avoid unexpected multiple solutions, I resolved to tell my students to use infinite bounds whenever solving for an unknown parameter.  It was a little dissatisfying to not be able to use my students’ “intuitive” lower bound of 0 for this problem, but at least they wouldn’t have to deal with unexpected, counterintuitive results.

Surprisingly, the permanent solution arrived weeks later when another student shared his fix for a similar problem when computing confidence interval bounds.

A PERMANENT FIX FROM AN UNEXPECTED SOURCE

I really don’t like the way almost all statistics textbooks provide complicated formulas for computing confidence intervals using standardized z- and t-distribution critical scores.  Ultimately a 95% confidence interval is nothing more than the bounds of the middle 95% of a probability distribution whose mean and standard deviation are defined by a sample from the overall population.  Where the problem above solved for an unknown mean, on a CAS, computing a confidence interval follows essentially the same reasoning to determine missing endpoints.

My theme in every math class I teach is to memorize as little as you can, and use what you know as widely as possible.  Applying this to AP Statistics, I never reveal the existence of confidence interval commands on calculators until we’re 1-2 weeks past their initial introduction.  This allows me to develop a solid understanding of confidence intervals using a variation on calculator commands they already know.

For example, assume you need a 95% confidence interval of the percentage of votes Bernie Sanders is likely to receive in Monday’s Iowa Caucus.  The CNN-ORC poll released January 21 showed Sanders leading Clinton 51% to 43% among 280 likely Democratic caucus-goers.  (Read the article for a glimpse at the much more complicated reality behind this statistic.)  In this sample, the proportion supporting Sanders is approximately normally distributed with a sample p=0.51 and sample standard deviation of p of $\sqrt((.51)(.49)/280)=0.0299$.  The 95% confidence interval is the defined by the bounds containing the middle 95% of the data of this normal distribution.

Using the earlier lesson, one student suggested finding the bounds on his CAS by focusing on the tails.

giving a confidence interval of (0.45, 0.57) for Sanders for Monday’s caucus, according to the method of the CNN-ORC poll from mid-January.  Using a CAS keeps my students focused on what a confidence interval actually means without burying them in the underlying computations.

That’s nice, but what if you needed a confidence interval for a sample mean?  Unfortunately, the t-distribution on the Nspire is completely standardized, so confidence intervals need to be built from critical t-values.  Like on a normal distribution, a 95% confidence interval is defined by the bounds containing the middle 95% of the data.  One student reasonably suggested the following for a 95% confidence interval with 23 degrees of freedom.  I really liked the explicit syntax definition of the confidence interval.

Alas, the CAS returned the input.  It couldn’t find the answer in that form.  Cognizant of the lessons learned above, I suggested reframing the query with an infinite bound.

That gave the proper endpoint, but I was again dissatisfied with the need to alter the input, even though I knew why.

That’s when another of my students spoke up to say that he got the solution to work with the initial commands by including a domain restriction.

Of course!  When more than one solution is possible, restrict the bounds to the solution range you want.  Then you can use the commands that make sense.

FIXING THE INITIAL APPROACH

That small fix finally gave me the solution to the earlier syntax issue with the golf ball problem.  There were two solutions to the initial problem, so if I bounded the output, they could use their intuitive approach and get the answer they needed.

If a mean of 288 yards and a standard deviation of 2.8 yards resulted in 12.7% of the area above 291.2, then it wouldn’t take much of a left shift in the mean to leave just 1% of the area above 291.2. Surely that unknown mean would be no lower than 3 standard deviations below the current 288, somewhere above 280 yards.  Adding that single restriction to my students’ original syntax solved their problem.

Perfection!

CONCLUSION

By encouraging a deep understanding of both the underlying statistical content AND of their CAS tool, students are increasingly able to find creative solutions using flexible methods and expressions intuitive to them.  And shouldn’t intellectual strength, creativity, and flexibility be the goals of every learning experience?

## Recentering a Normal Curve with CAS

Sometimes, knowing how to ask a question in a different way using appropriate tools can dramatically simplify a solution.  For context, I’ll use an AP Statistics question from the last decade about a fictitious railway.

THE QUESTION:

After two set-up questions, students were asked to compute how long to delay one train’s departure to create a very small chance of delay while waiting for a second train to arrive.  I’ll share an abbreviated version of the suggested solution before giving what I think is a much more elegant approach using the full power of CAS technology.

BACKGROUND:

Initially, students were told that X was the normally distributed time Train B took to travel to city C, and Y was the normally distributed time Train D took to travel to C.  The first question asked for the distribution of Y-X if the mean and standard deviation of X are respectively 170 and 20, and the mean and standard deviation of Y are 200 and 10, respectively.  Knowing how to transform normally distributed variables quickly gives that Y-X is normally distributed with mean 30 and standard deviation $\sqrt{500}$.

Due to damage to a part of the railroad, if Train B arrived at C before Train D, B would have to wait for D to clear the tracks before proceeding.  In part 2, you had to find the probability that B would wait for D.  Translating from English to math, if B arrives before D, then $X \le Y$.  So the probability of Train B waiting on Train D is equivalent to $P(0 \le Y-X)$.  Using the distribution information in part 1 and a statistics command on my Nspire, this probability is

FIX THE DELAY:

Under the given conditions, there’s about a 91.0% chance that Train B will have to wait at C for Train D to clear the tracks.  Clearly, that’s not a good railway management situation, setting up the final question.  Paraphrasing,

How long should Train B be delayed so that its probability of being delayed is only 0.01?

FORMAL PROPOSED SOLUTION:

A delay in Train B says the mean arrival time of Train D, Y, will remain unchanged at 200, while the mean of arrival time of Train B, X, is increased by some unknown amount.  Call that new mean of X, $\hat{X}=170+delay$.  That makes the new mean of the difference in arrival times

$Y - \hat{X} = 200-(170+delay) = 30-delay$

As this is just a translation, the distribution of $Y - \hat{X}$ is congruent to the distribution of $Y-X$, but recentered.  The standard deviation of both curves is $\sqrt{500}$.  You want to find the value of delay so that $P \left(0 \le Y - \hat{X} \right) = 0.01$.  That’s equivalent to knowing the location on the standard normal distribution where the area to the right is 0.01, or equivalently, the area to the left is 0.99.  One way that can be determined is with an inverse normal command.

The proposed solution used z-scores to suggest finding the value of delay by solving

$\displaystyle \frac{0-(30-delay)}{\sqrt{500}} = 2.32635$

A little algebra gives $delay=82.0187$, so the railway should delay Train B just a hair over 82 minutes.

CAS-ENHANCED ALTERNATIVE SOLUTION:

From part 2, the initial conditions suggest Train B has a 91.0% chance of delay, and part 3 asks for the amount of recentering required to change that probability to 0.01.  Rephrasing this as a CAS command (using TI-Nspire syntax), that’s equivalent to solving

Notice that this is precisely the command used in part 2, re-expressed as an equation with a variable adjustment to the mean.  And since I’m using a CAS, I recognize the left side of this equation as a function of delay, making it something that can easily be “solved”.

Notice that I got exactly the same solution without the algebraic manipulation of a rational expression.

My big point here is not that use of a CAS simplifies the algebra (that wasn’t that hard in the first place), but rather that DEEP KNOWLEDGE of the mathematical situation allows one to rephrase a question in a way that enables incredibly efficient use of the technology.  CAS aren’t replacements for basic algebra skills, they are enhancements for mathematical thinking.

I DON”T HAVE CAS IN MY CLASSROOM.  NOW WHAT????

The CAS solve command is certainly nice, but many teachers and students don’t yet have CAS access, even though it is 100% legal for the PSAT, SAT, and all AP math exams.   But that’s OK.  If you recognize the normCdf command as a function, you can essentially use a graph menu to accomplish the same end.

Too often, I think teachers and students think of normCdf and invNorm commands as nothing more than glorified “lookup commands”–essentially nothing more than electronic versions of the probability tables they replaced.  But, when one of the parameters is missing, replacing it with X makes it graphable.  In fact, whenever you have an equation that is difficult (or impossible to solve), graph both sides and find the intersection, just like a solution to a system of equations.  Using this strategy, graphing $y=normCdf(0,\infty,30-X,\sqrt{500})$ and $y=0.01$ and finding the intersection gives the required solution.

CONCLUSION

Whether you can access a CAS or not, think more deeply about what questions ask and find creative alternatives to symbolic manipulations.

## CAS and Normal Probability Distributions

My presentation this past Saturday at the 2015 T^3 International Conference in Dallas, TX was on the underappreciated applicability of CAS to statistics.  This post shares some of what I shared there from my first year teaching AP Statistics.

MOVING PAST OUTDATED PEDAGOGY

It’s been decades since we’ve required students to use tables of values to compute by hand trigonometric and radical values.  It seems odd to me that we continue to do exactly that today for so many statistics classes, including the AP.  While the College Board permits statistics-capable calculators, it still provides probability tables with every exam.  That messaging is clear:  it is still “acceptable” to teach statistics using outdated probability tables.

In this, my first year teaching AP Statistics, I decided it was time for my students and I to completely break from this lingering past.  My statistics classes this year have been 100% software-enabled.  Not one of my students has been required to use or even see any tables of probability values.

My classes also have been fortunate to have complete CAS availability on their laptops.  My school’s math department deliberately adopted the TI-Nspire platform in part because that software looks and operates exactly the same on tablet, computer, and handheld platforms.  We primarily use the computer-based version for learning because of the speed and visualization of the large “real estate” there.  We are shifting to school-owned handhelds in our last month before the AP Exam to gain practice on the platform required on the AP.

The remainder of this post shares ways my students and I have learned to apply the TI-Nspire CAS to some statistical questions around normal distributions.

FINDING NORMAL AREAS AND PROBABILITIES

Assume a manufacturer makes golf balls whose distances traveled under identical testing conditions are approximately normally distributed with a mean 295 yards with a standard deviation of 3 yards.  What is the probability that one such randomly selected ball travels more than 300 yards?

Traditional statistics courses teach students to transform the 300 yards into a z-score to look up in a probability table.  That approach obviously works, but with appropriate technology, I believe there will be far less need to use or even compute z-scores in much the same way that always converting logarithms to base-10 or base-to use logarithmic tables is anachronistic when using many modern scientific calculators.

TI calculators and other technologies allow computations of non-standard normal curves.  Notice the Nspire CAS calculation below the curve uses both bounds of the area of interest along with the mean and standard deviation of the distribution to accomplish the computation in a single step.

So the probability of a randomly selected ball from the population described above going more than 300 yards is 4.779%.

GOING BACKWARDS

Now assume the manufacturing process can control the mean distance traveled.  What mean should it use so that no more than 1% of the golf balls travel more than 300 yards?

Depending on the available normal probability tables, the traditional approach to this problem is again to work with z-scores.  A modified CAS version of this is shown below.

Therefore, the manufacturer should produce a ball that travels a mean 293.021 yards under the given conditions.

The approach is legitimate, and I shared it with my students.  Several of them ultimately chose a more efficient single line command:

But remember that the invNorm() and normCdf() commands on the Nspire are themselves functions, and so their internal parameters are available to solve commands.  A pure CAS, “forward solution” still incorporating only the normCdf() command to which my students were first introduced makes use of this to determine the missing center.

DIFFERENTIATING INSTRUCTION

While calculus techniques definitely are NOT part of the AP Statistics curriculum, I do have several students jointly enrolled in various calculus classes.  Some of these astutely noted the similarity between the area-based arguments above and the area under a curve techniques they were learning in their calculus classes.  Never being one to pass on a teaching moment, I pulled a few of these to the side to show them that the previous solutions also could have been derived via integration.

I can’t recall any instances of my students actually employing integrals to solve statistics problems this year, but just having the connection verified completely solidified the mathematics they were learning in my class.

CONFIDENCE INTERVALS

The mean lead level of 35 crows in a random sample from a region was 4.90 ppm and the standard deviation was 1.12 ppm.  Construct a 95 percent confidence interval for the mean lead level of crows in the region.

Many students–mine included–have difficulty comprehending confidence intervals and resort to “black box” confidence interval tools available in most (all?) statistics-capable calculators, including the TI-Nspire.

As n is greater than 30, I can compute the requested z-interval by filling in just four entries in a pop-up window and pressing Enter.

Convenient, for sure, but this approach doesn’t help the confused students understand that the confidence interval is nothing more than the bounds of the middle 95% of the normal pdf described in the problem, a fact crystallized by the application of the tools the students have been using for weeks by that point in the course.

Notice in the solve+normCdf() combination commands that the unknown this time was a bound and not the mean as was the case in the previous example.

EXTENDING THE RULE OF FOUR

I’ve used the “Rule of Four” in every math class I’ve taught for over two decades, explaining that every mathematical concept can be explained or expressed four different ways:  Numerically, Algebraically, Graphically (including graphs and geometric figures), and Verbally.  While not the contextual point of his quote, I often cite MIT’s Marvin Minsky here:

“You don’t understand anything until you learn it more than one way.”

Learning to translate between the four representations grants deeper understanding of concepts and often gives access to solutions in one form that may be difficult or impossible in other forms.

After my decades-long work with CAS, I now believe there is actually a 5th representation of mathematical ideas:  Tools.  Knowing how to translate a question into a form that your tool (in the case of CAS, the tool is computers) can manage or compute creates a different representation of the problem and requires deeper insights to manage the translation.

I knew some of my students this year had deeply embraced this “5th Way” when one showed me his alternative approach to the confidence interval question:

I found this solution particularly lovely for several reasons.

• The student knew about lists and statistical commands and on a whim tried combining them in a novel way to produce the desired solution.
• He found the confidence interval directly using a normal distribution command rather than the arguably more convenient black box confidence interval tool.  He also showed explicitly his understanding of the distribution of sample means by adjusting the given standard deviation for the sample size.
• Finally, while using a CAS sometimes involves getting answers in forms you didn’t expect, in this case, I think the CAS command and list output actually provide a cleaner, interval-looking result than the black box confidence interval command much more intuitively connected to the actual meaning of a confidence interval.
• While I haven’t tried it out, it seems to me that this approach also should work on non-CAS statistical calculators that can handle lists.

(a very minor disappointment, quickly overcome)

Returning to my multiple approaches, I tried using my student’s newfound approach using a normCdf() command.

Alas, my Nspire returned the very command I had entered, indicating that it didn’t understand the question I had posed.  While a bit disappointed that this approach didn’t work, I was actually excited to have discovered a boundary in the current programming of the Nspire.  Perhaps someday this approach will also work, but my students and I have many other directions we can exploit to find what we need.

Leaving the probability tables behind in their appropriate historical dust while fully embracing the power of modern classroom technology to enhance my students’ statistical learning and understanding, I’m convinced I made the right decision to start this school year.  They know more, understand the foundations of statistics better, and as a group feel much more confident and flexible.  Whether their scores on next month’s AP exam will reflect their growth, I can’t say, but they’ve definitely learned more statistics this year than any previous statistics class I’ve ever taught.

COMPLETE FILES FROM MY 2015 T3 PRESENTATION

If you are interested, you can download here the PowerPoint file for my entire Nspired Statistics and CAS presentation from last week’s 2015 T3 International Conference in Dallas, TX.  While not the point of this post, the presentation started with a non-calculus derivation/explanation of linear regressions.  Using some great feedback from Jeff McCalla, here is an Nspire CAS document creating the linear regression computation updated from what I presented in Dallas.  I hope you found this post and these files helpful, or at least thought-provoking.

## Calling for More CAS in Statistics

When you allow your students to solve problems in ways that make the most sense to them, interesting and unexpected results sometimes happen.  On a test in my senior, non-AP statistics course earlier this semester, we posed this.

A child is 40 inches tall, which places her in the top 10% of all children of similar age. The heights for children of this age form an approximately normal distribution with a mean of 38 inches. Based on this information, what is the standard deviation of the heights of all children of this age?

From the question, you likely deduced that we had been exploring normal distributions and the connection between areas under such curves and their related probabilities and percentiles.  Hoping to get students to think just a little bit, we decided to reverse a cookbook question (given or derive a z-score and compute probability) and asked instead for standard deviation.  My fellow teacher and I saw the question as a simple Algebra I-level manipulation, but our students found it a very challenging revision.  Only about 5% of the students actually solved it the way we thought.  The majority employed a valid (but not always justified) trial-and-error approach.  And then one of my students invoked what I thought to be a brilliant use of a CAS command that I should have imagined myself.  Unfortunately, it did not work out, even though it should.  I’m hoping future iterations of CAS software of all types will address this shortcoming.

What We Thought Would Happen

The problem information can be visually represented as shown below.

Given x-values, means, and standard deviations, our students had practice with many problems which gave the resulting area.  They had also been given areas under normal curves and worked backwards to z-scores which could be re-scaled and re-centered to corresponding points on any normal curve.  We hoped they would be able to apply what they knew about normal curves to make this a different, but relatively straightforward question.

Given the TI-nSpire software and calculators each of our students has, we’ve completely abandoned statistics tables.  Knowing that the given score was at the 90th percentile, the inverse normal TI-nSpire command quickly shows that this point corresponds to an approximate z-score of 1.28155.  Substituting this and the other givens into the x-value to z-score scaling relationship, $z=\frac{x-\overline{x}}{s}$ , leads to an equation with a single unknown which easily can be solved by hand or by CAS to find the unknown standard deviation, $s \approx 1.56061$ .  Just a scant handful of students actually employed this.

What the Majority Did

Recognizing the problem as a twist on their previous work, most invoked a trial-and-error approach.  From their work, I could see that most essentially established bounds around the potential standard deviation and employed an interval bisection approach (not that any actually formally named their technique).

If you know the bounds, mean, and standard deviation of a portion of a normal distribution, you can find the percentage area using the nSpire’s normal Cdf command.  Knowing that the percentage area was 0.1, most tried a standard deviation of 1, and saw that not enough area (0.02275) was captured.  Then they tried 2, and goth too much area (0.158655).  A reproduction of one student’s refinements leading to a standard deviation of $s \approx 1.56$ follows.

THE COOL PART:  Students who attempted this approach got to deal directly with the upper 10% of the area; they weren’t required to adapt this to the “left-side area” input requirement of the inverse normal command.  While this adjustment is certainly minor, being able to focus on the problem parameters–as defined–helped some of my students.

THE PROBLEM:  As a colleague at my school told me decades ago when I started teaching, “Remember that a solution to every math problem must meet two requirements.  Show that the solution(s) you find is (are) correct, and show that there are no other solutions.”

Given a fixed mean and x-value, it seems intuitive to me that there is a one-to-one correspondence between the standard deviation of the normal curve and the area to the right of that point.  This isn’t a class for which I’d expect rigorous proof of such an assertion, but I still hoped that some might address the generic possibility of multiple answers and attempt some explanation for why that can’t happen here.  None showed anything like that, and I’m pretty certain that not a single one of my students in this class considered this possibility.  They had found an answer that worked and walked away satisfied.  I tried talking with them afterwards, but I’m not sure how many fully understood the subtle logic and why it was mathematically important.

The Creative Solution

Exactly one student remembered that he had a CAS and that it could solve equations.  Tapping the normal Cdf command used by the majority of his peers, he set up and tried to solve an equation as shown below.

Sadly, this should have worked for my student, but it didn’t.  (He ultimately fell back on the trial-and-error approach.) The equation he wrote is the heart of the trial-and-error process, and there is a single solution.  I suspect the programmers at TI simply hadn’t thought about applying the CAS commands from one area of their software to the statistical functions in another part of their software.  Although I should have, I hadn’t thought about that either.

Following my student’s lead, I tried a couple other CAS approaches (solving systems, etc.) to no avail.  Then I shifted to a graphical approach.  Defining a function using the normal Cdf command, I was able to get a graph.

Graphing $y=0.1$ showed a single point of intersection which could then be computed numerically in the graph window to give the standard deviation from earlier.

What this says to me is that the CAS certainly has the ability to solve my student’s equation–it did so numerically in the graph screen–but for some reason this functionality is not currently available on the TI-nSpire CAS’s calculator screens.

Extensions

My statistics students just completed a unit on confidence intervals and inferential reasoning.  Rather than teaching them the embedded TI syntax for confidence intervals and hypothesis testing, I deliberately stayed with just the normal Cdf and inverse normal commands–nothing more.  A core belief of my teaching is

Memorize as little as possible and use it as broadly as possible.

By staying with these just two commands, I continued to reinforce what normal distributions are and do, concepts that some still find challenging.  What my student taught me was that perhaps I could limit these commands to just one, the normal Cdf.

For example, if you had a normal distribution with mean 38 and standard deviation 2, what x-value marks the 60th percentile?

Now that’s even more curious.  The solve command doesn’t work (even in an approximation mode), but now the numerical solve gives the solution confirmed by the inverse normal command.

What if you wanted the bounds on the same normal distribution that defined the middle 80% of the area?  As shown below, I failed to solve then when I asked the CAS to compute directly for any of the equivalent distance of the bounds from the mean (line 1), the z-scores (line 2), or the location of the upper x-value (line 3).

But reversing the problem to define the 10% area of the right tail does give the desired result (line 2) (note that nsolve does work even though solve still does not) with the solution confirmed by the final two lines.

Conclusion

Admittedly, there’s not lots of algebra involved in most statistics classes–LOADS of arithmetic and computation, but not much algebra.  I’m convinced, though, that more attention to some algebraic thinking could benefit students.  The different statistics packages out there do lots of amazing things, especially the TI-nSpire, but it would be very nice if these packages could align themselves better with CAS features to permit students to ask their questions more “naturally”.  After all, such support and scaffolding are key features that make CAS and all technology so attractive for those of us using them in the classroom.