Month: February 2012

Categories
Statistics

Explorations in Polling

Primary election season is here, and news reports are filled with sound bites from candidates, their supporters, and pundits all trying to get the edge by being the first with breaking news.  It’s also polling season, as every news organization seems to have their own poll, all designed to project the winners.  This provides a great opportunity to talk about some statistics concepts which often get buried in the high school curriculum: sampling, surveys, margin of error and confidence intervals.

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One nice resource I have used in my classes before is the site pollingreport.com. This site collects polls from many sources: news agencies, university organizations and polling companies. Students can search from a long menu of topics and examine the careful wording of survey questions, time-progression data and information on sample size and margin of error.
Having students select their own survey, and interpret the results, can lead to interesting class discussions. One problem with polls is that the results are often taken as absolute, rather than an estimate of a population. An interval plot can help remedy this, and get students thinking about that pesky margin of error, which is often buried, italicized, or shown in a smaller font than the rest of a poll’s results. Here’s an example of an interval plot, using the results of a poll from pollingreport.com:

Quinnipiac University Poll. Feb. 14-20, 2012. N=1,124 Republican and Republican-leaning registered voters nationwide. Margin of error ± 2.9.
“If the Republican primary for president were being held today, and the candidates were Newt Gingrich, Mitt Romney, Rick Santorum, and Ron Paul, for whom would you vote?”

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Some questions for discussion can then include:

  • How can these results be used?
  • What do you think would happen if we asked more people? Or if the election were held today?
  • What would it mean if intervals over-lapped each other?
  • How likely is it that nation-wide support for Rick Santorum is within the interval?

While confidence intervals don’t need to be defined formally, the concept of these intervals indicating plausible values for the population parameter can be discussed. The New York Times, in particular, does an excellent job of providing an accessible explanation for margin of error, such as this excerpt from a telephone poll summary:

In theory, in 19 cases out of 20, overall results based on such samples will differ by no more than three percentage points in either direction from what would have been obtained by seeking to interview all American adults. For smaller subgroups, the margin of sampling error is larger. Shifts in results between polls over time also have a larger sampling error.

Next, we can take a look at formulas for margin of error. One convenient formula found in some textbooks links margin of error directly to the sample size:

By going by to pollingreport.com, and pulling a sample of polls with different sample sizes, we can examine the accuracy of this short and snappy formula.  The scatterplot below uses sample size as the independent variable, and reported margin of error as the dependent variable.
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The formula seems to be a nice guide, and some polls clearly use more sophisticated formulas which generate more conservative margins of error.

Classes who wish to explore polling further can check out the New York Time polling blog, FiveThirtyEight, which provides more detailed analyses of polls and their historical accuracy.

Categories
Algebra

Slope and the ADA

The middle school in the district where I work is quite old.  Dedicated in 1959, and once serving as the district high school, the building is a Frankenstein of aging classrooms, newer additions, and inconsistent heat. One feature of the building is the network of ramps used to shuttle students from wing to wing, and supplies in and out.
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Recently, I worked with a team of 8th grade algebra I teachers to develop an activity which would utilize the many ramps, get kids moving and measuring, and reinforce slope as a measure of steepness. The teachers had great ideas for leading students through measurement activitites. My initial idea of having students choose points along the ramp, then measuring the rise and run between points, was discussed and improved. The teachers used blue painter’s tape to create guiding triangles along the bricks on the walls along two of the ramps. Another teacher noted that railings could be used to connect parallel lines to slopes, and triangles utilizing the railing were also provided.

Students measured the slope of two ramps using the provided triangles, then were led outside, where both a pedestrian ramp and a custodian’s ramp were measured.  The outside ramps were additional challenges, as no guiding tape marks were provided.  Wacthing student reactions and approaches to these ramps was intriguing.  Some students attempted to use the bricks on the building to trace their own triangles, while another group discovered that the level ground along the freight ramp could be used as the “run”.

After the activity, the class discussed and compared their results.  In one class, the unusual steepness of one ramp in our building was questioned, and related to the legal limits of handicapped ramps.  The class agreed that the ramp seems to be an original part of the building, and that an elevator had been installed alongside the ramp for our disabled friends.  Further discussion could include the requirements of the Americans with Disabilities Act, which contains the following requirements for ramps:

The least possible slope shall be used for any ramp. The maximum slope of a ramp in new construction shall be 1:12. The maximum rise for any run shall be 30 in (760 mm) . Curb ramps and ramps to be constructed on existing sites or in existing buildings or facilities may have slopes and rises as allowed in 4.1.6(3)(a)  if space limitations prohibit the use of a 1:12 slope or less.

As a follow-up, students found pictures of objects or places which they felt represented interesting slopes.  Geometer’s Sketchpad was then used to measure and compare the slopes in their pictures:
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Categories
Algebra

Who Takes 5 Hours to Mow a Lawn?

Some units and chapters in algebra lend themselves naturally to interesting openers. Interesting scenarios to discuss slopes, systems of equations or quadratic functions are abundant. Finding examples for topics like radicals and complex numbers or rational expressions can be a bit more of a challenge. Addition and subtraction of rational expressions mean that shared work problems can’t be far behind, like this nugget from algebra.com:

One good use for rational equations is the shared work problem. This solution would be of great help in scheduling employees. For example, If Bob can mow a lawn in 3 hours and Joe can do it in 5 hours, how long would it take them together?

A few thoughts come to mind:

  • I’m doubting that the personnel schedulers at WalMart or Jiffy Lube are using rational expressions to schedule their employees.
  • How many of our kids would guess 8 hours, or even 4 hours, as their initial guess?
  • Joe needs to stop lollygagging on the job.

I set out to make a video to encourage discussion of these problems. In a first attempt, my sister and nephew were recruited to each build a Lego tower separately, then together.
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Working together had little effect on the overall time, as the partnership tripped over each other digging into the bucket for Legos, and had trouble coordinating the overall tower construction. This leads to a nice discussion of the assumed independence of the two volunteers in these problems, but made for a pretty bad video.

In the video below, teachers Christine and John were recruited to staple index cards to a stack of 50 “top secret” papers. A shared work ending was also produced. But in a version that was later eliminated, Christine passed papers to John, who then stapled. In order to maintain independence, a new ending was shot where they worked separately, yet simultaneously.

Christine’s final time was 4:50, while John’s final time was 4:29
To find the ideal shared time, we let x = the number of seconds required to complete the job together.

  • Christine’s rate is 1 / 290 of the job completed per second
  • John’s rate is 1 / 269 of the job completed per second

Since we want one job to be completed, this leads to the equation:

Solving for x yields an ideal solution of 2:19, so the partnership’s time of 2:10 is not too surprising.  The subjects admitted that they were a bit more competitive to do well working together than when they were separated.  Also, my quick appearance during the shared portion on the video is due to the team needing more index cards, and not any funny business!  What would happen if Christine showed up a minute late?  How long would it take them to complete 2000 cards?

Hopefully, we can encourage some discussion and debate, and move away from Joe and his 5-hour lawns.