Math – Applications for Living VI

Statistics in the news!!

The report (from the BBC) cites a study which says “55% of Syrians think President Assad should stay”.  The source is http://www.bbc.co.uk/news/magazine-17155349

This is a case where the journalist actually did a good job with the statistics.  The quote above comes from an online survey done with about 1000 respondents in the middle east and north Africa.  The article describes several statistical problems with the conclusion.  Among them:

• Syrians live (generally) in Syria … the report did not state how many were actual Syrians or lived there.  One reference in the report allows an estimate of about 100.
• One thousand is a modest number for a survey — this one covers an entire region.
• One hundred is statistically insufficient to measure the opinions of a country.
• Few Syrians have internet access; since it was an internet survey, the Syrians who did respond are not likely to be representative of all Syrians.

Curiously, our Math — Applications for Living class (Math119) yesterday covered a ‘rule of thumb’ for the margin of error for polls like this.  For most polls, the quick little formula 1/√n (reciprocal of the square root of the sample size) is surprisingly accurate.   For the 100 Syrians actually included in the results, the margin of error is 10% .. the true population parameter would be between 45% and 65% (most likely!).

The sad part of this story is that the original story on this survey did not provide this more complete context for the results.  Take a look at the BBC report for more information on that.  One can only hope that the bad use of statistics does not contribute to an already bad situation.

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Math — Applications for Living V

The class (Math119, called “Math — Applications for Living”) is now covering quite a bit of statistics, and I thought I would share a problem from yesterday’s class that incorporates ‘measures of typical values’ (aka ‘average’). 

So, here is the situation described:  “A small local company has 8 workers, and here are their hourly rates of pay:   $9, $9, $9, $10, $11, $18, $36. What is the average hourly pay?”

In this case, I had students work on this problem in pairs; they had directions for finding the mean, median and mode.  The big question was “Which average reflects ‘typical’?”

This was a good situation to show the weakness of the mean as an average or typical value; those outliers create false impressions.  The group actually thought that the mode was the best average because 3 workers had this pay … even though it was the lowest.  Numerically, the median was the best and we talked a bit about the pros and cons of each average. 

Essentially, this work on the ‘average’ supports the cynical statistician view of the world — we don’t have the answer, all we have are hints at something that might (or might not) be true.  Fortunately, this same class gave a chance to talk about distributions of data, and begin ‘distributional thinking’.  The students got the idea that we should try to represent a set of data with one number.

Some students in class had already noticed that the median was used for some things (like home prices, and the net wealth discussion — see http://www.pewsocialtrends.org/2011/07/26/wealth-gaps-rise-to-record-highs-between-whites-blacks-hispanics/).  It was also clear that the word ‘average’ used so often does not state which one — too often, it is the mean (as in ‘the average number of televisions per household is 2.4’).

The class is going to move on to other statistical topics, some of which have more exciting uses in life.  The one above might be of interest, or at least be enjoyable to read.

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Math – Applications for Living IV

I’ve seen the ads, often on the back of a semi-trailer, where companies say that they will pay so much per mile or so much per mile (and perhaps mention that drivers get to be home on weekends).  I can’t bring mathematics to the weekend issue for drivers, but I can  bring math to their pay system.

A typical rate of fuel consumption for the ‘big rigs’ is 8 miles per gallon (this is a little high, but is nicer for calculation!).   A truck’s speed is supposed to be 60 mi/hr in my state, and the average fuel price is $3.749 per gallon.  How much does fuel cost per hour?

    60 mi/hr * 1 gal/8 mi  * $3.749/gal  = $28.12  (rounded)

I have a problem like this on today’s test in my quantitative reasoning course (only it’s for a car, since not many of my students drive a semi).   If you are curious, a typical car would have  an hourly fuel cost of around $7 … we could get in to the cost per pound per hour, which adjusts for the much larger capacity of the semi for hauling stuff.  However, we can be sure that the average semi is loaded with far more than 4 times what a car carries.

Back to the start of this post … if a company pays semi drivers per hour, the driver has (normally) this fuel cost of roughly $30 per hour.  Now, when I see ads that say “$40 per hour to start”, I know that the real income is closer to $10 per hour.  If the pay is ‘per mile’, the calculation is a little simpler (1 gal/ 8 mi * $3.749/gal, which is something like $0.47 per mile).

Within our class, we are using problems like these to become more flexible in our proportional reasoning — a given rate can be represented in two fractions, with our choice being determined by looking at what we start from and what we need.
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Math – Applications for Living III

In class this week, we talked about ‘precision’.  Even though many of our math classes ignore this topic, students relate to it reasonably well.

One example — find the area of a rectangle that is 3.6 meters wide and 4.2 meters long.  The correct answer is ’15 square meters’, since each measurement has only 2 significant digits.  Calculating ‘15.12’ is only part of the story.  How this 15 square meters is used depends on the purpose for finding the area.  If we are estimating the amount of time needed to paint the area, it is fairly safe to work with 15 square meters … however, if we are buying the paint, we might do better with 20 square meters (1 significant digit).

Several students in class have been dealing with the concept of significant digits in their science class as well … isn’t it nice when people can see an immediate use for what we cover in math class?

The topic of significant digits is a natural whenever we cover geometry.  However, we tend to do a bad job with this; I suspect that we are too concerned about ‘keeping things simple enough’.  One of the larger errors on our part is the treatment of π.  In many books and courses, students are told to use ‘3.12’ as the value of this number regardless of the precision of the other numbers involved.  As you know, the correct process is to use all available digits and round the final answer to the appropriate number of digits.  The irony is that almost all students have access to the value of π to 10 or more digits (calculator or computer).  Let’s start doing good mathematics by having students use the built-in constant instead of the (always inappropriate) approximation.  [It’s always inappropriate because we are not supposed to round intermediate values.]

Another example from class: “A city has a deficit of 43.8 million dollars.  How much per person is this if the city has a population of 136,500?”   As a division, we can calculate any number of digits; many students would ‘naturally’ round the result to the nearest cent ($320.88), though this is not the correct value.  We often say ’round money to the nearest cent’, and this is quite appropriate with interest calculations.  However, it may not be appropriate in many other applications.

The topic of ‘significant digits’ (precision) is appropriate for most math classes, and is accessible to almost all students.

 
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