371chapter10f2011a - Chapter 10 Describing A Numerical...

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Chapter 10 Describing A Numerical Response 10.1 Pictures Thus far in these notes, the response has been either a dichotomy or a count that follows the Poisson or Binomial distribution. In this chapter we extend our work to counts that are neither Poisson nor Binomial and to responses that are measurements. Suppose the subjects are students in this class. Below are some examples of numerical re- sponses. Counting: Number of points on homework to date; number of credits this semester; number of persons living in current household. Measuring: Height; weight; age. As often happens in life, the boundary between these options can be blurry. For example, consider annual income. Literally, annual income is determined by counting the number of dollars earned in the year, but economists and other researchers tendtotrea ti tasameasuremen t . The general guideline is that if a count variable has many values in a population, and no one value dominates others in terms of relative frequency, it is usually mathematically more convenient to treat the variable as a measurement. Two important words are: precise and accurate .A c c u r a t em e a n sc l o s e t o t h e t r u t h .F o r example, if I state that my dog Casey lived for 15.5 years, tha tisaccu ra te . I fIs ta tetha tmy grandfather Wardrop lived to be 110, that is highly inaccurate. Precise is most useful for measurements. If I state: Yesterday I ran one mile in 250.376 seconds, this is incredibly precise (to the nearest one-thousandths of a second), but ridiculously inaccurate. If I say I ran it ‘In less than one hour’ it is accurate, but not the least precise. Here is a good general guideline for science: measurements should be precise enough to create variation in our population or subjects of interest, but there is no need to get carried away with it! Precise is somewhat meaningless for counts that take on smallvalues.Forexample,itisaccu- rate to say that 2 cats live in my house. It is no more precise to say I have 2.000 cats! 103
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Table 10.1: Sorted Speeds, in MPH, by Time, of 100 Cars. Speeds at 6:00 pm 26 26 27 27 27 27 28 28 28 28 28 28 28 28 28 28 28 28 28 29 29 29 29 29 29 29 29 29 29 29 30 30 30 30 30 30 30 30 31 31 31 31 32 33 33 33 34 34 35 43 Speeds at 11:00 pm 27 28 30 30 30 31 31 31 32 32 32 32 32 32 32 33 33 33 33 33 33 33 34 34 34 34 34 34 35 35 35 35 36 36 36 37 37 37 37 37 37 38 38 39 39 40 40 40 40 40 For large counts, precision does become meaningful. For example, if forced to guess, I would say that there are 300 million people living in the US. I suspect that this is somewhat accurate, but clearly I am not being very precise. 10.1.1 Dot Plot We begin with an example of measurement data, taken from a student project in my Statistics 301 class. On a spring evening, a Milwaukee police of±cer measured the speeds of 100 automobiles. The data were collected on a street in a “warehouse district” withaspeedlimitof25MPH .Fiftycars were measured between roughly 5:45 and 6:15 pm, referred to below as 6:00 pm. The remaining 50 cars were measured between roughly 10:40 and 11:20 pm, referred to below as 11:00 pm.
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371chapter10f2011a - Chapter 10 Describing A Numerical...

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