Visualizing a Sampling DistributionProbability HistogramsThe Mean and Standard Deviation of R1The Family of Normal CurvesUsing a Normal Curve to obtain a fancy math approximationComputingAreas Under any Normal CurveUsing a Website to Perform the Sum of Ranks TestReading Minitab Output; Comparing Approximation MethodsSummaryPractice ProblemsSolutions to Practice ProblemsHomework ProblemsChapter 7Visualizing a Sampling DistributionLet’s revi ew what we have learned about sampling distributions. We have considered samplingdistributions for the test of means (test statistic is U) and the sum of ranks test (test statistic isR1). We have learned, in principle, how to find an exact sampling distribution. I say in p rinciplebecause if the number of possible assignments is large, then it is impractical to attempt to obtainan exact sampling distribution.We have learned an excellent way t o approximate a sampl ing distribution, namely a computersimulatio n experiment with m = 10, 000 run s. We can calculate a nearly certain interval to assessthe precision of any given approximation and, if we are not happy with the precision, we can obtainbetter precision simply by increasing th e value of m. Computer simulations are a powerful tooland I am mo re than a bit sad that they were not easy to perform when I was a stu dent many decadesago. (We had to walk uphill, through the snow, just to get to the large building that housed thecomputer and then we had to punch zillions of cards before we could submit o ur programs.)Before computer simulations were practical, or even before computers existed, statisticians andscientists obtai ned approximations to sampling distributions by usin g what I will call fancy mathtechniques. We will be using several fancy math methods in these notes.Fancy math meth ods have severe limitations. For many situations they give poor approxima-tions and, unlike a computer simulation, y ou cannot improve a fancy math approximation simplyby increasing the value of m; there is nothing that plays the role of m in a fancy math approxima-tion. Also, there is nothing like the nearly certain interval that will tell us the likely precision of afancy math approximati on.Nevertheless, fancy math approximations are very impo rtant and can be quite us eful ; here aretwo reasons why:1. Do not think o f computer simulations and fancy math as an either/or situation. We can, andoften will, use th em together in a probl em. For example, a simple fancy math argument willoften show that one computer simulatio n experiment can be applied to many—sometimesan infinite number of—sit uations. We will see many examples of this phenomenon later inthese Course Notes.2. Being educated is not about acqui ri ng l ots and lots of facts. It is more about seeing how lotsand lots of facts relate to each other or reveal an elegant structure in the world. Computer141Table 7.1: The sampling distribution of R1for Cathy’s CRD.r1P (R1= r1) r1P (R1= r1)6 0.05 11 0.157 0.05 12 0.1580.10 13 0.109 0.15 14 0.0510 0.15 15 0.05simulatio n s are very good at helpin g us acquire facts, whereas fancy math helps us see howthese facts fit togeth er.Fancy m at h results can be very difficult to prove and these proofs are not appropriate for thiscourse. Many of these results, however, can be mo tivated with pict ures. This begs the question:Which pictures? The answer: Pictures of sampling distributions.Thus, our first goal in this chapter is to learn how to draw a particular picture, called theprobability histogram, of a sampling distribution.7.1 Prob ability Histogram sAs the name suggest, a probability histogram is similar to the histograms we learned about inChapter 2. For example, just like a histogram for data, a probability histogram is comprised ofrectangles on t he number line. There are some important differences, however. First, a motivationfor our histograms in Chapter 2 was to group data values in order to obtain a better picture. Bycontrast, we never group values in a probability histogram. Second, without grouping, we don’tneed an endpoint convention for a probability histogram and, as a result, we will have a new wayto place/locate its rectangles.The total area of th e rectangles in a probability his togram equals 1, which is a feature sharedby density histo grams o f Chapter 2. The reason? Density hi stograms use area to represent relativefrequencies of data; hence, their total area is on e. Probability histograms use area to representprobabilities; hence, their total area equals the total probability, one.Table 7.1 presents the sampl ing distribution of R1for Cathy’s study of running. (Remember:There were no ties in Cathy’s six response values.) This table was presented in Chapter 6. Itsprobability histogram is presented in Figure 7.1. Lo ok at it briefly and then read my descriptionbelow of h ow it was created.First, some terminology. Thus far in these Course Notes our sampling distributions have beenfor test statistics, either U or R1. In general, we talk about a sampling distribution for a randomvariable X, wi th observed value x. Here is the idea behind the term random variab le. We sayvariable because we are interested in some feature t hat has the potential t o vary. We say randombecause the values that the feature might yield are described by probabiliti es. Both of ou r test142Figure 7.1: The probability histogram for the sampling distri bution in Table 7.1.6 7 8 9 1011 121314150.050.100.15statistics are special cases of random variables and, hence, are covered by the metho d describedbelow.1. On a horizontal number line, mark all possible values, x, o f the random variable X.For the samp ling distribution in Table 7.1 these values of x (r1) are 6, 7, 8, . . . 1 5 and theyare marked in Figure 7.1.2. Determine the value of δ (lower case Greek delta) for the random variable of interest. T henumber δ is the smallest distance between any two consecutive values of the random variable.For the sampl ing dist ri bution in Table 7.1, the distance between consecutive values is al-ways 1; hence, δ = 1.3. Above each value of x, draw a rectangle, with its center at x, its base equal to δ and its heigh tequal to P (X = x)/δ.In the current example, δ = 1, making the height of each rectangle equal to the probabilit yof its center value.For a probability histogram the area of a rectangle equals the p robability of its center value, be-cause:Area of rectangle centered at x = Base × Height = δ ×P
View Full Document
Unlocking...