Chapter 7 Visualizing a Sampling Distribution Let s review what we have learned about sampling distributions We have considered sampling distributions for the test of means test statistic is U and the sum of ranks test test statistic is R1 We have learned in principle how to find an exact sampling distribution I say in principle because if the number of possible assignments is large then it is impractical to attempt to obtain an exact sampling distribution We have learned an excellent way to approximate a sampling distribution namely a computer simulation experiment with m 10 000 runs We can calculate a nearly certain interval to assess the precision of any given approximation and if we are not happy with the precision we can obtain better precision simply by increasing the value of m Computer simulations are a powerful tool and I am more than a bit sad that they were not easy to perform when I was a student many decades ago We had to walk uphill through the snow just to get to the large building that housed the computer and then we had to punch zillions of cards before we could submit our programs Before computer simulations were practical or even before computers existed statisticians and scientists obtained approximations to sampling distributions by using what I will call fancy math techniques We will be using several fancy math methods in these notes Fancy math methods have severe limitations For many situations they give poor approximations and unlike a computer simulation you cannot improve a fancy math approximation simply by increasing the value of m there is nothing that plays the role of m in a fancy math approximation Also there is nothing like the nearly certain interval that will tell us the likely precision of a fancy math approximation Nevertheless fancy math approximations are very important and can be quite useful here are two reasons why 1 Do not think of computer simulations and fancy math as an either or situation We can and often will use them together in a problem For example a simple fancy math argument will often show that one computer simulation experiment can be applied to many sometimes an infinite number of situations We will see many examples of this phenomenon later in these Course Notes 2 Being educated is not about acquiring lots and lots of facts It is more about seeing how lots and lots of facts relate to each other or reveal an elegant structure in the world Computer 141 Table 7 1 The sampling distribution of R1 for Cathy s CRD r1 P R1 r1 6 0 05 7 0 05 8 0 10 9 0 15 10 0 15 r1 P R1 r1 11 0 15 12 0 15 13 0 10 14 0 05 15 0 05 simulations are very good at helping us acquire facts whereas fancy math helps us see how these facts fit together Fancy math results can be very difficult to prove and these proofs are not appropriate for this course Many of these results however can be motivated with pictures 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 the probability histogram of a sampling distribution 7 1 Probability Histograms As the name suggest a probability histogram is similar to the histograms we learned about in Chapter 2 For example just like a histogram for data a probability histogram is comprised of rectangles on the number line There are some important differences however First a motivation for our histograms in Chapter 2 was to group data values in order to obtain a better picture By contrast we never group values in a probability histogram Second without grouping we don t need an endpoint convention for a probability histogram and as a result we will have a new way to place locate its rectangles The total area of the rectangles in a probability histogram equals 1 which is a feature shared by density histograms of Chapter 2 The reason Density histograms use area to represent relative frequencies of data hence their total area is one Probability histograms use area to represent probabilities hence their total area equals the total probability one Table 7 1 presents the sampling distribution of R1 for Cathy s study of running Remember There were no ties in Cathy s six response values This table was presented in Chapter 6 Its probability histogram is presented in Figure 7 1 Look at it briefly and then read my description below of how it was created First some terminology Thus far in these Course Notes our sampling distributions have been for test statistics either U or R1 In general we talk about a sampling distribution for a random variable X with observed value x Here is the idea behind the term random variable We say variable because we are interested in some feature that has the potential to vary We say random because the values that the feature might yield are described by probabilities Both of our test 142 Figure 7 1 The probability histogram for the sampling distribution in Table 7 1 0 15 0 10 0 05 6 7 8 9 10 11 12 13 14 15 statistics are special cases of random variables and hence are covered by the method described below 1 On a horizontal number line mark all possible values x of the random variable X For the sampling distribution in Table 7 1 these values of x r1 are 6 7 8 15 and they are marked in Figure 7 1 2 Determine the value of lower case Greek delta for the random variable of interest The number is the smallest distance between any two consecutive values of the random variable For the sampling distribution in Table 7 1 the distance between consecutive values is always 1 hence 1 3 Above each value of x draw a rectangle with its center at x its base equal to and its height equal to P X x In the current example 1 making the height of each rectangle equal to the probability of its center value For a probability histogram the area of a rectangle equals the probability of its center value because Area of rectangle centered at x Base Height P X x P X x In the previous chapter we studied the sum of ranks test with test statistic R1 In all ways mathematical this test statistic is much easier to study if there are no ties in the data I will now show how the presence of one tie affects the probability histogram Example 7 1 A small CRD with two values tied Table 7 2 presents the sampling distribution of R1 for a balanced CRD with a total of n 6 units with one particular pair of tied observations the two smallest observations are tied and the other four observations are not Thus the ranks are 1 5 1 5 3 4 5 and 6
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