Exercise in Summation Skills ANSWERS In this class, we assume that you are able to work with summation algebra, and that you are comfortable with basic mathematical notation. Work through this handout to familiarize yourself with the notation and calculations. If you have questions or have difficulty completing this worksheet, please talk to the instructor or TA. We will use a small data set with 7 observations. For each unit i, values for each of 3 variables (x, y, z) are recorded. Note that for this data set, i = 1, 2, … , 7. Also notice that for i = 1, y1 = 7 and for i = 7, x7 = -4. i 1 2 3 4 5 6 7 yi 7 8 6 5 4 2 3 xi 3 3 3 3 -4 -4 -4 zi 4 2 3 1 5 2 4 Exercise #1: sum the x’s and z’s; recall that you can add negative numbers (i.e., subtraction). 2107171==∑∑== iiiizx Exercise #2: What is the sum when n = 7 and k = 1/7. 171771===∑=inknk Exercise #3: What well-known measure of central tendency would obtain if 711==nk? Mean or average Exercise #4: What is the mean (average) of the x’s (x) and the z’s (z)? 070171===∑=iixnx3721171===∑=iiznz Exercise #5: What would you obtain if k = -5; that is, if ∑=−+71))5((iiy = ∑=−71)5(iiy? ∑∑∑===−=+=+niiniiniinynkyky1115)( Exercise #6: Obtain ∑=−71)(iixx and ∑=−71)(iizz. Subtracting the mean from each data point is called “centering the data” or “correcting for the mean.” ∑∑===−=−=−71710)0(70)(iiiixnxxx ∑∑===−=−=−71710)3(721)(iiiiznzzz Exercise #7: Show that ∑=71iiiyx= 42. ∑=71iiiyx= (7 + 8 + 6 + 5)(3) + (4 + 2 + 3)(-4) = 78 – 36 = 42Exercise #8: What is ∑=niiizx1and ∑=niiiyz1? 141−=∑=niiizx 1031=∑=niiiyz Exercise #9: Show that 271712∑ ∑= =⎥⎦⎤⎢⎣⎡≠i iiixx. xi2= 4(9) + 3(16) = 84i=17!xii=17!"#$%&'2= 02= 0 Exercise #10: What is ∑=−712)(iixx and what is ∑=−712)(iizz?? ∑∑==−=−7122712)(iiiixnxxx= 84-7(0) = 84 ∑=−712)(iizz= 75-7(9) = 12 In 10, the sum of squares corrected for the mean (or sum of squared deviations from the mean) is always nonnegative, which means that the sum is equal to or larger than zero (or never
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