Mathematical Statistics with Applications
Mathematical Statistics with Applications
7th Edition
ISBN: 9780495110811
Author: Dennis Wackerly, William Mendenhall, Richard L. Scheaffer
Publisher: Cengage Learning
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Chapter 13, Problem 88SE
To determine

Show that Total SS= SST+SSB+SSE.

Expert Solution & Answer
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Explanation of Solution

The formulas for sum of squares are as follows:

TSS =i=1kj=1b(YijY¯)2SST =bi=1k(Y¯i.Y¯)2SSB =kj=1b(Y¯.jY¯)2SSE=i=1kj=1b(YijY¯i.Y¯.jY¯)2.

The total sum of squares is calculated is as follows:

TSS =i=1kj=1b(YijY¯i.+Y¯i.Y¯.j+Y¯.jY¯+Y¯Y¯)2=i=1kj=1b((Y¯i.Y¯)+(Y¯.jY¯)+YijY¯i.Y¯.j+Y¯)2=[i=1kj=1b(Y¯i.Y¯)2+i=1kj=1b(Y¯.jY¯)2+i=1kj=1b(YijY¯i.Y¯.j+Y¯)2+2i=1kj=1b(Y¯i.Y¯)(Y¯.jY¯)+2i=1kj=1b(Y¯i.Y¯)(YijY¯i.Y¯.j+Y¯)+2i=1kj=1b(Y¯.jY¯)(YijY¯i.Y¯.j+Y¯)]

Consider

j=1b(Y¯.jY¯)=[j=1b1k(i=1kYij)b×i=1kj=1bYijbk]=[1ki=1kj=1bYij1ki=1kj=1bYij]=0

i=1k(Y¯.iY¯)=[i=1k1b(j=1bYij)k×i=1kj=1bYijbk]=[1bi=1kj=1bYij1bi=1kj=1bYij]=0

i=1k(YijY¯i.Y¯.j+Y¯)=i=1k(Yij1bj=1bYij1ki=1kYij1bki=1kj=1bYij)=i=1kYij1bi=1kj=1bYijkki=1kYij+1bi=1kj=1bYij=0

Thus, the required TSS is as follows:

TSS =[i=1kj=1b(Y¯i.Y¯)2+i=1kj=1b(Y¯.jY¯)2+i=1kj=1b(YijY¯i.Y¯.j+Y¯)2+2i=1kj=1b(Y¯i.Y¯)(Y¯.jY¯)+2i=1kj=1b(Y¯i.Y¯)(YijY¯i.Y¯.j+Y¯)+2i=1kj=1b(Y¯.jY¯)(YijY¯i.Y¯.j+Y¯)]=[i=1kj=1b(Y¯i.Y¯)2+i=1kj=1b(Y¯.jY¯)2+i=1kj=1b(YijY¯i.Y¯.j+Y¯)2+2i=1k(Y¯i.Y¯)i=1k(Y¯.jY¯)+2j=1b(Y¯i.Y¯)i=1k(YijY¯i.Y¯.j+Y¯)+2j=1k(Y¯.jY¯)i=1b(YijY¯i.Y¯.j+Y¯)]=i=1kj=1b(Y¯i.Y¯)2+i=1kj=1b(Y¯.jY¯)2+i=1kj=1b(YijY¯i.Y¯.j+Y¯)2+0+0+0=i=1kj=1b(Y¯i.Y¯)2+i=1kj=1b(Y¯.jY¯)2+i=1kj=1b(YijY¯i.Y¯.j+Y¯)2=bi=1k(Y¯i.Y¯)2+kj=1k(Y¯.jY¯)2+i=1kj=1b(YijY¯i.Y¯.j+Y¯)2=SST+SSB+SSE

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Chapter 13 Solutions

Mathematical Statistics with Applications

Ch. 13.4 - It is believed that women in the postmenopausal...Ch. 13.4 - If vegetables intended for human consumption...Ch. 13.4 - One portion of the research described in a paper...Ch. 13.4 - The Florida Game and Fish Commission desires to...Ch. 13.4 - Prob. 15ECh. 13.4 - An experiment was conducted to examine the effect...Ch. 13.5 - Prob. 17ECh. 13.5 - Refer to Exercise 13.17 and consider YiYi for i ...Ch. 13.5 - Refer to the statistical model for the one-way...Ch. 13.7 - Refer to Examples 13.2 and 13.3. a Use the portion...Ch. 13.7 - Refer to Examples 13.2 and 13.4. a Use the portion...Ch. 13.7 - a Based on your answers to Exercises 13.20 and...Ch. 13.7 - Refer to Exercise 13.7. a Construct a 95%...Ch. 13.7 - Prob. 24ECh. 13.7 - Prob. 25ECh. 13.7 - Prob. 26ECh. 13.7 - Prob. 27ECh. 13.7 - Prob. 28ECh. 13.7 - Prob. 29ECh. 13.7 - Prob. 30ECh. 13.7 - Prob. 31ECh. 13.7 - Prob. 32ECh. 13.7 - Prob. 33ECh. 13.7 - Prob. 34ECh. 13.7 - Prob. 35ECh. 13.8 - Prob. 36ECh. 13.8 - Prob. 37ECh. 13.8 - Prob. 38ECh. 13.8 - Prob. 39ECh. 13.8 - Prob. 40ECh. 13.9 - Prob. 41ECh. 13.9 - The accompanying table presents data on yields...Ch. 13.9 - Refer to Exercise 13.42. Why was a randomized...Ch. 13.9 - Prob. 44ECh. 13.9 - Prob. 45ECh. 13.9 - Prob. 46ECh. 13.9 - Prob. 47ECh. 13.9 - Prob. 48ECh. 13.9 - Prob. 49ECh. 13.9 - Prob. 50ECh. 13.9 - Prob. 51ECh. 13.10 - Prob. 52ECh. 13.10 - Prob. 53ECh. 13.10 - Prob. 54ECh. 13.10 - Refer to Exercise 13.46. Construct a 95%...Ch. 13.10 - Prob. 56ECh. 13.10 - Prob. 57ECh. 13.11 - Prob. 58ECh. 13.11 - Prob. 59ECh. 13.11 - Prob. 60ECh. 13.11 - Prob. 61ECh. 13.11 - Prob. 62ECh. 13.12 - Prob. 63ECh. 13.12 - Prob. 64ECh. 13.12 - Prob. 65ECh. 13.12 - Prob. 66ECh. 13.12 - Prob. 67ECh. 13.12 - Prob. 68ECh. 13.13 - Prob. 69ECh. 13.13 - Prob. 70ECh. 13.13 - Refer to Exercise 13.42. Answer part (a) by...Ch. 13.13 - Refer to Exercise 13.45. Answer part (b) by...Ch. 13 - Prob. 73SECh. 13 - Prob. 74SECh. 13 - Prob. 75SECh. 13 - Prob. 77SECh. 13 - A study was initiated to investigate the effect of...Ch. 13 - Prob. 79SECh. 13 - A dealer has in stock three cars (models A, B, and...Ch. 13 - In the hope of attracting more riders, a city...Ch. 13 - Prob. 84SECh. 13 - Prob. 85SECh. 13 - Prob. 86SECh. 13 - Prob. 87SECh. 13 - Prob. 88SECh. 13 - Prob. 89SECh. 13 - Prob. 90SECh. 13 - Prob. 92SECh. 13 - Prob. 94SE
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