Spreadsheet Problem Solving

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1 CO Emmssos for the US, Class meetg #6 Moday, Sept 14 th CO Emssos (MMT Carbo) y = 1.3x Year GEEN 1300 Itroducto to Egeerg Computg Spreadsheet Problem Solvg array formulas vector-matrx calculatos fttg models to data straght-le regresso Homework #3 due Wedesday 1 Vector-Matrx Calculatos Excel called Array Formulas Set up example for case study of f(x) vs x Case study by copyg formula f x x 1 l x Copy dow usg relatve addressg rage B:B11 amed x 1

2 Alterate approach usg array formula Select C:C11 as destato for array formula (ot just C) Type formula terms of array x (ot just B) Fsh wth Ctrl-Shft-Eter (ot just the Eter key) Notce that Excel surrouds the array formula wth braces { } Excel does ths automatcally Here, oe formula aloe cotrols the calculato -- the formula copy example, 10 formulas 3 Use of array formulas for vector-matrx calculatos Vector addto Ctrl- Shft- Eter Matrx fuctos: MMULT matrx multplcato MINVERSE matrx verse MDETERM matrx determat TRANSPOSE matrx traspose all requre the use of array formulas ad Ctrl-Shft-Eter 4

3 bass: Matrx multplcato b11 b1 a11 a1 a13 a11b11 a1b1 a13b31 a11b1 a1b a13b3 b1 b a1 a a 3 a1b11 ab1 a3b31 a1b1 ab a3b 3 b31 b 3 x 3 3 x x er dmesos must agree xk tmes kxm gves xm 5 Use of array formulas for vector-matrx calculatos Matrx multplcato x x 3 Matrx traspose 3 x Ctrl-Shft-Eter Ctrl-Shft-Eter 6 3

4 Use of array formulas for vector-matrx calculatos Matrx determat Ctrl-Shft-Eter chage to 9 determat s "effectvely" zero -- matrx s sgular 7 Use of array formulas for vector-matrx calculatos 1 Matrx verse recall H H I Ctrl-Shft-Eter Erroeous result whe H s sgular 8 4

5 Use of array formulas for vector-matrx calculatos Most commo applcato -- solvg a set of lear equatos x1 x 3x x1 0 4x1 5x 6x x 1 7x1 8x 8x x 3 1 Soluto: Axb 1 1 A A x A b A x b 1 I x A b 1 x A b 9 Use of array formulas for vector-matrx calculatos Most commo applcato -- solvg a set of lear equatos Ctrl-Shft-Eter Note: o result f set of equatos s sgular 10 5

6 Straght-le Lear Regresso y 1 y = ax + b Model y y 11 e 11 ŷy 11 x x 11 For each data pot, there s a error betwee that pot ad the model le. Fttg the model has to do wth mmzg these errors. 11 Fdg the model parameters that gve the best ft For the straght-le model, the model parameters are the slope (a) ad the tercept (b). The problem s the to fd the values of a ad b that gve the best ft. What s meat by the best ft? The stadard measure of goodess of ft s the sum of squares of the errors: ˆ SSE y y 1 ŷ ax b So, the problem reduces to fdg the mmum of SSE by adjustg a ad b. 1 6

7 Fttg a straght-le model to data The mmzato of SSE ca be solved by calculus to gve formulas for the best values of a ad b: xy x y a x x 1 1 y b a 1 1 x ad Excel solves problems lke ths wth ether formulas or bult- tools (Data Aalyss Regresso & Tredle). 13 Example: straght-le ft 14 7

8 Trasfer the data to a Excel spreadsheet ad create a graph 150 CO Emmssos for the US, (MMT Carbo) CO Emssos Year 15 Calculatg the slope ad tercept usg Excel formulas x y x y a x x 1 1 y b a 1 1 x 16 8

9 The formulas behd the umbers xy x y a x x 1 1 y b a 1 1 x 17 Usg the model straght-le equato to compute the predctos: Here, a array formula s used. Oe could also use a sgle formula D3 ad copy t dow. 18 9

10 ad copy these to the graph, dsplayg as a straght le CO Emmssos for the US, CO Emssos (MMT Carbo) y = 1.3x Year 0 10

11 Usg a alterate, shortcut approach Start wth a smple graph of the data Tredle Select the data seres by clckg o a marker Rght-clck o a selected data pot to get the cotext-sestve meu Select Add Tredle... opto 1 The Format Tredle dalog box Lear selected by default OK for ths problem Set for Dsplay equato o chart 11

12 Set Le Color to black Set Le Wdth to pt ad Dash type as show Clck Close 3 Ital form of graph wth straght-le added 1550 CO Emmssos for the US, y = 1.315x CO Emssos (MMT Carbo) Fx up equato dsplay Year 4 1

13 1550 CO Emmssos for the US, arbo) CO Emssos (MMT Ca y = 1.3x Year Looks just lke before, but we got there qucker But ether of these approaches gves us much formato 5 about the model, how good t s, etc. I the ext class, ad lab ths week, we wll troduce you to the Data Aalyss Regresso tool. It provdes much more formato regardg the regresso calculato l ad s a more robust tool to use whe fttg curves to data. 6 13

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