WORKSHEET STORED BY MINITAB VERSION 6.1 1997 910 Release 9.1EAD 1############Species ####D@###,, D###@###<#######<#######<###`#<#######<#######<#######<#######<## #H.,G####E### ##<#######<#######<###-F#######<#######<#######<#######<###.E###, #B### #<#######<#######<###(E###mD###n##<#######<#######<### h.-D###E### #-D#####<#######<###mE###,##<#######<#######<#######<####MnF### ##<#######<#######<#######<####.D###@#E###m,G#######<#######<###`(#<###### #<#######<#######<#######<###(M.D### <#######<#######<#######<###M.E###- <#######<#######<#######<####-D#####<#######<#######<#######<### L.E###`,E####D#####-E######<###J.mD###L.F###n#<#######<#######<###@ lF###n#E###mD### ###<#######<###`)D### LE#######<#######<#######<####)mF###L#<#######<#######<#######<###nF### ##<#######<#######<#######<###@*LLD### ##<#######<#######<#######<###` D#######<#######<#######<#######<###@)F### L##<#######<#######<#######<###`#E####.MG### #<#######<#######<###@H.lD####-mF### NF###`###<#######<###@*#<#######<#######<#######<#######<###? D#######<#######<#######<#######<####* <#######<#######<#######<#######<########8#######body wt # ?######}@.=*B@j##;@# #3?#@#####@#####?cg@#####H@######$@!_ff @#####@#####i@######O@#####@#####\\@##@33#@#####A@HpQ?YPO? #######@#####K@######Y@#z#J@####=@j,# ?##`d;? ######h@########8#######brain wt#?33 @#####pz@#####]@#####\\@#######@######I@#####@#####0z@#####x@#####\\@#9@### ##@@#####`y@#####@#####P@#####Q@#####`f@######L@######?# ? `?33(@#####e@#####c@#####{@#####Pc@!_ff? #######@#####f@########8#######logbody ?HV#@># ?H##?Dc##/q? #I  E#@# ?#@\\ #5-#@#`##@######? ? ##@9###@B# ? _#@#@#@?O?#b#Xw_\\6N#_w? 6 ]@#######?z? _##@L9###@# ?H##?Dc##/q?^1(*/F# ? #@\\ #5-#@#`##@######? ? ##@9###@B#^@#_? 1(*/F?O?#b#Xw_\\6N#_w?6 ]@#######?z? ^1(*/FL9###@Math Applications>Minitab. Use File>Open Worksheet to load the Minitab file u:\\msu\\course\\stt\\201\\s\\brain.mtw, which contains the body weight (in kg) and brain weight (in g) of several animals. Look at column C1 in the worksheet and notice that it contains names of the animals of very different size. Consider what would happen if you tried to plot these weights on a number line with units in inches. A guinea pig's weight would plot at 1 inch, a cat's at 3.3 inches, and a horse's at 521 inches (or about 14.5 yards). 1. If the body weight of Brachiosaurus is plotted on the number line with units in inches, how many yards would the point be from the origin? (Recall that there are 36 inches in one yard.) ____________ Because of the different scales of magnitude, the data have been transformed by taking logarithms (base 10). Use your calculator or knowledge of logarithms to find the logs of the following numbers: .01, 1.0, 100, 10000. Log10(0.01) = _______Log10(1.00) = ____________Log10(100) = _____________Log10(10000) = __________ Notice that the logs of these numbers give their order of magnitude, and they don't vary too much. From now on we concentrate only on the transformed data, given in the columns labeled logbody, logbrain, delogbody, and delogbrain. These last two columns have data with the three dinosaurs deleted. Use Graph>Scatter Plot to get a plot of logbrain versus logbody. Choose Simple plot, select logbrain as Y variable, and logbody as X variable. The scatterplot will appear in a graph window. It should look similar to the graph below. If it does not, please raise your hand and have the lab instructor check your screen before proceeding further. Scatterplot of logbrain vs logbody 4 logbrain 3 2 1 0 -2 -1 0 1 2 logbody 3 4 5 2. What kind of association do you see on the scatterplot: positive, negative, or no association? _____________ Find the points on the scatterplot that represent dinosaurs, and circle them. Use Stat>Basic Stat>Correlation to get the correlation between logbody and logbrain: select logbody and logbrain as the variables, uncheck \"display p-values\" box, and do not check \"store matrix\" box. Record the correlation coefficient: 3. rbetween logbody and logbrain = ______________ Use Stat>Regression>Regression>Fit Regression Model to find the regression equation for logbrain variable (response variable y) vs logbody (continuous predictor variable x). 4. What is the slope of the regression line reported by Minitab? _______________(Scroll through the output to Regression Equation part). Use Stat>Basic Statistics>Display Descriptive Statistics to get the descriptive statistics of logbody and logbrain. Recall that logbody is the x-variable and logbrain is the y-variable. Record their means below: 5. x __________ 6. y __________. Plot the point ( x , y ) on the scatter plot (this is not one of the data points displayed since the means are not of logbody and logbrain do necessarily correspond to one particular animal). Use the regression equation reported by Minitab, and plug in the value of x as logbody value in the equation. 7. The resulting value of y (logbrain) is _______. Is your answer to #7 the same as answer to #6, up to rounding error? Since the coordinates of the point ( x , y ) satisfy the regression equation, this point is called a pin-point or a balance point of the regression line. Draw the regression line on the scatterplot provided above, following these steps. First, plot the balance point ( x , y ) . To get the coordinates of another point on the line, choose an x-value and substitute that value into the regression equation and get the corresponding y-value. Write the coordinates in the table below: x y Note that the point you got does not have to be one of the data points. Plot this second point on the scatterplot, and finally, draw the line determined by these two points. This is a plot of the regression line. We will now use the regression equation to predict brain weight of animals based on their body weight. 8. For a rabbit, the value of logbogy in the data column is x= _________. 9. The actual observed value of logbrain for a rabbit is y=_____________. 10. Using the regression equation and plugging in the value of x for a rabbit into it, the predicted value of logbrain is y ________________. Round your answer to 5 decimal places. 11. The error of prediction (also called residual) using logbrain versus logbody equation for a rabbit is y y _________________. Round your answer to 5 decimal places. In Minitab's output for regression, find the table called Analysis of Variance. In this table, the row Error and Column Adj SS (Adjusted Sum of Squares) give the SSE, sum of squares of errors of prediction. This quantity, SSE, equals the sum of squares of errors of prediction for all animals in the data. In #11, you computed one of such errors, for a rabbit. 12. For the regression of logbrain versus logbody, SSE=_______________. We now repeat the regression analysis but with the three "dinosaur" points removed. The logged data with dinosaurs deleted appears in the columns, "delogbody" and "delogbrain". Use Stat>Basic Stat>Correlation, select delogbody and delogbrain as the variables to find the correlation between delogbody and delogbrain: 13. rbetween delogbody and delogbrain = ______________. How does this correlation coefficient compare to the one with dinosaurs included from #3? Use Stat>Regression>Regression>Fit Regression model to find the regression equation for delogbrain (response variable y) vs delogbody (predictor variable x). We will now use this equation to predict the delogbrain for a rabbit. 14. Using the regression equation of delogbrain versus delogbody, and plugging in the value of x for a rabbit into it, the predicted value of delogbrain is y ________________. Round your answer to 5 decimal places. 15. The error of prediction (also called residual) using delogbrain versus delogbody equation for a rabbit is y y ________________. Round your answer to 5 decimal places. Compare your answers to #11 and #15. Which one is smaller in absolute value? 16. For the regression of delogbrain versus delogbody, SSE=_____________. Compare your answers to # 12 and #16. Which one is smaller (i.e. prediction is better)?________________ To see why the error of prediction changes when dinosaurs are deleted from the data, we will use Minitab to plot both lines, with and without dinosaurs, on the scatterplot. Use Graph>Scatterplot, choose option With Regression. In the table of Y and X variables, enter logbrain as Y and logbody as X in the first row, and in the second row enter delogbrain as Y and delogbody as X. Click on Multiple Graphs and select the option Overlaid on the same graph. Click OK and again OK. The graph with two lines plotted in different colors will appear. 17. Which of the two regression lines (dinosaurs included or dinosaurs excluded) passes closer to most points on the scatterplot? Circle one: dinosaurs included / dinosaurs excluded. In Minitab's output of Model Summary, find the R2 values for the two regressions. In the regression output, Minitab uses notation R-sq for R2, and it gives the percentage of variation in the y-variable that is explained by the x-variable. So the higher the R2 value, the better the x-variable is as a predictor of the y-variable for a given data set. When submitting your answers to questions 18 and 19 to LON-CAPA, enter just the numbers between 0 and 100 without % symbol. 18. R2 = ___________ (dinosaurs included) 19. R2 = ______________ (dinosaurs excluded) Which R2 is larger?_________________________ Does your answer agree with answer to #17? ___________ The text discusses r 2 , the squared correlation coefficient. R2 reported by Minitab is simply r 2 expressed as a percentage: R 2 r 2 100% . Check this using delogbrain versus delogbody data. From #13: 20. The squared correlation coefficient for delogbrain versus delogbody r 2 =____________. Compare the number you obtained to your answer to #19