ࡱ> M x8bjbj== WWW1>l.......B8<B9|$6% 18888888$'; G= 9E.9"99 9..6e994..6898M~1..3 0cB>m23{909"2=m`=3BB....This handout contains: Notes on how to use a T! 83 to do linear regression. A copy of the transparencies about education vs. income. A copy of the transparencies about trends in job safety. A copy of the transparencies about the population of St. Louis. A discussion of r, the correlation coefficient and the interpretation of r2, sometimes called the coefficient of determination. Two regression examples using real data. Notes on how to use a TI 83 to do linear regression: The T! 83 can be used to do linear regression. Perhaps the easiest way to find out how to do it is to have someone who knows how show you. Using Google to search on TI 83, linear regression yields over 21,000 hits. The first few pages of results include dozens of different step by step guides on how to do this. Ive included the one from Tompkins Cortland Community College in Dryden, NY. It seems OK and it explains how to Turn on diagnostics. If you do that then the correlation coefficient and its square will both show up automatically along with the values of a and b. The page at www.mtsu.edu/~math141/regress.html also seemed quite clear. If you are going this route it is probably best to find one set of instructions and stay with them because there are lots of minor variations in how to proceed. Here is www.acad.sunytccc.edu/instruct/sbrown/ti83/regress.htm  HYPERLINK "http://www.tc3.edu/default.asp" TC3!  HYPERLINK "http://www.acad.sunytccc.edu/instruct/sbrown/" StanBrown!  HYPERLINK "http://www.acad.sunytccc.edu/instruct/sbrown/ti83/" TI-83/84/89! Correlation&RegressionrevisedSep29,2003Scatter Plot, Correlation, and Regression ontheTI-83/84 Copyright 20022004 StanBrown,  HYPERLINK "http://oakroadsystems.com/" OakRoadSystems Summary: When you have a set of (x,y) data points and want to find the best equation to describe them, you are performing a regression. The Statistics Manual (pages 201-202) has you first do correlation and regression analysis, then plot the points and the line. I think it's faster and more logical to combine the steps for the two procedures, as shown on this page. Students in almost any math course can also use these techniques. Dial (x)02356Temp, F (y)6"1"3"10"16When you need to do a regression on a data set like this one, there are three steps to follow on your TI-83/84: Plot the points as a scatter plot. Perform the regression. Plot the regression line. Step 0. Setup Set floating point mode.[MODE] [INCLUDEPICTURE "../../../Documents%20and%20Settings/rr/Desktop/regress_files/ticrsrdn.gif" \* MERGEFORMATINET ] [ENTER]Go to the home screen[2nd] [QUIT] [CLEAR]Turn on diagnostics. (Look for D above the [x-1] key.)[2nd] [CATALOG] [D] Scroll down to DiagnosticOn [ENTER] [ENTER]The calculator will remember these settings when you turn it off: next time you can start with Step 1. Step 1. Make the Scatter Plot Before you even run a regression, you should first plot the points and see whether they seem to lie along a straight line. If the distribution is obviously not a straight line, don't do a linear regression. (Some other form of regression might still be appropriate, but that is outside the scope of this course.) Turn off other plots.[Y=] Move cursor to each highlighted = sign or Plot number and press [ENTER] to deactivate.Enter the numbers.[STAT] [1] selects the list-edit screen.Move cursor onto the label L1 at top of first column, then [CLEAR] [ENTER] erases the list. Enter the x values.Move cursor onto the label L2 at top of second column, then [CLEAR] [ENTER] erases the list. Enter the y values.Set up the scatter plot.[2nd] [STATPLOT] [1] [ENTER] turns Plot 1 on.[INCLUDEPICTURE "../../../Documents%20and%20Settings/rr/Desktop/regress_files/ticrsrdn.gif" \* MERGEFORMATINET ] [ENTER] selects scatter plot.[INCLUDEPICTURE "../../../Documents%20and%20Settings/rr/Desktop/regress_files/ticrsrdn.gif" \* MERGEFORMATINET ] [2nd] [L1] ties list 1 to the x axis.[INCLUDEPICTURE "../../../Documents%20and%20Settings/rr/Desktop/regress_files/ticrsrdn.gif" \* MERGEFORMATINET ] [2nd] [L2] ties list 2 to the y axis.Plot the points.[ZOOM] [9] automatically adjusts the window frame to fit the data, but does not adjust the grid spacing.(optional) [WINDOW], set Xscl=1 and Yscl=5, then [GRAPH] to redisplay it.Step 2. Perform the Regression Set up to calculate statistics.[STAT] [INCLUDEPICTURE "../../../Documents%20and%20Settings/rr/Desktop/regress_files/ticrsrrt.gif" \* MERGEFORMATINET ] [4] pastes LinReg(ax+b) to home screen.[2nd] [L1] [,] [2nd] [L2] defines L1 as x values and L2 as y values.Set up to store regression equation.[,] [VARS] [INCLUDEPICTURE "../../../Documents%20and%20Settings/rr/Desktop/regress_files/ticrsrrt.gif" \* MERGEFORMATINET ] [1] [1] pastes Y1 into the LinReg command.Make it so![ENTER] shows correlation and regression statistics and pastes the regression equation into Y1.INCLUDEPICTURE "../../../Documents%20and%20Settings/rr/Desktop/regress_files/regress1.gif" \* MERGEFORMATINET Write down a (slope), b (y intercept), r (correlation coefficient; r* is our symbol). Round a and b to two more decimal places than your actual y values have; remember that final rounding should be done only at the end of calculations. Round r* to two decimal places unless it's very close to 1 or to 0. a = -3.52 b = 6.46 r* = -0.992 Step 3. Display the Regression Line Show line with original data points.[GRAPH]INCLUDEPICTURE "../../../Documents%20and%20Settings/rr/Desktop/regress_files/regress2.gif" \* MERGEFORMATINET  The text of the transparencies about education vs. income: These notes drawn from Statistics, by Freedman, Pisani, Purves and Adhikari. Does education pay? Figure 1 shows the relationship between income and education, for a representative sample of 637 California men age 25-29 in 1988. The summary statistics are: average education is 12.5 years, average income is $19,700 The regression estimates for average income at each educational level fall along the regression line shown in the figure. The line slopes up, showing that on the average, income does go up with education.  INCLUDEPICTURE "http://www.analytictech.com/mb313/images/regres1.gif" \* MERGEFORMATINET  Figure 1. The regression line. The scatter diagram shows income and education for a representative sample of 637 California men age 25-29 in 1988. The slope of the regression line is $1,400 per year. So far, it looks like education does pay off for the men, at the rate of $1,400 per year. predicted income = ($1,400 per year) x education + $2,200. Example 1. For 676 California women age 25-29 in 1988, there is a relationship between income and education; data are from the Current Population Survey. The relationship can be summarized as follows. average education 12 years average income $11,600 predicted income = ($1,200 per year) x (education) - $2,800. Part (b). Substituting 8 years for education gives ($1,200 per year) x (8 years) - $2,800 = $6,800. This completes the solution. Despite the negative intercept, the predictions are quite reasonable -- for most of the women. In this example, the slope is $1,200 per year. Associated with each extra year of education, there is an increase of $1,200 in income, on the average. The phrase "associated with" sounds like it is talking around some difficulty, and here is the issue: Are income differences caused by differences in educational level, or do both reflect the common influence of some third variable? The phrase "associated with" was invented to let statisticians talk about regressions without having to commit themselves on this sort of point. Often, the slope is used to predict how y will respond, if someone intervenes and changes x. This is legitimate when the data come from a controlled experiment. However, with observational studies the inference is often shaky because of confounding. Take example 1. On the average, the women who finished college (16 years of education) earned about $4,800 more than women who just finished high school (12 years). With an observational study, the slope and intercept of the regression line are only descriptive statistics. They say how the average value of one variable is related to values of another variable, in the population being observed. The slope cannot be relied on to predict how Y would respond if the investigator changes the value of X.  Here is population data for St. Louis: Population of St. LouisYearPopulation1970618,3061980450,7071990395,8572000346,7682003332,223   Industry:All workersYearAnnual1992621719926217199363311993633119946632199466321995627519956275199662021996620219976238199762381998605519986055199960541999605420005920200059202001590020015900(p,n)p : preliminaryn : Excludes Sept. 11th terrorist attacks   The correlation coefficient: The correlation coefficient, traditionally denoted r, is a number which can be computed from the linear regression data which is a measure of how close the data points are to the regression line. The value is always between -1 and 1, the negative values if the slope of the regression line is negative, positive values corresponding the positive slope. In these cases X and Y are said to be negatively correlated or positively correlated respectively. The formula is complicated but the calculator doesnt mind: HYPERLINK "http://davidmlane.com/hyperstat/A67908.html"    HYPERLINK "http://davidmlane.com/hyperstat/A67908.html"   HYPERLINK "http://davidmlane.com/hyperstat/A67908.html"   The values close to 1 and -1 show up in situations where the data points are close to the line (X and Y are highly correlated or strongly correlated). Values of r near 0 when the regression line doesnt do a good job at all or approximating the data points (X and Y are weakly correlated or uncorrelated). One nice feature of the correlation coefficient is that it doesnt depend on the units of measurement. For instance the correlation coefficient between height and weight for students in this class could be computed using inches and pounds or using meters and kilos; the same value would show up in both cases. There is an interpretation commonly given to r2: r2 is the fraction of the variability in the Y variable that you can accurately predict if you know the value of the value of the X variable. Although this is only literally true in certain highly very specific circumstances, it is commonly used as an aid in interpreting things. In particular it emphasizes the fact that mid-range correlation coefficients, for instance r. = .3 arent especially informative; they only allow you to predict about 9% of the variation in Y. A caveat: Correlation is NOT the same as causation. Saying that you can predict a certain amount of the variation in Y if you know the values of X Is not the same as saying that you can explain some of the variation of Y if you know the values of X and it is certainly not the same as saying that a specified change in X will cause a certain change in Y. These issues tend not to be a problem when the X variable is time. In this case the regression line is often called the trend line. There is a clear understanding that extrapolating a trend line to make a forecast is very different from explaining why things will change. In the example below with the St. Louis population you dont think that the population has been going down because time has passed. The population is going down because more people are moving out than are moving in. However the education and income data is the type of example that invites confusion. I dont happen to know the correlation coefficient for that particular data set but r = .37 is the value for a similar data set. Lets suppose that that is the correct value here too. So r2 = .137. This does mean that the residuals have about 14% less variability than the original set of income data. But that is not the same as saying that the schooling explains higher income levels. As a policy maker you shouldnt be willing to use this data to argue that we can improve income levels by having people get more education. That may well improve incomes but this data is only the weakest support for the prediction that it will. 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