1.4 - Linear Regression and MS Excel

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1 1.4 - Linear Regression and MS Excel Regression is an analytic technique for determining the relationship between a dependent variable and an independent variable. When the two variables have a linear correlation, you can develop a simple mathematical model for the relationship between the two variables by finding a line of best fit. You can then use the equation for this line to make predictions by interpolation (estimating between data points) and extrapolation (estimating beyond the range of the data).

2 Example One Suppose a university would like to construct a mathematical model to predict first year marks for incoming students based on their achievement in grade 12. A comparison of these marks from a random sample of first year students is shown below. 1. Construct a scatter plot (using MS Excel) of this data. 2. Classify the linear correlation as negative or positive, based on the scatter plot. 3. Construct a line of best fit (using MS Excel). 4. Use the linear model to predict (interpolate) the first year average for a student who had an 82 average in grade Use the linear model to predict (extrapolate) the grade 12 average for a student with a first year average of 65.

3 Example One

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6 Correlation Coefficient (r) This coefficient gives a quantitative measure of the strength of a linear correlation. In other words, the correlation coefficient indicates how closely the data points cluster around the line of best fit. The coefficient always has values in the range from -1 to 1. It is important to be aware that increasing the number of data points used in determining a correlation improves the accuracy of the mathematical model.

7 Coefficient of Determination (r 2 ) A number from 0 to +1 that gives the relative strength of the relationship between two variables. For example, if r 2 = 0.44, this means that 44% of the variation of the dependent variable is due to variation in the independent variable. Residual Values The vertical distance between a data point and the line of best fit is called the residual value. If the model is a good fit, the residuals should be fairly small and show both positive and negative values. Residual values should be graphed with the line of best fit.

8 Example Two Use the data from example one to create a residual plot. Use the scale of the residual plot as well as the location of the points to comment on whether the line of best fit chosen is a good fit.

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10 Outliers Outliers can skew a regression analysis, but they could also simply indicate that the data really do have large variations. A comprehensive analysis of a set of data should look for outliers, examine their possible causes and their effect on the analysis, and discuss whether they should be excluded from the calculations. Note: Outliers have less effect on larger samples.

11 Example Three To evaluate the performance of one of its instructors, a driving school tabulates the number of hours of instruction and the driving-test scores for the instructor s students. a) What assumption is the management of the driving school making? Is this assumption reasonable? b) Create the scatter plot in MS Excel. Use the scatter plot to determine whether it suggests that the instructor is an effective teacher. c) Comment on any data that seems unusual. d) Determine the effect of any outlier on your analysis.

12 Example Three