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Go to Exercise #9 on Class Handout #5:. 9 .

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  1. Go to Exercise #9 on Class Handout #5: 9. Open the SPSS data file Job Satisfaction. In the “NORMALITY AND DATA TRANSFORMATION” section of the textbook, read the subsections “Normality” and “Data Transformation”. Then, follow the instructions in the subsection “Practical Example” beginning on page 60 to obtain Tables 2.10, 2.11, and 2.12, and Figures 2.8, 2.9, 2.10. Compare the syntax file commands generated by the output with those shown in the textbook.

  2. Go to Exercise #1 on Class Handout #6: Do each of the following for the line Y = 0.5 + 0.9X : 1. (a) (b) (c) Find the value of Y for each of the values of X listed: X = 0 X = 1 X = 2 X = 3 X = 4 X = 5 Y = Y = Y = Y = Y = Y = 0.5 1.4 2.3 3.2 4.1 5.0 5 On the graph, label the horizontal axis from 0 to 5, label the vertical axis from 0 to 5, and draw a graph of the line. 4 3 Write an interpretation of the slope of the line. 2 Each time X increases by 1 (one) unit, Y increases by 0.9 units. 1 0 1 2 3 4 5

  3. Do each of the following for the line Y = 10 – 1.5X : 2. (a) (b) (c) Find the value of Y for each of the values of X listed: X = 0 X = 1 X = 2 X = 3 X = 4 X = 5 Y = Y = Y = Y = Y = Y = 10 8.5 7.0 5.5 4.0 2.5 10 On the graph, label the horizontal axis from 0 to 10, label the vertical axis from 0 to 10, and draw a graph of the line. 8 6 4 Write an interpretation of the slope of the line. 2 Each time X increases by 1 (one) unit, Y decreases by 1.5 units (i.e., Y increases by –1.5 units). 0 2 4 6 8 10

  4. Class Handout #6 (Chapter 3) Definitions Simple Linear Regression dependent (response) variable (Y) a variable we predict (model) from one or more other variables independent (explanatory) variable (X) a variable from which predictions about a dependent (response) variable are made; an independent (explanatory) variable can also be called a predictor variable simple linear regression data a sample of n pairs of observations of an independent variable X and a dependent variable Y : (x1 , y1) (x2 , y2) … (xn , yn) observational regression data regression data where the values of the independent variables are not set prior to observing the dependent variable experimental regression data regression data where the values of the independent variables are controlled using a designed experiment (set prior) to observing the dependent variable Go to Exercise #3 on Class Handout #6:

  5. 3. An appliance store conducts a 5-month experiment to study the prediction of sales revenue from advertising expenditure. MONTH ADVERTISING SALES NUMBER EXPENDITURE EXPEDITURE ($100s) ($1000s) 1 1 1 2 2 1 3 3 2 4 4 2 5 5 4 (a) Identify the dependent (response) variable and the independent (explanatory) variable for a regression analysis. The dependent (response) variable is Y = ”sales revenue”, and the independent (explanatory) variable is X = “advertising expenditure”. (b) Does the data appear to be observational or experimental? Since the advertising expenditures do not look random, it appears that the data is experimental.

  6. scatter plot a graphical display for simple linear regression data where the independent variable is scaled on a horizontal axis and the dependent variable is scaled on a vertical axis; dots are used to represent each data point. least squares line the equation of the line which minimizes the sum of the squared vertical distances between the data points and the line We shall use Y to represent an observed value of the dependent (response) variable, we shall use Y to represent a predicted value of the dependent (response) variable, and we shall of course use Y to represent the mean of all observed values of the dependent (response) variable. ^ ^ The (estimated) unstandardized linear equation is where Y = a + bX ^ a is the intercept, which is the (predicted) value Y when X = 0 (Note: This predicted value often is not meaningful in practice.), and b is the slope, which is the (estimated) amount Y changes on average with each increase of one unit in the variable X. The (estimated) standardized linear equation is where ZY = ZX • (beta) is the standardized regression coefficient, • ZX is the z-score for the value of X, • and • ZY is the z-score for the predicted value of Y.

  7. 3. An appliance store conducts a 5-month experiment to study the prediction of sales revenue from advertising expenditure. MONTH ADVERTISING SALES NUMBER EXPENDITURE EXPEDITURE ($100s) ($1000s) 1 1 1 2 2 1 3 3 2 4 4 2 5 5 4 (a) Identify the dependent (response) variable and the independent (explanatory) variable for a regression analysis. The dependent (response) variable is Y = ”sales revenue”, and the independent (explanatory) variable is X = “advertising expenditure”. (b) Does the data appear to be observational or experimental? Since the advertising expenditures do not look random, it appears that the data is experimental. (c) It can be proven that the slope and intercept of the least squares line can be obtained from the bivariate using the following formulas: (x–x)(y–y) b = a = y–bx (x–x)2

  8. Use these formulas to find the equation of the least squares line. 5 n = x = y = 3 2 (x–x)2 = (1  3)2 + (2  3)2 + (3  3)2 + (4  3)2 + (5  3)2 = 10 (x–x)(y–y) = (1  3)(1  2) + (2  3)(1  2) + (3  3)(2  2) + (4  3)(2  2) + (5  3)(4  2) = 7 (x–x)(y–y) 7 — = 0.7 10 = 2 – (0.7)(3) = – 0.1 b = a = y–bx = (x–x)2 ^ The least squares line can be written y = – 0.1 + 0.7x . (d) Use the least squares line to predict sales revenue resulting from an advertising expenditure of $250. ^ y = – 0.1 + 0.7(2.5) = 1.65 thousand dollars = $1,650

  9. This exercise makes use of the data in Exercise #1 of Class Handout #1. The data is stored in the SPSS data file survey. The prediction of yearly income ($1000s) from age is of interest, and the 30 individuals selected for the data set are treated as a random sample for simple linear regression. A 0.05 significance level is selected for all hypothesis testing. 4. (a) Identify the dependent (response) variable and the independent (explanatory) variable for a regression analysis. The dependent (response) variable is Y = “yearly income ($1000s)”, and the independent (explanatory) variable is “age”. (b) Does the data appear to be observational or experimental? Since the ages look random, it appears that the data is observational. We shall continue with this exercise next class. (c) In the document titled Using SPSS Version 19.0, use SPSS with the section titled Performing a simple linear regression with bivariate data, with checks of linearity, homoscedasticity, and normality assumptions to do each of the following: Follow the instructions in the first five steps to graph the least squares line on a scatter plot; then decide whether or not the linearity assumption appears to be satisfied.

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