Regression Analysis - Lecture notes 5 PDF

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modeling linear trends Section 4.3 Regression Line A tool for making predictions about future observed values Has the form y = a + bx, where a is the y-intercept and b is the slope Usually generated using appropriate technology Example: Regression Equation

A The scatterplot shows a fairly strong positive linear trend. The regression equation has a slope of 2.16 and a y-intercept of 30.46. The positive trend indicates that players who hit more home runs tend to have more RBIs. • Example: Using a Regression Equation The scatterplot shows a negative linear trend. As age of car increases, value tends to decrease. The regression equation is: predicted value = 21375 – 1215 age • Example: Using the Regression Equation predicted value = 21375 – 1215 age Use the regression equation to predict the value of a car that is 12 years old. predicted value = 21375 – 1215 age predicted value = 21375 – 1215(12) predicted value =$6795 • Finding the Regression Equation • To find the regression equation using technology, follow the same steps as for finding the correlation coefficient. • Example The table below shows the heights and weights for six women. Find the regression equation that describes the relationship between height and weight. NOTE: We previously determined that this data followed a linear trend, so it is appropriate to find the regression equation. • Finding the Regression Equation Using StatCrunch Enter the data into StatCrunch. STAT > Regression > Simple Linear Select the x-variable, select the y-variable, select COMPUTE. • StatCrunch Output • Simple linear regression results: Dependent Variable: Weight Independent Variable: Height Weight = -442.88235 + 9.0294118 Height Sample size: 6 R (correlation coefficient) = 0.88093363 R-sq = 0.77604407 • Example: Using the Regression Equation

Weight = -442.882 + 9.03 Height Use the regression equation to predict the weight of a woman who is 65 inches tall. Weight = -442.882 + 9.03 Height Weight = -442.882 + 9.03 (65) Weight = 144.07 inches • Notes about the Regression Equation • Order matters. If x and y are switched, the regression equation will change. • We use the x-variable to make predictions about the y-variable, so the x-variable is called the explanatory or predictor variable. It is also called the independent variable. • The y-variable is the response or predicted variable. It is also called the dependent variable. • Example The table below shows the heights and weights for six women. Find the regression equation that describes the relationship between height and weight. This time use weight as the predictor or explanatory variable (x) and height as the predicted or response variable (y). • Example Simple linear regression results: Dependent Variable: Height Independent Variable: Weight Height = 52.397256 + 0.085946249 Weight Sample size: 6 R (correlation coefficient) = 0.88093363 R-sq = 0.77604407 Note: r (correlation coefficient) remains the same; however, the regression equation is different from our previous result. • Interpreting the Slope of the Regression Equation • Slope tells us how much the y-variable changes when the x-variable is increased by 1 unit. • A slope close to 0 means there is no linear relationship between x and y. • Example: Interpreting the Slope Weight = -442.882 + 9.03 Height The slope of this line is 9.03. The y-variable is weight and the x-variable is height. Interpretation: For every additional inch in height, weight tends to increase by 9.03 pounds. Every increase of 1 inch in height is associated with an increase in weight of 9.03 pounds. • Example: Interpreting Slope In a previous example on the association between age of car and value of car, the regression equation was: predicted value = 21375 – 1215 age Interpret the slope of the regression equation. Slope = -1215, x-variable is age, y-variable is value. Interpretation: For each additional year of age, value of car tends to decrease by $1215. Each additional year of age is associated with a decrease of $1215 in value.

Interpreting the y-Intercept of the Regression Equation The y-intercept is the predicted value when x is 0. The y-intercept is meaningful only if it makes sense for x to equal 0. Example: Interpreting the y-Intercept In a previous example on the association between age of car and value of car, the regression equation was: predicted value = 21375 – 1215 age Interpret the y-intercept of the equation, if appropriate. y-intercept = 21375. It is the predicted value when x (age) is 0. In other words, when the car is new, its value is $21,375. • Example: Interpreting the y-Intercept In a previous example on the association between height and weight in women, the regression equation was: Weight = -442.882 + 9.03 Height • • • •

Interpret the y-intercept, if appropriate. y-intercept = -442.882. It is the predicted value for weight if x (height) is 0. It is impossible to weigh -442 pounds and it is impossible for a woman to be 0 inches tall, so in this case the y-intercept is meaningless. • Evaluating the Linear Model • Section 4.4 • Cautionary Notes Regarding Regression • Don’t use linear models to describe non-linear associations. Always look at a scatterplot first! • Correlation is not causation! An association between two variables is not sufficient evidence to conclude that a cause-and-effect relationship exists between the variables. • Beware of outliers that can have a big effect on r. Always check the scatterplot for outliers first. • Don’t extrapolate! Don’t make predictions beyond the range of the data, because we are not sure that the linear trend will continue beyond the range of the data. • Example: Extrapolation In a previous example we found there was a strong linear relationship between heights and weights in women, and the regression equation is Weight = -442.882 + 9.03 Height. What weight does this equation predict for a woman who is 36 inches tall? • Example: Extrapolation Weight = -442.882 + 9.03 Height Weight = -442.882 + 9.03(36) = -117.8 pounds Note: The range of the data was for women 61 to 68 inches tall. It is not appropriate to use the regression equation to predict the height for a 36 inch tall woman since 36 is beyond the range of the data (extrapolation). • Coefficient of Determination: r2

The square of r, the correlation coefficient Usually converted to a percentage, so always between 0% and 100% Measures how much variation in the response variable is explained by the explanatory variable • The larger r2, the smaller the amount of variation or scatter about the regression line. • Example: r2 For the data on car age and predicted value, r = -0.778. Compute and interpret r2. r2 = (-0.778)2 = .605, so r2 = 60.5%. Car age explains about 60.5% of the variation in car value. • • •...