Correlation and Regression

Transcription

1 Correlation and Regression 1 Association between Categorical Variables 2 2 Association between Quantitative Variables 3 3 Prediction 8 1

2 1 Association between Categorical Variables The response (or dependent) variable is the outcome variable we expect to be affected by changes in the explanatory (or independent) variable. Two variables are associated if certain values of one variable tend to occur with certain values of the other. Examples: Example 1 Pesticides, p. 95. Also, contingency tables and conditional proportions. 2

3 2 Association between Quantitative Variables Scatterplots display relationships between two quantitative variables. The explanatory (or independent) variable is plotted on the horizontal axis. The response (or dependent)) variable is plotted on the vertical axis. 3

4 Two variables are positively associated if above-average observations of both variables occur together. Two variables are negatively associated if above-average observations of one variable occur with below-average observations of the other and vice versa. 4

5 Example 2 First Exam Grade vs. Final Exam Grade Third Exam Grade Final Exam Grade Example 3 Age and Price of Corvettes Age (years) Price ($100) Example 4 Find variable pairs that are positively associated, negatively associate, and seemingly unassociated for the student data. 5

6 Correlation Coefficient r The correlation coefficient r measures the direction and strength of the linear association between two quantitative variables. r = 1 n 1 1 n 1 ( ) ( ) xi x yi y i s x z xi z yi i s y The correlation coefficient is always between -1 and 1: 1 r 1 The sign of r indicates positive or negative correlation. The magnitude of r indicates the strength of the correlation. Calculating r by hand is tedious, so we usually use technology. 6

7 Example 5 Guess the r values for the scatterplots on slide 4. Example 6 Below are retail regular unleaded gasoline prices and daily high temperatures for four days. By hand, make a scatterplot and calculate r using the lists in your calculator. Price ($) High Temp ( F )

8 3 Prediction Review of Linear Equations Linear equations in one variable have the form y = b 1 x + b 0 x is the independent (or explanatory) variable y is the dependent (or response) variable b 0 is the y-intercept (where the line intersects the y-axis) b 1 is the slope (the change in the dependent variable divided by the change in the independent variable) 8

9 Example 7 Graph the following equations: y = 2x + 3 3(y + 1) = 2x 1/2 y = x/2 y = 3x y = 4 x = 5 9

10 Regression Lines A regression line is the line that best fits a data set and helps approximate relationships between variables. The y-values on the regression line that correspond to the x-values from the data set are denoted ŷ i. The error (or residual) associated with the data point (x i, y i ) is the quantity y i ŷ i 10

11 The least-squares regression line is the line that minimizes the sum of squared errors i (y i ŷ i ) 2. Its equation is where ŷ = b 1 x + b 0 b 1 = r s y s x and b 0 = y b 1 x These calculations are tedious so we usually use technology. Good least-squares regression lines are used to predict outcomes of one quantitative variable based on observations of another. A prediction of y using an x-value outside the range of x-value observations may not be reasonable and is called extrapolation. Note that changing x by one standard deviation results in changing y by r standard deviations. ŷ always passes through the point (x, y). 11

12 Example 8 The following data are observations of classroom temperatures and mean test scores for five sections of MATH 1530: Temp ( ) Score (%) Find the least-squares regression line with your calculator and Minitab. Could you use the regression line to predict the mean test score when the temperature is 70 o F? 72.5 o F? 12

13 Outliers and Influential Observations An outlier is a data point that lies far from the regression line relative to the other points. An influential observation is a data point whose removal from the set will considerably change the regression line. Example 9 The following are ages (in years) and prices (in $100) of twelve Corvettes: Age Price Find the least-squares regression line equation, r, and identify any outliers and influential observations. Example Presidential Election 13

14 Coefficient of Determination r 2 The coefficient of determination r 2 is defined to be the proportion of variation in the observed values of the response variable explained by the regression line: r 2 i (ŷ i y) 2 variance of predicted values ŷ i (y = i y) 2 variance of observed values y It turns out that the coefficient of determination is equal to the square of the correlation coefficient. This is the reason for using the notation r 2. Example 11 Calculate r 2 for Example

15 Cautions in Analyzing Associations Using a regression model to predict values of the response variable based on observations of the explanatory variable that are outside the region of observed values is called extrapolation. Outliers are data points far from the regression line. Influential observations are data points whose removal would cause a dramatic change in the regression line. Example: Association does not imply causation! Lurking variables influence the association between the variables of main interest. Example: Ice Cream Causes Drowning! Confounding. 15