+ Statistical Methods in

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1 + Statistical Methods in Practice STAT/MATH Discovering Statistics 2nd Edition Daniel T. Larose Dr. A. B. W. Manage Associate Professor of Mathematics & Statistics Department of Mathematics & Statistics Sam Houston State University Chapter 4: Correlation and Regression Lecture PowerPoint Slides + Chapter 4 Overview 3 + The Big Picture Scatterplots and Correlation 4.2 Introduction to Regression 4.3 Further Topics in Regression Where we are coming from and where we are headed Chapter 3 showed us methods for summarizing data using descriptive statistics, but only one variable at a time. In Chapter 4, we learn how to analyze the relationship between two quantitative variables using scatterplots, correlation, and regression. In Chapter 5, we will learn about probability, which we will need in order to perform statistical inference : Scatterplots and Correlation Objectives: Construct and interpret scatterplots for two quantitative variables. Calculate and interpret the correlation coefficient. Determine whether a linear correlation exists between two variables. 5 6 Scatterplots Whenever you are examining the relationship between two quantitative variables, your best bet is to start with a scatterplot. A scatterplot is used to summarize the relationship between two quantitative variables that have ben measured on the same element. A scatterplot is a graph of points (x,y), each of which represents one observation from the data set. One of the variables is measured along the horizontal axis and is called the x variable. The other variable is measured along the vertical axis and is called the y variable. Lot x=square footage (100s of sq ft) y=sales price ($1000s) Harding St Newton Ave Stacy Ct Eastern Ave Second St Sunnybrook Rd Ahlstrand Rd Eastern Ave

2 Scatterplots The relationship between two quantitative variables can take many different forms. Four of the most common are: Positive linear relationship: As x increases, y also tends to increase. 7 Correlation Coefficient Scatterplots provide a visual description of the relationship between two quantitative variables. The correlation coefficient is a numerical measure for quantifying the linear relationship between two quantitative variables. 8 Negative linear relationship: As x increases, y tends to decrease. The correlation coefficient r measures the strength and direction of the linear relationship between two variables. The correlation coefficient r is å(x - x )( y - y) r = (n -1)s x s y No apparent relationship: As x increases, y tends to remain unchanged. where s x is the sample standard deviation of the x data values, and s y is the sample standard deviation of the y data values. Nonlinear relationship: The x and y variable are related, but not in a way that can be approximated using a straight line. + Equivalent Computational Formula for Calculating the Correlation Coefficient r Calculating Correlation Coefficient 10 r xy xy / n x x / n y y / n x x 41.1 n y y 50.6 n Calculating Correlation Coefficient xx sx n x x y y s y yy n r n 1 sxsy Properties of r 1. The correlation coefficient r is always -1 r When r = +1, a perfect positive relationship exists between x and y. 3. Values of r near +1 indicate a positive relationship between x and y. The closer r gets to +1, the stronger the evidence for a positive The variables are said to be positively associated. As x increases, y tends to increase. 4. When r = -1, a perfect negative relationship exists between x and y. 5. Values of r near -1 indicate a negative relationship between x and y. The closer r gets to -1, the stronger the evidence for a negative The variables are said to be negatively associated. As x increases, y tends to decrease. 6. Values of r near 0 indicate there is no linear relationship between x and y. The closer r gets to 0, the weaker the evidence for a linear The variables are not linearly associated. A nonlinear relationship may exist between x and y. 12 2

3 Properties of r 13 Test for Linear Correlation There is a simple comparison test that will tell us whether the correlation coefficient is strong enough to conclude that the variables are correlated. 15 Comparison Test for Linear Correlation 1. Find the absolute value r of the correlation coefficient. 2. Turn to the Table of Critical Values for the Correlation Coefficient and select the row corresponding to the sample size n. 3. Compare r to the critical value from the Table. If r > critical value, you can conclude x and y are linearly correlated. If r > 0, they are positively correlated. If r < 0, they are negatively correlated. If r is not greater than critical value, then x and y are not linearly correlated. 3

4 + 4.2: Introduction to Regression Objectives: Understand and calculate the range of a data set. Explain in my own words what a deviation is. Calculate the variance and the standard deviation for a population or a sample The Regression Line Section 4.1 introduced the correlation coefficient. In this section, we learn how to approximate the linear relationship between two numerical variables using the regression line and regression equation. City x = low temp y = high temp Minneapolis Boston Chicago Philadelphia Cincinnati Wash., DC Las Vegas Memphis Dallas Miami We write the equation of the regression line as ˆ y b 1 x b 0 The Regression Line Equation of the Regression Line The equation of the regression line that approximates the relationship between x and y is where the regression coefficients are the slope, b 1, and the y intercept, b 0. The equations of these coefficients are (x x )(y y ) b 1 (x x ) 2 ˆ y b 1 x b 0 b 0 y (b 1 x ) 21 Interpreting Slope and y-intercept In statistics, we interpret the slope of the regression line as the estimated change in y per unit increase in x. The y-intercept is interpreted as the estimated value of y when x equals 0. b 1 = 0.9. For each increase of 1F in low temp, the estimated high temp increases by 0.9F. b 0 = 20. When the low temp is 0F, the estimated high temp is 20F. 22 Note: The hat over the y (pronounced y-hat ) indicates this is an estimate of y and not necessarily an actual value of y. The slope b 1 and the correlation coefficient r always have the same sign. b 1 is positive if and only if r is positive. b 1 is negative if and only if r is negative. Predictions and Prediction Error 23 Predictions and Prediction Error 24 We can use the regression equation to make estimates or prediction. For any value of x, the predicted value of y lies on the regression line. Example: Low Temp = 50F y ˆ 0.9x (50) The prediction error, or residual, measures how far the predicted y-hat value is from the actual value of y observed in the data set. The prediction error may be positive or negative. Positive prediction error: The data value lies above the regression line, so the observed value is greater than the predicted value for the given value of x. Negative prediction error: The data value lies below the regression line, so the observed value is less than the predicted value for the given value of x. Prediction error equal to zero: The data value lies directly on the regression line, so the observed value of y is exactly equal to what is predicted for the given value of x. Note: The predicted high temp for a city with a low temp of 50F is 65F. Dallas had a low temp of 50F and an actual high temp of 70F. 4

5 Cautions with Regression 25 Coefficient of Determination r 2 26 The correlation coefficient and regression line are both sensitive to extreme values. The coefficient of determination r 2 measures the goodness of fit of the regression equation to the data. We interpret r 2 as the proportion of the variability in y that is accounted for by the linear relationship between y and x. The values that r 2 can take are 0 r 2 1. Extrapolation consists of using the regression equation to make estimates or predictions based on x- values that are outside the range of the x-values in the data set. 5