4.1 Introduction. 4.2 The Scatter Diagram. Chapter 4 Linear Correlation and Regression Analysis

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1 4.1 Introduction Correlation is a technique that measures the strength (or the degree) of the relationship between two variables. For example, we could measure how strong the relationship is between people s heights and their weights. Regression is a statistical technique that produces a model of the relationship (correlation) between the two variables. A chart that defines a person s ideal weight for a given height is constructed through the statistical technique called regression. Linear Correlation will help us determine if there is a relationship, and how strong, or weak, that relationship is. Possible correlation questions: Is there a relationship between a person s income and intelligence? Is there a relationship between a country s food supply and mortality rate? Is there a relationship between the average length of schooling for citizens in a country and the country s life expectancy? 4.2 The Scatter Diagram Example on The Scatter Diagram pg Consider a sample of 12 randomly selected females attending Nassau Community College. We measure each female s height and weight. Height and weight are the two continuous variables. We ll label the height variable x and the weight variable y. For each female, we have a pair of numbers, height and weight, x and y. The pair of numbers can also be written as (x,y) which is called an ordered pair. The ordered pair (63, 123) would indicate that this student has height 63 inches and weight 123 pounds. A scatter diagram is a graph representing the ordered pairs of data on a set of axes. We start with two lines, a horizontal and vertical line, to represent the two axes. The x-axis (horizontal) represents the x-values; these are the heights. The x-axis is labeled height The y-axis (vertical) represents the y-values; these are the weights. The y-axis is labeled weight. Do you think that the scatter diagram shows a relationship between a female s height and her weight? 1

2 Visual inspection of a scatter diagram can help to determine whether there is an apparent relationship (correlation) between the two variables and what type of relationship this is. 3 basic types of relationships: Positive correlation Negative correlation No linear correlation A positive correlation between two variables, x and y, occurs when high measurements for the x variable tend to be associated with high measurements for the y variable, and low measurements for the x variable tend to be associated with low measurements for the y variable. A negative correlation between to variables, x and y, occurs when high measurements for the x variable tend to be associated with low measurements for the y variable, and low measurements for the x variable tend to be associated with high measurements for the y variable. No linear correlation means there is no linear relationship between the two variables. That is high and low measurements for the two variables are not associated in any predictable straight line pattern. The female height/weight example is an example of. The appearance of positive correlation is one in which the points move up towards the right of the scatter diagram. If we approximate a line through the dots of the scatter diagram, we can see that they follow a straight-line path. A linear relationship has a graph is forms a line. A negative correlation between two variables, x and y, occurs when high measurements for the x variable tend to be associated with low measurements for the y-variable and Low measurements for the x-variable tend to be associated with high measurements for the y-variable. No linear correlation means there is no linear relationship between the two variables. That is, high and low measurements for the two variables are not associated in any predictable straight line pattern. 2

3 How to produce a scatter diagram on the calculator Example 4.2 pg. 174 Use the sample data to construct a scatter diagram on your calculator. Indicate the type of correlation, if any exist, and explain why. 1. Put the x values into L1 on your calculator. (STAT EDIT) 2. Put the y-values into L2. DO NOT SORT the lists! 3. Turn on STAT PLOT (2 nd Stat Plot) (make sure only one stat plot is on) 4. Choose the scatter diagram (first graph) from the Type menu. 5. Xlist should be the list containing the x values. 6. Ylist should be the list containing the y values 7. Clear out data from Y= 8. Click Zoom 9: Zoom Stat Do these variables, x and y have a positive correlation, negative correlation, or no linear correlation? Review Example 4.1 on pg. 173 and Example 4.3 on pg. 175 in the Text. 4.3 The Coefficient of Linear Correlation When a scatter diagram seems to indicate that there is a linear correlation between two variables, our next step is to measure the strength of the relationship between the two variables. By a linear correlation, we mean how closely the points of a scatter diagram closely approximate a straight-line pattern. The closer the points of a scatter diagram approximate a straight-line pattern, the stronger the linear correlation between the two variables. The strength of a linear correlation between the two variables can be numerically measured by Pearson s correlation coefficient, r. To measure how close the points on a scatter diagram come to forming a straight line, we use the following formula: r, is Pearson s Correlation Coefficient, or just correlation coefficient, and it measures the strength of a linear relationship between two variables for a sample. x represents the data values for the first variable y represents the data values for the second variable n represents the number of pairs of data values The values for r can range from -1 to 1 ( 1 r 1). 3

4 Interpreting the Values of r A value of r = 1 represents the strongest positive linear correlation possible and it indicates a perfect positive linear correlation. This means that all the points of a scatter diagram will lie on a straight line which is sloping upward from left to right. A value of r = -1 represents the strongest negative linear correlation possible and it indicates a perfect negative linear correlation. This means that all the points of a scatter diagram will lie on a straight line which is sloping downward from left to right. A value of r = 0 represents no linear correlation between the two variables. Correlation Coefficient on the Calculator Before starting, you must turn Diagnostics On. Once turned on, you won t have to adjust this setting again unless you reset your calculator. 2 nd Catalog D DiagnosticsOn Enter Enter Example 4.5 pg. 178 Use the sample data in the table to calculate the sample correlation coefficient, r. 1. Put the x values in L1 and the y values in L2. 2. Press STAT CALC 4: LinReg(ax+b) ENTER 3. Enter the two lists separated by a comma. (L1, L2) 4. Enter The correlation coefficient, r, is. Remember, the correlation coefficient is a number between -1 and 1 and represents how strong a linear relationship the two variables have. The closer the number is to 1, the stronger the positive linear relationship. The closer to -1, the stronger the negative linear relationship. Review Example 4.4 on pg. 177 in the Text. 4.5 The Coefficient of Determination An important statistical measure that can be calculated from the correlation coefficient, r, is called the coefficient of determination. This statistical measure is used to explain the degree of influence that one variable called the independent variable has on the other variable called the dependent variable. The coefficient of determination measures the proportion of the variance of the dependent variable y that can be accounted for by the variance of the independent variable x. It is calculated by squaring the correlation coefficient, r. Coefficient of Determination = r 2 4

5 Example: Real World Application (not in textbook) Use the data in the table to calculate the correlation coefficient, r, to measure the strength of the relationship between the two variables. Country Average length of schooling (in years) x Australia Bolivia Botswana China Ethiopia Iraq Mexico India Romania Rwanda South Africa Spain Sweden United States Life expectancy y The correlation coefficient is. What type of correlation is this? (see scatter diagram) First: Identify x: Identify y: We can see that the dots are moving as we look at this diagram from left to right. But it is not a perfect correlation because the dots do not form a straight line. Very, very rarely will real-world variables form a perfect linear relationship. To draw the Scatter Plot: 1. Put the x values into L1 on your calculator. (STAT EDIT) 2. Put the y-values into L2. DO NOT SORT the lists! 3. Turn on STAT PLOT (2 nd Stat Plot) (make sure only one stat plot is on) 4. Choose the scatter diagram (first graph) from the Type menu. 5. Xlist should be the list containing the x values. 6. Ylist should be the list containing the y values 7. Clear out data from Y= 8. Click Zoom 9: Zoom Stat To find the Equation of the Regression Line: 1. Put the data into two lists. 2. Press STAT CALC 4: LinReg(ax+b) ENTER 3. Enter the lists separated by a comma. (L1, L2) To graph the line with the scatter diagram: 1. Set stat plot to scatter diagram with the two lists with x and y data. 2. Press Y= at the top left of your calculator and enter the linear equation. X is the button below MODE key. 3. Press GRAPH button at the top right of your calculator. 5

6 We have shown that there is a positive linear correlation between the average length of schooling and life expectancy of a country s population. But there are also other factors that influence the life expectancy that exist outside of our data. The degree of influence that one variable (schooling) has on another variable (life expectancy) can be found with a number called the coefficient of determination, r 2. In other words, how much of an influence does average schooling length have on life expectancy? The answer to this question will be a percentage. Simply put, how much does y (life expectancy) depend on x (average length of schooling)? We find the coefficient of determination by squaring the coefficient of correlation, r. To interpret the meaning of the coefficient determination, we can form the following general explanation: % of the variability in (dependent variable y) can be accounted for by the variability in (independent variable x). So for this application problem, explain/interpret r 2 : The coefficient of determination, r 2 =, suggests that there is some other reasons why a country s life expectancy is a certain amount. Since the coefficient of determination is, we may conclude that the remaining of variability is due to other unexplained factors. The unexplained amount is out of the scope of the problem. We can just accept that there are other factors that contribute to the variable life expectancy. A note of caution regarding the interpretation of correlation results Two variables may have a significant linear relationship, but it doesn t imply that there is a cause-andeffect relationship. In other words, the presence of one variable does not (necessarily) cause the presence of the variable. For example, the number of storks nesting in various European towns in the early 1900 s and the number of human babies born in the same towns during this period have a very high correlation. However, we can t conclude that an increase in storks will cause an increase in babies (or vice versa). A significant linear correlation should not be interpreted to mean that a change in one variable caused a change in the other variable. Rather, changes in one variable are accompanied by changes in the other variable. 6

7 4.6 Linear Regression Analysis Once a significant linear correlation has been established between two variables, a linear model can be developed to predict a value for the dependent variable given a value for the independent variable. To determine the linear model that will generate a close estimate of the actual y value, we obtain a line that best fits all sample points on the scatter diagram. The best fitting line is called the regression line. A strong positive correlation has been shown to exist between high school students standardized test results and success the first year of college as measured by the students GPAs. By creating a linear model, we can predict the 1 st year college success of a student with particular standardized test score. Linear regression analysis provides us with a linear model (an equation) that can be used to predict the value of the y variable (college GPA) given the value of the x variable (standardized test scores). The predicted value for y may not be exactly correct, but it will be a close estimate. The line that is created is the best fit line between the points that is positioned closely among all the sample points. The line that is created is called the regression line. Regression Line Formula y a bx where: y is the predicted value of y (the dependent variable), given the value of x (the independent variable). and a and b are the regression coefficients. We will be using the graphing calculator to obtain the equation. 7

8 Example: Going back to the Real World Application In the real world application, we saw that a positive linear correlation exists between a country s average schooling length and life expectancy. What if we wanted to estimate a country s life expectancy by simply knowing the average length of schooling? Knowing that there is a significant linear correlation from the sample data, we can create a line that best fits the sample data. Then we can use the line to estimate other values for countries not part of the sample. A linear model (equation of a line) can be developed to predict a value for the dependent variable (y) given a value for the dependent variable (x). 1. Use the sample to develop a regression line to prediction the life expectancy given the average length of schooling of a country. Find the regression line for the ordered pairs, length of schooling and life expectancy. 2. Use this line to predict the life expectancy for a country whose average length of schooling is: a. 15 years b. 17 years 3. Graph the scatter diagram and regression line together. Review Example 4.7 on pg. 184, Example 4.8 on pg. 187, and Example 4.9 on pg. 188 in the Text. 8