REGRESSION AND CORRELATION

10.1 SIMPLE LINEAR REGRESSION 10.2 SCATTER DIAGRAM 10.3 GRAPHICAL METHOD FOR DETERMINING REGRESSION 10.4 LEAST SQUARE METHOD 10.5 COEFFICIENT DETERMINATION 10.7 TEST OF SIGNIFICANCE 10.6 CORRELATION 10.8 ANALYSIS OF VARIANCE (ANOVA)

INTRODUCTION TO REGRESSION  Regression – is a statistical procedure for establishing the relationship between 2 or more variables.  This is done by fitting a linear equation to the observed data.  The regression line is then used by the researcher to see the trend and make prediction of values for the data.  There are 2 types of relationship:  Simple ( 2 variables)  Multiple (more than 2 variables)

10.1 THE SIMPLE LINEAR REGRESSION MODEL  is an equation that describes a dependent variable (Y) in terms of an independent variable (X) plus random error where, = intercept of the line with the Y-axis = slope of the line = random error  Random error, is the difference of data point from the deterministic value.  This regression line is estimated from the data collected by fitting a straight line to the data set and getting the equation of the straight line,

Example 10.1: 1) A nutritionist studying weight loss programs might wants to find out if reducing intake of carbohydrate can help a person reduce weight. a)X is the carbohydrate intake (independent variable). b)Y is the weight (dependent variable). 2) An entrepreneur might want to know whether increasing the cost of packaging his new product will have an effect on the sales volume. a)X is cost b)Y is sales volume

10.2 SCATTER DIAGRAM  A scatter plot is a graph or ordered pairs (x,y).  The purpose of scatter plot – to describe the nature of the relationships between independent variable, X and dependent variable, Y in visual way.  The independent variable, x is plotted on the horizontal axis and the dependent variable, y is plotted on the vertical axis.

n Positive Linear Relationship E(y)E(y)E(y)E(y) x Slope β 1 is positive Regression line Intercept β 0  10.2 SCATTER DIAGRAM

n Negative Linear Relationship E(y)E(y)E(y)E(y) x Slope is negative β 1 Regression line Intercept β 0  10.2 SCATTER DIAGRAM

n No Relationship E(y)E(y)E(y)E(y) x Slope β 1 is 0 Regression line Intercept β 0  10.2 SCATTER DIAGRAM

 A linear regression can be develop by freehand plot of the data. Example 10.2: The given table contains values for 2 variables, X and Y. Plot the given data and make a freehand estimated regression line GRAPHICAL METHOD FOR DETERMINING REGRESSION

The least squares method is commonly used to determine values for and that ensure a best fit for the estimated regression line to the sample data points The straight line fitted to the data set is the line: 10.4 LEAST SQUARES METHOD

i) y-Intercept for the Estimated Regression Equation,  and Theorem 3.1:  Given the sample data, the coefficients of the least squares line are: 10.4 LEAST SQUARES METHOD are the mean of x and y respectively.

10.4 LEAST SQUARES METHOD ii) Slope for the Estimated Regression Equation, Where,

10.4 LEAST SQUARES METHOD Given any value of the predicted value of the dependent variable, can be found by substituting into the equation

Example 10.2: Students score in history The data below represent scores obtained by ten primary school students before and after they were taken on a tour to the museum (which is supposed to increase their interest in history) Before,x After, y a)Fit a linear regression model with “before” as the explanatory variable and “after” as the dependent variable. b)Predict the score a student would obtain “after” if he scored 60 marks “before”.

 The coefficient of determination is a measure of the variation of the dependent variable (Y) that is explained by the regression line and the independent variable (X).  The symbol for the coefficient of determination is or.  If =0.90, then =0.81. It means that 81% of the variation in the dependent variable (Y) is accounted for by the variations in the independent variable (X).  The rest of the variation, 0.19 or 19%, is unexplained and called the coefficient of nondetermination.  Formula for the coefficient of nondetermination is 10.5 COEFFICIENT OF DETERMINATION( )

 Relationship Among SST, SSR, SSE where: SST = total sum of squares SST = total sum of squares SSR = sum of squares due to regression SSR = sum of squares due to regression SSE = sum of squares due to error SSE = sum of squares due to error SST = SSR + SSE n The coefficient of determination is: where: SSR = sum of squares due to regression SST = total sum of squares 10.5 COEFFICIENT OF DETERMINATION( )

Example ) If =0.919, find the value for and explain the value. Solution : = It means that 84% of the variation in the dependent variable (Y) is explained by the variations in the independent variable (X).

 Correlation measures the strength of a linear relationship between the two variables.  Also known as Pearson’s product moment coefficient of correlation.  The symbol for the sample coefficient of correlation is, population.  Formula :  The coefficient of correlation is the square root of the coefficient of determination CORRELATION (r)

Properties of :   Values of close to 1 implies there is a strong positive linear relationship between x and y.  Values of close to -1 implies there is a strong negative linear relationship between x and y.  Values of close to O implies little or no linear relationship between x and y.

Students score in history Refer Example 10.2: Students score in history c)Calculate the value of r and interpret its meaningSolution: Thus, there is a strong positive linear relationship between score obtain before (x) and after (y). Thus, there is a strong positive linear relationship between score obtain before (x) and after (y).

 To determine whether X provides information in predicting Y, we proceed with testing the hypothesis.  Two test are commonly used: i) ii) 10.7 TEST OF SIGNIFICANCE t Test F Test

1) t-Test 1. Determine the hypotheses. 2. Compute Critical Value/ level of significance. 3. Compute the test statistic.  p-value ( no linear r/ship) (exist linear

1) t-Test 4. Determine the Rejection Rule. Reject H 0 if : t t p -value < α There is a significant relationship between variable X and Y. 5.Conclusion.

2) F-Test 1. Determine the hypotheses. 2. Specify the level of significance. 3. Compute the test statistic. F α  wi F α  with 1 degree of freedom (df) in the numerator, n-2 degrees of freedom (df) in the denominator F = MSR/MSE ( no linear r/ship) (exist linear r/ship)

There is a significant relationship between variable X and Y. 5.Conclusion. 2) F-Test 4. Determine the Rejection Rule. Reject H 0 if : p -value < α F test > F 1,n-1,n-2

Students score in history Refer Example 10.2: Students score in history d) Test to determine if their scores before and after the trip is related. Use α=0.05 Solution: 1. ( no linear r/ship) (exist linear r/ship) (exist linear r/ship)2.3.

4. Rejection Rule: 5. Conclusion: Thus, we reject H 0. The score before (x) is linear relationship to the score after (y) the trip. Thus, we reject H 0. The score before (x) is linear relationship to the score after (y) the trip.

 ANOVA is a procedure used to test the null-hypothesis that the means ( ) of three or more populations are equal.  The value of the test statistic F for an ANOVA test is calculated as: F=Variance between samples Variance within samples F=MSR MSE  To calculate MSR and MSE, first compute the regression sum of squares (SSR) and the error sum of squares (SSE) ANALYSIS OF VARIANCE (ANOVA)

General form of ANOVA table: ANOVA Test 1) Hypothesis: 2) Select the distribution to use: F-distribution 3) Calculate the value of the test statistic: F 4) Determine rejection and non rejection regions: 5) Make a decision: Reject Ho/ accept H ANALYSIS OF VARIANCE (ANOVA) Source of Variation Degrees of Freedom(df) Sum of Squares Mean SquaresValue of the Test Statistic Regression1SSR=B1SxyMSR=SSR/1 F=MSR MSE Errorn-2SSE=SST-SSRMSE=SSE/n-2 Totaln-1SST=Syy

Example 10.4 The manufacturer of Cardio Glide exercise equipment wants to study the relationship between the number of months since the glide was purchased and the length of time the equipment was used last week. 1)Determine the regression equation. 2)At, test whether there is a linear relationship between the variables

Solution (1): Regression equation:

Solution (2): 1) Hypothesis: 1) F-distribution table: 2) Test Statistic: F = MSR/MSE = or using p-value approach: significant value = ) Rejection region: Since F statistic > F table (17.303> ), we reject H0 or since p-value ( )we reject H0 5)Thus, there is a linear relationship between the variables (month X and hours Y).