Testing a Hypothesis on the Correlation Coefficient Simple Linear Regression.pptx

Testing a Hypothesis on the Correlation Coefficient Simple Linear Regression.pptx

1. S Establish if there is a relationship between two variables. Forecast new observations.

2. This is the variable whose values we want to explain or forecast. Its values depend on something else We denote it as y This is the variable that explains the other one. Its values are independent We denote it as x Dependent Variable Independent Variable

3. Linear Equation You may remember one of these: y = a + bx y = mx + b In the stats world, we use a different notation: y = + 1 x  

4. Linear Equation Example y = + 1 x  

5. Simple Linear Regression Model = Regression is a powerful method in research. It is a method of analyzing the variability of a dependent variable by resorting the information available on one or more independent variables. Where: = is the value of the dependent variable = the Y-intercept = the slope of the regression line = is the value of the independent variable = is the random error term  

6. Method of Least Squares  

7. Example: The Chief of Admission Office of the MHS wanted to determine if System Admission and Scholarship Examination (SASE) scores is a good indicator of the grade point average (GPA) of the 16 academic scholars selected at random from freshmen class. Their GPA and SASE scores are shown: Question: How can one predict and estimate GPA from SASE score? Student GPA (Y) SASE Score (X) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1.52 1.85 1.86 1.79 1.67 2.96 2.05 2.79 2.63 2.71 2.12 1.94 2.11 1.95 2.59 2.45 85 76 69 75 106 61 70 59 56 54 62 73 64 57 92 85

8. Step 1

9. GPA (Y) SASE Score (X) XY Y 2 X 2 1.52 1.85 1.86 1.79 1.67 2.96 2.05 2.79 2.63 2.71 2.12 1.94 2.11 1.95 2.59 2.45 85 76 69 75 106 61 70 59 56 54 62 73 64 57 92 85 129.20 140.60 128.34 134.25 177.02 180.56 143.50 164.61 147.28 146.34 131.44 141.62 135.04 111.15 238.28 208.25 2.31 3.42 3.46 3.20 2.79 8.76 4.20 7.78 6.92 7.34 4.49 3.76 4.45 3.80 6.71 6.00 7225 5776 4761 5625 11236 3721 4900 3481 3136 2916 3844 5329 4096 3249 8464 7225 1.999 2.124 2.222 2.138 1.707 2.333 2.208 2.360 2.402 2.430 2.319 2.166 2.291 2.388 1.902 1.999 34.99 1144 2457.48 79.42 84984 GPA (Y) SASE Score (X) XY Y 2 X 2 1.52 1.85 1.86 1.79 1.67 2.96 2.05 2.79 2.63 2.71 2.12 1.94 2.11 1.95 2.59 2.45 85 76 69 75 106 61 70 59 56 54 62 73 64 57 92 85 129.20 140.60 128.34 134.25 177.02 180.56 143.50 164.61 147.28 146.34 131.44 141.62 135.04 111.15 238.28 208.25 2.31 3.42 3.46 3.20 2.79 8.76 4.20 7.78 6.92 7.34 4.49 3.76 4.45 3.80 6.71 6.00 7225 5776 4761 5625 11236 3721 4900 3481 3136 2916 3844 5329 4096 3249 8464 7225 1.999 2.124 2.222 2.138 1.707 2.333 2.208 2.360 2.402 2.430 2.319 2.166 2.291 2.388 1.902 1.999 34.99 1144 2457.48 79.42 84984 = 2.187   = 71.50         = = 3.187       Step 2  

10. The scatter plot and the regression line describing SASE score and GPA of some academic scholars is shown in the figure below.

11. In linear regression, the concern is the significance of the slope parameter, . If the slope is significant, then there exists linear relationship between X and Y, which means that knowledge of the X values will significantly improve one’s prediction.   Testing H o : = 0 with the F - Test . The hypothesis of ‘zero regression slope’ in the analysis of variance states “there is no linear relationship between the independent variable and dependent variable”. The analysis of variance denoted by ANOVA summarizes the sources of variation of the dependent variable, namely regression and error.  

12. Source of Variation Sum of Squares df Mean Square F Regression SSR 1 MSR MSR/MSE Error SSE n-2 MSE Total SST n-1 ANOVA table for Simple Linear Regression SSR: sum of squares due to regression “ explained variation” SSE: sum of squares due to error “unexplained variation” SST: sum of squares total “total variation”

13. The computational formulae for the different sum of squares are given below: SSE = SSR = SST – SSE F =  

14. Solving the F - test for data on SASE score and GPA and following the steps in hypothesis testing, we proceed as follows: H o : There is no linear relationship between SASE score and grade point average. H 1 : There is linear relationship between SASE score and grade point average. 2. Level of Significance: 𝛼 = 0.05 3. Test Statistics: Simple Linear Regression, F = 𝑀𝑆𝑅/𝑀𝑆𝐸 4. Critical Region: Reject H o if F c ≥ 4.60 5. Computation SST = 79.42 - = 79.42 – 76.519 = 2.901 SSE = = 2.285 SSR = 2.901 – 2.285 = 0.616   Sources of Variation SS df MS F c Regression 0.616 1 0.616 3.78 Error 2.285 14 0.163 Total 2.901 15 6. Statistical Decision: Do not reject H o 7. Interpretation/conclusion: On the basis of this data, we have no sufficient evidence that there is significant relationship between SASE score and grade point average. GPA (Y) SASE Score (X) XY Y 2 X 2 1.52 1.85 1.86 1.79 1.67 2.96 2.05 2.79 2.63 2.71 2.12 1.94 2.11 1.95 2.59 2.45 85 76 69 75 106 61 70 59 56 54 62 73 64 57 92 85 129.20 140.60 128.34 134.25 177.02 180.56 143.50 164.61 147.28 146.34 131.44 141.62 135.04 111.15 238.28 208.25 2.31 3.42 3.46 3.20 2.79 8.76 4.20 7.78 6.92 7.34 4.49 3.76 4.45 3.80 6.71 6.00 7225 5776 4761 5625 11236 3721 4900 3481 3136 2916 3844 5329 4096 3249 8464 7225 1.999 2.124 2.222 2.138 1.707 2.333 2.208 2.360 2.402 2.430 2.319 2.166 2.291 2.388 1.902 1.999 34.99 1144 2457.48 79.42 84984 GPA (Y) SASE Score (X) XY Y 2 X 2 1.52 1.85 1.86 1.79 1.67 2.96 2.05 2.79 2.63 2.71 2.12 1.94 2.11 1.95 2.59 2.45 85 76 69 75 106 61 70 59 56 54 62 73 64 57 92 85 129.20 140.60 128.34 134.25 177.02 180.56 143.50 164.61 147.28 146.34 131.44 141.62 135.04 111.15 238.28 208.25 2.31 3.42 3.46 3.20 2.79 8.76 4.20 7.78 6.92 7.34 4.49 3.76 4.45 3.80 6.71 6.00 7225 5776 4761 5625 11236 3721 4900 3481 3136 2916 3844 5329 4096 3249 8464 7225 1.999 2.124 2.222 2.138 1.707 2.333 2.208 2.360 2.402 2.430 2.319 2.166 2.291 2.388 1.902 1.999 34.99 1144 2457.48 79.42 84984

15. Thank You!