Virtual COMSATS Inferential Statistics Lecture27 Ossam Chohan Assistant

Virtual COMSATS Inferential Statistics Lecture-27 Ossam Chohan Assistant Professor CIIT Abbottabad 1

Recap of previous lectures • Introduction of Correlation and Regression. – – – –. Simple correlation and its significance in research. Solution to problems. Properties of correlation. Scatter plot. Coefficient of determination. Regression analysis. Simple regression model. Probable error and standard error. 2

Objective of lecture-27 • Introduction of Correlation and Regression. – – Simple Regression model. Hypothesis testing for correlation coefficient. Hypothesis testing for regression coefficients. Some practice Problems.

Assessment Problem-26 • Given the data Fertili 0. 3 zer (X) 0. 6 0. 9 1. 2 1. 5 1. 8 2. 1 2. 4 Yield (Y) 15 30 35 25 30 50 45 10 • Assuming normality, calculate the 95% confidence interval for the (a) value of α (b) the value of β. 4

Problem-33 • Hypothesis Testing for β. • Estimate a regression line from the following data of height (X) and Weight (Y) of 12 persons: • Test the hypothesis that the population regression coefficient β=0, height and weight are independent at 5% level of significance. 5

Problem-33 Height (X) Weight (Y) 60 110, 135, 120 62 120, 140, 135 64 150, 145 70 170, 185, 160 Cont… 6

Problem-33 Solution 7

Problem-33 Solution 8

Problem-33 Solution 9

Hypothesis testing for correlation coefficient • There are two situations of testing coefficient of correlation. – When population correlation coefficient ρ is having some value. – When population correlation coefficient ρ is having value 0. 10

Problem-34 • (i). Test the hypothesis that ρ=0. 7, if a sample of 50 gave r=0. 6. • (ii). A value of r of 0. 6 is calculated from a random sample of 39 pairs of observations from a bi variate normal population. Is this value of r consistent with the hypothesis that ρ=0. 4 11

Problem-34 Solution 12

Cont… 13

Problem-35 • A random sample of 20 pairs of observations gives a coefficient of correlation of 0. 45. Test the hypothesis at the 0. 05 level of significance that the correlation coefficient ρ in the population is zero. 14

Problem-35 Solution 15

Problem-36 • A random sample of 28 pairs from a certain bivariate normal population gave r=0. 6, another random sample of 23 pairs from another bivariate population gave r=0. 4. Test at 5% level, the hypothesis H 0: ρ1=ρ2 against H 0: ρ1≠ρ2. 16

Problem-36 Solution 17

Assessment Problem-27 • A sample of 10 pairs of observations yields a correlation coefficient of 0. 7. Is it reasonable to suppose that such a value would arise from a population where the coefficient is 0. 85. – Is the value of 0. 7 itself significant? – Another sample of 12 pairs of observations shows a coefficient of 0. 90. Is this likely to be from the same population at the first? 18

Assessment Problem-28 • Two independent samples have 28 and 29 pairs of observations with correlation coefficients 0. 55 and 0. 75 respectively. Are these values of r consistent with the hypothesis that the samples have been drawn from the same population? 19

Assessment Problem-29 • Find the linear regression equation from the following data: X 65 50 55 65 55 70 65 70 50 55 Y 85 74 76 90 85 87 94 91 81 91 76 74 • Assuming normality, test the hypothesis – H 0: β = 0 against H 1: β ≠ 0; – H 0: α = 32 against H 1: α ≠ 32; – At 0. 01 level of significance. 20