Virtual COMSATS Inferential Statistics Lecture-26 Ossam Chohan Assistant Professor CIIT Abbottabad

Recap of previous lectures We are working on hypothesis testing. And we discussed following topics in our last lectures: Simple Correlation and its significance in research. Solution to problems. Properties of Correlation. Scatter Plot. Coefficient of determination.

Objective of lecture-26 Introduction of Correlation and Regression. Regression Analysis. Simple Regression model. Probable error and standard error. Hypothesis testing for correlation coefficient.

Regression Analysis Predict the value of a dependent variable based on the value of at least one independent variable. Study the impact of changes in dependent variable on the basis of changes in independent variable. Independent Variable: Variable that have impact on dependent variable. Dependent variable: The variable in which we are interested.

Simplest form of Regression model There must be one dependent variable in each model. There should be single independent variable. Relationship must be linear. There must be logical relations between x and y, that is changes in y are because of x.

Discussion on Regression model

Discussion on Regression model

Population Linear Regression Random Error term, or residual Population Slope Coefficient Population y intercept Independent Variable Dependent Variable

Linear Regression Assumptions Error values are normally distributed for any given value of x The probability distribution of the errors has constant variance The underlying relationship between the x variable and the y variable is linear Error values (ε) are statistically independent The probability distribution of the errors is normal

Estimated Regression Model Estimated (or predicted) y value Estimate of the regression intercept Estimate of the regression slope Independent variable The individual random error terms ei have a mean of zero

How to calculate unknown bs

Problem-28 For the problem-27, fit a simple regression model. And interpret the results. The test-Retest method is one way of establishing the reliability of a test. The test is administered and then, at a later date, the same test is re-administered to the same individuals. Fit a simple linear regression model for two sets of scores. First score 75 87 60 98 80 68 84 47 72 Second score 90 52 94 78 53 70

Problem-28 Solution

Problem-28 Solution

Where is Inferential Statistics????

Probable error and standard error Probable error of correlation coefficient indicates extent to which its value depends on the conditions of random samples. PEr = 0.6745 SEr If r is the correlation coefficient t in a sample of n pairs of observations, then the standard error SEr of r is given by SEr= (1-r2)/√n The range within which population coefficient of correlation is expected to fall would be: ρ= r ± PEr

Problem-29 If r=0.8 and n=25, then find the limits within which population correlation coefficient will fall:

Hypothesis testing for population correlation coefficient (small samples) IS there any significant correlation between x and y. Problem-30 A random sample of 27 pairs of observations from a normal population gives a correlation coefficinet of 0.42. is it likely that the variables in the population are uncorrelated?

Problem-30 Solution

Hypothesis testing for population correlation coefficient (large samples)

Hypothesis testing for population correlation coefficient (large samples)

Discussion Cont…

Problem-30 What is the probability that a correlation coefficient of 0.75 or less arises in a sample of 30 pairs of observations from a normal population in which the true correlation is 0.90. Apply Fisher’s z-transformation

Problem-30 Solution

Problem-31 Test the significance of the correlation r= 0.5 from a sample of size 18 against hypothesized population correlation 0.70. We need to apply z-transformation

Problem-31 Solution

CI Estimation of Population Regression Co-efficient-β Sample estimate of β is b. The sampling distribution of b is normally distributed with a mean β, and standard deviation (standard error) δY.X /√Σ(X- )2 That is, the variable Z=(b- β)/ δY.X /√Σ(X- )2 is standard normal variable. What about population standard deviation

CI Estimation of Population Regression Co-efficient-β

Cont…

Problem-32 Use the following data to construct 95% confidence interval for the (a) value of α (b) the value of β. X 2 3 4 5 6 Y 8 18 22 24 13 10 16

Problem-32 Solution

Problem-32 Solution

Assessment Problem-26 Given the data Assuming normality, calculate the 95% confidence interval for the (a) value of α (b) the value of β. Fertilizer (X) 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 Yield (Y) 10 15 30 35 25 50 45