Lecture 6/7 (BIVARIATE REGRESSION ANALYSIS)

Question 1

REGRESSION ANALYSIS

Answer 1

The process of constructing a mathematical model or function that can be used to predict or determine one variable by another variable.

Question 2

CORRELATION

Answer 2

A measure of the degree of relatedness of two variables.

Question 3

COEFFICIENT OF CORRELATION (r)

Answer 3

Applicable only if both variables being analysed have at least an interval level of data.

The term r is a measure of the linear correlation of two variables.
The number range from -1 to 0 to +1.
The closer it is to +1, the higher the correlation between the dependent and the independent variables.
r<0 - negative correlation
r> 0 positive correlation
r=0 no correlation

Question 4

PEARSON PRODUCT MOMENT CORRELATION COEFFICIENT

Answer 4

Formula in booklet.

r =SSxy / sqrt (SSx)(SSy)

Question 5

BIVARIATE (TWO VARIABLES) LINEAR REGRESSION MODEL

Answer 5

The most elementary regression model.

Question 6

DEPENDENT VARIABLE

Answer 6

The variable to be predicted, usually Y.

Question 7

INDEPENDENT VARIABLE

Answer 7

The predictor or explanatory variable. Usually X.

Question 8

DETERMINISTIC REGRESSION MODEL

Answer 8

y = β0 + β1x
β0 and β1 are population parameters
They are estimated by sample statistics b0 and b1

Question 9

PROBABILISTIC REGRESSION MODEL

Answer 9

y = β0 + β1 + ͼ

Question 10

EQUATION OF THE SIMPLE REGRESSION LINE

Answer 10

Yhat = b0 + b1x

b0 = sample intercept
b1 = sample slope
yhat = predicted value of y

Question 11

LEAST SQUARES REGRESSION ANALYSIS

Answer 11

A process whereby a regression model is developed by producing the minimum sum of the squared error values.
The vertical distance from each point to the line is the error of prediction.
The least squares regression line is the regression line that results in the smallest sum of errors squared.

Question 12

What is the formula for b1 and b0?

Answer 12

b1 = SSxy/SSxx
b0 = ybar - b1 * xbar

Question 13

RESIDUAL

Answer 13

The difference between the actual value and the value predicted by the regression model (y-hat); the error of the regression model in predicting each value of the dependent variable.

Question 14

ASSUMPTIONS OF THE SIMPLE REGRESSION ANALYSIS

Answer 14

1. The model is linear.
2. The error terms have constant variances. (homoskedasticity)
3. The error terms are independent.
   The error terms are normally distributed.

Question 15

RESIDUAL PLOT

Answer 15

A graph in which the residuals for a particular regression model are plotted along with their associated value of x.

Question 16

STANDARD ERROR OF THE ESTIMATE

Answer 16

Residuals represent errors of estimation for individual points.
A more useful measurement of error is the standard error of the estimate.
The standard error of the estimate, denoted Se, is a standard deviation of the error of the regression model.

Se = sqrt (SSE/n-2)
SSE formula in booklet

Question 17

COEFFICIENT OF DETERMINATION

Answer 17

r^2
Is the proportion of variability of the dependent variable (y) accounted for or explained by the independent variable (x).
Ranges from, 0 to 1.
An r^2 of zero means that the predictor accounts for none of the variability of the dependent variable and that there is no regression prediction of y by x.
An r^2 of 1 means perfect prediction of y by c and that 100% of the variability of y is accounted for by x.

Question 18

FORMULA FOR COEFFICIENT OF DETERMINATION

Answer 18

SSyy = explained variation + unexplained variation
SSyy = SSR + SSE
r^2 = SSR/SSyy
=1 - SSE/SSyy

Question 19

HYPOTHESIS TESTS FOR THE SLOPE OF THE REGRESSION MODEL

Answer 19

A hypothesis test can be conducted on the sample slope of the regression model to determine whether the population slope is significantly different from zero.

Using the non regression model (the model) as a worst case, the researcher can analyse the regression line to determine whether is adds a more significant amount of predictability of y than does the model.
As the slope of the regression line diverges from zero, the regression model is adding predictability that the line is not generating.
Testing the slope of the regression line to determine whether the slope is different from zero is important.
If it is not different from zero, the regression line is doing nothing more than the average line of y predicting y.

Question 20

What is the hypothesis test for the slope of the regression model?

Answer 20

H0 : β1 = 0 
H1 : β1 =/ 0 
OR
H0 : β  0
OR
H0 : β -> 0 
H1 : β < 0
t test:
t = (b1 - β1) / Sb
WHERE:
Sb = Se/sqrt(SSxx)
Se = sqrt(SSE/n-2)
β1 = the hypothesised slope
df = n -2