Simple Linear Regression

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- SCATTER DIAGRAM

A scatter diagram also called a scatter plot, is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a test of data. If the points are color coded one can increase the number of displayed variables to three. the data is displayed as a collection of points, each having the value of one variable determining the position on the vertical axis.

A scatter plot can be used either when one continuous variable that is under the control of the experimenter and the other depends on it or when both continuous variables are independent.

The plot can suggest various kinds of correlations between variables with a certain confidence interval. The scatter diagram pairs of numerical data, with one variable on each axis, to look for a relationship between them. If the variables are correlated, the points will fall along a line or curve. The better the correlation, the tighter the points will hug the line.

- THE MEANING OF ORDINARY LEAST SQUARES AND IT EQUATION, ESTIMATOR AND THEIR PARAMETER.

The ordinary least-squares method (OLS) is a technique for fitting the ‘‘best’’ straight line to the sample of XY observations. It involves minimizing the sum of the squared (vertical) deviations of points from the line:

Min(Yi – Ŷi)2……………….. (1)[pic 5]

Where Yi refers to actual observations, and Ŷi refers to the corresponding fitted value, so that

Yi – Ŷi = ei, the residual. This gives the following two normal equations:

[pic 6]…………………………(iii)

where n is the number of observations and 0 and i are estimators of the true parameters bo and bi. [pic 7][pic 8]

Solve simultaneously, We obtain[pic 9]

……………………(iv)

Therefore, to obtain bo from eqn (i)

Yi = n0 + I Xi[pic 10][pic 11][pic 12][pic 13]

Divide all sides by (n)

[pic 14]

Ῡi = 0 + Ii[pic 15][pic 16][pic 17]

0 = Ῡi - ii[pic 18][pic 19][pic 20]

Alternatively, i can be this[pic 21]

[pic 22]

Where; xi = (X-)[pic 23]

yi = (Y-Ῡ)

variance (x) =2x = (X-)2 note: i ( the slope of the estimated regression line)[pic 24][pic 25][pic 26]

0 (the Y intercept), Ŷ (the estimated regression equation)[pic 27]

Test of significance of parameters estimate (i.e. bo , bi … bn)

To do such, the variance of o and i is required[pic 30][pic 28][pic 29]

since 2u is unknown, the residual variance S2 is used as an (unbiased) estimate of 2u .[pic 31][pic 32]

[pic 33]where ei = (Y – Ŷi)

k = number of parameter estimates.

Thus; unbiased estimate of the variance 0 and i are:[pic 34][pic 35]

[pic 36]

Therefore, using the t – test

[pic 37][pic 38] Where bo , bi , equals o.