Module 11 Examples

Simple Linear Regression

x	1	2	4	5	8	Mean = 4 =(1+2+4+5+8)/ 5
y	20	19	34	30	47	Mean = 30 =(29+19+34+30+47)/ 5

Find the Regression line for the points

Formula:

y = a + bx

Calculate b

	(1-4)(20-30) +	(2-4)(19-30) +	(4-4)(34-30) +	(5-4)(30-30) +	(8-4)(47-30)
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	(1-4)^2+	(2-4)^2+	(4-4)^2+	(5-4)^2+	(8-4)^2

	(-3)(-10) +	(-2)(-11) +	(0)( 4) +	(1)(0) +	(4)(17)
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	(-3)^2+	(-2)^2+	(0)^2+	(1)^2+	(4)^2

	30+	22 +	0 +	0 +	68
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	9 +	4 +	0 +	1 +	16

	120
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	30

.b = 4

Calculate a

30 –4(4)

30 –16

.a =14

Calculate y

y = a + bx

y = a + bx

14 + 4x

.y =14 + 4x

The regression line is y= 14 + 4x

Calculate the regression Coefficient for a and b:

	(20-30)^2+	(19-30)^2+	(34-30)^2+	(30-30)^2+	(47-30)^2
	(-10)^2+	(-11)^2+	(4)^2+	(0)^2+	(17)^2
	100 +	121 +	16 +	0 +	289

_{Total Variation =} = 526

r = .9553

The correlation coefficient is close to 1, indicating a strong positive correlation Exactly the same result can be achieved using the formula for R²

x	Fitted y value y = a + bx y =14 + 4x	Explained variation
1	14 + 4(1) =18	(18-30)^2=144
2	14 + 4(2) =22	(22-30)^2= 64
4	14 + 4(4) =30	(30-30)^2= 0
5	14 + 4(5) =34	(34-30)^2= 16
8	14 + 4(8) = 46	(46-30)^2=256
		Explained variation=480

R² = Explained variation / Total variation

R² = .9125

Squaring r = r²

Precisely equivalent to the result obtained form the formula for R²

^{Check Residual}

x	y	Fitted y value y = a + bx y =14 + 4x	Residual Act y – fit. y
1	20	14 + 4(1) = 18	20 – 18 = 2
2	19	14 + 4(2) = 22	19 – 22 = -3
4	34	14 + 4(4) = 30	34 – 30 = 4
5	30	14 + 4(5) = 34	30 – 34 = -4
8	47	14 + 4(8) = 46	47 – 46= 1

Find the linear regression line and correlation on HB10 Calculator

Find the linear regression line for the following table of numbers.  Also find the correlation.

1: Clear Memory	(CLS)
2: Enter Data	1 20 2 19 4   34 5 30  8 47
3: Calculate the mean of x and mean of y	7 K (SWAP)  (mean of y)	mean x = 4 mean y = 30
4: Compute the slope of the regression line	0 5 (=14) K (SWAP) (= 4)	y =14 + 4x
5: Compute the y-intercept of the regression line.	0 5 (=14)
6: Compute the correlation.	4 K (SWAP)	R^2 =.9553 r = .9553

    You should get a slope of 14, a y-intercept of 4, and a correlation of 0.9553, The regression line would be: y = 14 +4x