xYDependent (response) Variable

Independent (control) Variable

Random Error

X Yx1 y1

x2 y2

… …xn yn

Raw Raw data:data:

nixY iii ,...,2,1 ,

Assumption:•i ‘s are independent normally distributed random variables with mean 0 and (common) variance 2.

Estimator for :xx

xy

SS

b̂

Estimator : xbYa ̂

Estimator for the Y value for given x: bxay ˆ

. ,

,1

,1

,1

1111

2

11

2

1

2

2

11

2

1

2

xbyaSS

b

yxn

yxyyxxS

yn

yyyS

xn

xxxS

xx

xy

n

ii

n

ii

n

iii

n

iiixy

n

ii

n

ii

n

iiyy

n

ii

n

ii

n

iixx

• Estimators for parameters unbiased?• Confidence intervals for the true

parameters?• Confidence interval for the Y(x) value:• Estimate E(Y(x))?• Hypothesis Testing?• Quality of the model? Better model?

(Sec. 11.2)

Air Velocity

x (cm/s)

Evaporation

Coefficient y

(mm2/s)20 0.1860 0.37

100 0.35140 0.78180 0.56220 0.75260 1.18300 1.36340 1.17380 1.65

• (Linear) relationship exists?• Linear relationship?• Prediction for x = 190 cm/s• Error?

  x y x^2 y^2 xy  20 0.18 400 0.0324 3.6  60 0.37 3600 0.1369 22.2  100 0.35 10000 0.1225 35  140 0.78 19600 0.6084 109.2  180 0.56 32400 0.3136 100.8  220 0.75 48400 0.5625 165  260 1.18 67600 1.3924 306.8  300 1.36 90000 1.8496 408  340 1.17 115600 1.3689 397.8  380 1.65 144400 2.7225 627

Sum 2000 8.35 532000 9.1097 2175.4

Sxx 132000  Slope b0.00382

9Syy 2.13745  Intercept a 0.069242Sxy 505.4     

xbxay 00383.00692.0ˆ

80.0)190(00383.00692.0)190(ˆ

y

nibxay iii ,...,2,1 ,ˆ Residuals:

Residual sum of squares (or Error Sum of Squares):

xx

xyyy

n

iii

n

ii S

SSbxaySSE

2

1

2

1

2ˆ

Estimator for 2:

22

nSSEse

Standard Error of the Estimate:

2nSSEse

xx

xy

SS

b ̂

Distribution of b (as a statistic):)2(~)(

ntSsb

xxe

• Confidence interval for the true population slope is

• To test the null hypothesis H0: = 0, use the statistic (with df = n – 2.)

• Only when the null hypothesis H0: = 0 is rejected, the independent variable x can be included in the model.

xxe Ssntb 1)2(2/

xxe

Ss

bt )( 0

Distribution of a (as a statistic):

• Confidence interval for the true population intercept is

• To test the null hypothesis H0: = 0, use the statistic (with df = n – 2.)

• Only when the null hypothesis H0: = 0 is rejected, the nonzero intercept can be included in the model.

xx

e Sx

nsnta

2

2/1)2(

20 )(

xnS

nSs

atxx

xx

e

xbYa ̂

)2(~)(2

ntxnS

nSsa

xx

xx

e

Inference about E[Y(x)] = Inference about E[Y(x)] = + + x, the mean of x, the mean of the response value for given x.the response value for given x.

.1

,2

2

xxSxx

nbxaVar

xbxaE

Confidence interval for E[Y(x)] is

xx

e Sxx

nstbxa

2

2/1

where t-distribution has df = n – 2.

To predict the “future” Y(x), we use the interval

xx

e Sxx

nstbxa

2

2/11

where t-distribution has df = n – 2.

Note:•Confidence interval for Y(x) is wider than the confidence interval for E[Y(x)];•Width of the confidence intervals for Y(x) and E[Y(x)] depend on the x value. (Limit of prediction.)

Excel Outputs:SUMMARY OUTPUT

Regression StatisticsMultiple R 0.95148137R Square 0.9053168Adjusted R Square 0.8934814Standard Error 0.15905212Observations 10

es

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 0.06924242 0.100973708 0.685747071 0.5122514 -0.163603363 0.302088211x 0.00382879 0.000437777 8.745986838 2.286E-05 0.002819273 0.004838302

Estimators a and b

xx

e Sx

ns

21

xxe Ss 1

T-score in Hypothesis

Tests

P-values in Hypothesis

Tests

95% Confidence

Intervals for and

• Compare with the critical t-value(s);

• Compare with the value;

• Check if the interval contains zero.