Fundamentals of Probability and Statistics for Engineers

Answer: in this case, we have n 14. The quantities of interest are

The substitution of these values into Equations (11.7), (11.8), and (11.33) gives

The estimated regression line together with the data are shown in F igure 11.3.
The estimated standard deviation is 13 g cm^2 ,andthe
1 -band is also shown in the figure.

11.1.4 Confidence Intervals for R egression Coefficients

In addition to point estimators for the slope and intercept in linear regression, it
is also easy to construct confidence intervals for them and for x, the mean
of Y, under certain distributional assumptions. In what follows, let us assume
that Y is normally distributed according to N( x,^2 ). Since estimators
and x are linear functions of the sample of Y, they are also normal
random variables. Let us note that, when sample size n is large, and
are expected to follow normal distributions as a consequence of the
central limit theorem (Section 7.2.1), no matter how Y is distributed.
We follow our development in Section 9.3.2 in establishing the desired
confidence limits. Based on our experience in Section 9.3.2, the following are
not difficult to verify:

Linear Models and Linear Regression 347

ˆ

xˆ

1

n

Xn

iˆ 1

xiˆ

1

14

… 2 ‡ 2 : 5 ‡

‡ 20 †ˆ 11 : 11 ;

yˆ

1

n

Xn

iˆ 1

yiˆ

1

14

… 9 : 1 ‡ 19 : 2 ‡

‡ 130 : 8 †ˆ 57 : 59 ;

Xn

iˆ 1

…xix†^2 ˆ 546 : 09 ;

Xn

iˆ 1

…yiy†^2 ˆ 17 ; 179 : 54 ;

Xn

iˆ 1

…xix†…yiy†ˆ 2862 : 12 :

^ˆ^2862 :^12

546 : 09

ˆ 5 : 24 ;

^ˆ 57 : 59  5 : 24 … 11 : 11 †ˆ 0 : 63 ;

b^2 ˆ^1

12

‰ 17 ; 179 : 54 … 5 : 24 †^2 … 546 : 09 †Šˆ 182 : 10 :

^ˆ

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,



‡

‡ A^,

B^ A^‡^

A^,B^,

A^‡Bx^

, B

182 : 10 ˆ13 49g cm: