Page 26 - Mechanical Engineers' Handbook (Volume 2)

3 Statistics in the Measurement Process 15

n t(v,0.10) a t/ n
2 1 6.31 3.65
3 2 2.92 1.46
5 4 2.13 0.87

a From a t-statistic table. 9
Thus a sample of three tires is sufficient to establish the tire life within 10% at a 90%
level of confidence.

3.7 Goodness of Fit
Statistical methods can be used to fit a curve to a given data set. In general, the least-squares
principle is used to minimize the sum of the squares of deviations away from the curve to
be fitted. The deviations from an assumed curve y ƒ(x) are due to errors in y,in x,or in
both y and x. In most cases the errors in the independent variable x are much smaller than
the dependent variable y. Therefore, only the errors in y are considered for the least-squares
curve. The goodness of fit of an assumed curve is defined by the correlation coefficient r,
where
(y y) 2 (y y) 2
i
r (y y) 2 1 (y y) 2
i
i
ˆ 2 y,x
1 ˆ 2 y (31)
where (y y) 2 total variation (variation about mean)
i
2
(y y) unexplained variation (variation about regression)
i
(y y) 2 explained variation (variation based on assumed regression equation)
ˆ y estimated population standard deviation of y variable
ˆ y,x standard error of estimate of y on x
When the correlation coefficient r is zero, the data cannot be explained by the assumed
curve. However, when r is close to 1, the variation of y with respect to x can be explained
by the assumed curve and a good correlation is indicated between the variables x and y.
The probabilistic statement for the goodness-of-fit test is given by
P[r r] 1 (32)
calc
where r calc is calculated from Eq. (31) and the null and alternate hypotheses are as follows:

H : No correlation of assumed regression equation with data.
0
H : Correlation of regression equation with data.
1
The goodness-of-fit for a straight line is determined by comparing r calc with the value of r
obtained at n 2 degrees of freedom at a selected confidence level from tables. If r calc
r, the null hypothesis is rejected and a significant fit of the data within the confidence level
specified is inferred. However, if r calc r, the null hypothesis cannot be rejected and no
correlation of the curve fit with the data is inferred at the chosen confidence level.

Example 5 Goodness-of-Fit Test. The given x–y data were fitted to a curve y a bx
by the method of least-squares linear regression. Determine the goodness of fit at a 5%
significance level (95% confidence level):