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74 4 Bivariate Statistics

subplot(1,2,1), bar(v,n_syn,'r')
subplot(1,2,2), bar(v,n_exp,'b')

Visual inspection of the bar plots reveals similarities between the data sets.
Hence, the χ -test can be used to test the hypothesis that the residuals follow
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a gaussian distribution.

chi2 = sum((n_exp - n_syn).^2 ./n_syn)
chi2 =
2.3465

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2
The critical χ can be calculated by using chi2inv. The χ test requires the
degrees of freedom df, which is the number of classes reduced by one and
the number of parameters estimated. In our example, we test for a gaussian
distribution with two parameters, mean and standard deviation. Therefore the
degrees of freedom is df=6-(1+2)=3. We test at a 95% signifi cance level:
chi2inv(0.95,3)

ans =
7.8147
The critical χ of 7.8147 is well above the measured χ of 2.3465. It is not
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possible to reject the null hypothesis. Hence, we conclude that our residuals
follow a gaussian distribution and the bivariate data set is well described by
the linear model.

4.6 Bootstrap Estimates of the Regression Coeffi cients

We use the bootstrap method to obtain a better estimate of the regression
coefficients. Again, we use the function bootstrp with 1000 samples

(Fig. 4.6).
p_bootstrp = bootstrp(1000,'polyfit',meters,age,1);
The statistics of the fi rst coefficient, i.e., the slope of the regression line is

hist(p_bootstrp(:,1),15)

mean(p_bootstrp(:,1))
ans =
5.6023

std(p_bootstrp(:,1))