Chi-square statistic for bolt diameter variance

Question

Bolt diameters should have variance no greater than $0.0002$ cm $^2$ . A sample of $12$ bolts gives sample variance $0.0003$ . To test $H_0:\sigma^2=0.0002$ against $H_a:\sigma^2>0.0002$ at level $5\%$ , assuming normal diameters, what is the value of the test statistic?

Short answer

$16.5$ . A variance test uses $\chi^2=\dfrac{(n-1)s^2}{\sigma_0^2}=\dfrac{11\times0.0003}{0.0002}=16.5$ , compared with a chi-square distribution on $11$ degrees of freedom.

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Lessons behind this question

Maximum Likelihood Estimation: Score and Log-Likelihood
Independence
Covariance and Correlation

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