Chapter 8, Problem 76SE

Chapter 7 presented a CI for the variance σ² of a normal population distribution. The key result there was that the rv χ² = (n − 1)S²/σ² has a chi-squared distribution with n − 1 df. Consider the null hypothesis $H_0:{\sigma }^2={\sigma }_0^2$ (equivalently, σ = σ₀). Then when H₀ is true, the test statistic χ² = (n − 1)S²/ ${\sigma }_0^2$ has a chi-squared distribution with n − 1 df. If the relevant alternative is $H_{\text{a}}:{\sigma }^2>{\sigma }_0^2$ the P-value is the area under the χ² curve with n − 1 df to the right of the calculated χ² value. To ensure reasonably uniform characteristics for a particular application, it is desired that the true standard deviation of the softening point of a certain type of petroleum pitch be at most .50°C. The softening points of ten different specimens were determined, yielding a sample standard deviation of .58°C. Does this strongly contradict the uniformity specification? Test the appropriate hypotheses using α = .01. [Hint: Consult Table A.11.]