When the population distribution is normal, it can be shown that the variable X^{2} ( 1)s^{2} /^{2} has a chi- squared distribution with 1 df. This can be used as a basis for testing H_{0}: _{0}, as follows: Replace ^{2} in X^{2} by its hypothesized value^{2}_{0} to obtain a test statistic. If the alternative hypothesis is Ha: _{0}, the P-value is the area under the 1 df chi-squared curve to the right of the calculated X^{2} (an upper-tailed test).

a. 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. Assume that the distribution from which the observations were selected is normal. Does the data contradict the uniformity specification? State and test the appropriate hypotheses using .01.

b. Suppose that the investigator who performed the experiment described in part (a) had wished to test H0: .70 versus Ha: .70. Can this test be carried out using the chi-squared table in this book? Why or why not?