When the population distributi

When the population distribution is normal, it can be shown that the variable X2  (  1)s2 /2 has a chi- squared distribution with   1 df. This can be used as a basis for testing H0:   0, as follows: Replace 2 in X2 by its hypothesized value20 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 X2 (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?

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When the population distributi

When the population distribution is normal and n is large, the statistic s has approximately a normal distribution with  Use this fact to develop a large-sample two-sided confidence interval formula for . Then calculate a 95% confidence interval for the true standard deviation of the fracture strength distribution based on the data given in Exercise 16 (the cited paper gave compelling evidence in support of assuming normality).

Exercise 16

The article “Evaluating Tunnel Kiln Performance” (Amer. Ceramic Soc. Bull., August 1997: 59–63) gave the following summary information for fracture strengths (MPa) of n = 169 ceramic bars fired in a particular kiln:   89.10, s  3.73.

a. Calculate a two-sided confidence interval for true average fracture strength using a confidence level of 95%. Does it appear that true average fracture strength has been precisely estimated?

b. Suppose the investigators had believed a priori that the population standard deviation was about 4 MPa. Based on this supposition, how large a sample would have been required to estimate ­ to within . MPa with 95% confidence?

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