Problem 1: Pencil lead is a ceramic material consisting of clay and graphite; the clay provides the ceramic matrix suppo

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Problem 1 Pencil Lead Is A Ceramic Material Consisting Of Clay And Graphite The Clay Provides The Ceramic Matrix Suppo 1 (99.15 KiB) Viewed 31 times

Problem 1 Pencil Lead Is A Ceramic Material Consisting Of Clay And Graphite The Clay Provides The Ceramic Matrix Suppo 2 (11.3 KiB) Viewed 31 times

Problem 1: Pencil lead is a ceramic material consisting of clay and graphite; the clay provides the ceramic matrix supporting the graphite. Pencil lead formulation focuses on the ratio of clay to graphite: the more the clay, the firmer the grade and the lighter the mark made by the pencil. One measure of the ratio of clay to graphite is the "ash" content. Inspectors take a random sample of n = 40 pencils from a production process, extract the lead, and burn off all the "volatiles" (essentially the carbon) leaving the ash. The ash content, denoted by X, is then weighed and reported as a percentage. A high ash content indicates too much clay in the formulation; a low ash content indicates too much graphite. The data are below. (Source: Statistical Methods for Engineers By G. Geoffrey Vining, Scott Kowalski) 43.0 42.5 40.2 41.5 42.4 40.7 42.2 42.6 42.5 42.6 41.0 41.0 40.7 41.9 41.3 41.7 40.4 42.0 42.5 42.9 42.6 42.3 42.1 41.8 41.9 43.0 41.7 43.2 43.4 38.3 41.3 41.3 41.7 41.7 41.8 41.3 40.1 40.4 41.6 40.3 (a) It is believed that the mean ash content is 42.5 percent. At the a= 0.1 significance level, formulate a hypothesis test to investigate this assertion. To gain full credit, you should provide the following: i. Identify the parameter of interest, and its meaning in the context of this problem. ii. State and check the modeling assumptions that we have discussed in class. iii. State the null and alternative hypotheses, the test statistic, the rejection region, and the p-value. iv. Draw a conclusion based on your test statistic and the rejection region. Also use the p-value to draw a conclusion. Do these decisions correspond? If not go back and check your work. v. Interpret your conclusion in the context of the problem. (b) Calculate a 90 percent confidence interval for the population mean and interpret your interval. Explain clearly (in words) what u represents in the context of this problem. Page 1
(c) Could we have used the 90 percent confidence interval calculated in (b) to make a decision about the hypothesis test conducted in (a)? Discuss, in depth, why or why not this can be done.