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1.  Coil winding machines (Cont’d). A plant contains a large number of coil winding machines. A production analyst studied a certain characteristic of the wound coils produced by these machines by selecting four machines at random and then choosing 10 coils at random from the day’s output of each selected machine. The results are stored in the file “coil.txt” as well as given in the table below:

coil (j)

a. Estimate the error variance σ2 with a 90% confidence interval. Interpret your interval estimate.

b. Obtain a point estimate for the variance of winding machine efects σu(2) .

c. Obtain an approximate 90% confidence intervals for σu(2) using the Satterthwaite procedure.

d. Obtain the (estimated) best linear unbiased predictors for the efects of the four machines used in this study on this characteristic of the wound coils.

2. Miles per gallon. An automobile manufacturer wished to study the efects of diferences between drivers and diferences between cars on gasoline consumption. Four drivers were randomly se- lected; and five cars of the same model with manual transmission were randomly selected from the assembly line. Each driver drove each car twice over a 40-mile test course and the miles per gallon were recorded. The data are saved in ‘mpg.txt”, where the 1st column is for mpg, the 2nd column is for driver index, the 3rd column is for car index, and the last column is for replicate index.

a. Identify factor(s) in this study. For each factor, state the number of factor levels and whether it should be treated as random or fixed.

b. Which model should be used to analyze this data? Explain.

c. Test whether or not there are driver and car interaction efects at the significance level 0.05. State the null and alternative hypotheses, the decision rule based on the critical value approach, and the conclusion. What is the p-value of the test?

d. Test whether or not driver- and car- main efects are present, respectively, at the significance level 0.01. For each test, state the null and alternative hypotheses, the decision rule based on the p-value approach, and the conclusion.

3. Imitation pearls. Preliminary research on the production of imitation pearls entailed studying the efect of the number of coats of a special lacquer applied to an opalescent plastic bead used as the base of the pearl on the market value of the pearl. Four batches, each with 12 beads, were used in the study, and it is desired to also consider the batch efect on the market value. The three levels of the number of coats (6, 8, and 10 coats) were fixed in advance, while the four batches were randomly sampled from the bead production process. The market value of each pearl was determined by a panel of experts in terms of a coded number.  The data are saved in the file “pearl.txt”, where the 1st column is for the market value, the 2nd column is for number of coats index, the 3rd column is for batch index, and the last column is for replicate index.

a. Identify factor(s) in this study. For each factor, state the number of factor levels and whether it should be treated as random or fixed.

b. Which model should be used to analyze this data? Explain.

c. Test whether or not there are number of coats and batch interaction efects at the significance level 0.05.  State the null and alternative hypotheses, the decision rule based on the critical value approach, and the conclusion. What is the p-value of the test?

d. Test whether or not number of coats- and batch- main efects are present, respectively, at the significance level 0.05.  For each test, state the null and alternative hypotheses, the decision rule based on the p-value approach, and the conclusion.

4. Miles per gallon (Cont’d). An automobile manufacturer wished to study the efects of difer- ences between drivers and diferences between cars on gasoline consumption. Four drivers were randomly selected; and five cars of the same model with manual transmission were randomly selected from the assembly line. Each driver drove each car twice over a 40-mile test course and the miles per gallon were recorded. The data are saved in ‘mpg.txt”, where the 1st column is for mpg, the 2nd column is for driver index, the 3rd column is for car index, and the last column is for replicate index.