An engineer is designing a battery for use in a device that will be subjected to extreme variations in temperature (Montgomery 1991). He considers a two-way factorial with plate material and temperature as the two factors but has a large number of feasible choices for plate material and temperatures. Suppose that three plate materials and three temperatures were chosen, both at random, for use in the study. Four batteries were tested at each combination of plate material and temperature, and the resulting 36 tests are run in a random sequence. The data, observed battery life, are in the following table.

Material Temperature type 15 ◦F 70 ◦F 125 ◦F 1 130 155 34 40 20 70 74 180 80 75 82 58 2 150 188 136 122 25 70 159 126 106 115 58 45 3 138 110 174 120 96 104 168 160 150 139 82 60 Use a SAS proc mixed program to analyze these data. Provide answers to the following questions:

a. Write the appropriate model for the analysis of these data, explaining each term in the model and the assumptions made about these.

b. Construct an analysis of variance table needed to analyze this data.

Include a column of expected mean squares.

c. Use the ANOVA table to test whether temperature, type of material, or their interaction contributes significantly to the variation in battery life using α = 0.1.

d. Estimate significant variance components by providing point estimates and by calculating 95% confidence intervals.