The process engineer for a large paint manufacturer is concerned about the consistency of an ingredient in the paint that determines the ability of the paint to resist fading. The paint has a specification of 5% by weight of the ingredient. She designed the following study to assess the consistency. Ten batches of paint, each consisting of 500 1-liter containers of paint, are randomly selected from the previous week€™s production. From each of the 10 batches, five containers of paint are selected, and a determination of the percentage of the ingredient is made. The following table contains the percentages from the 50 determinations.

a. Write a random-effects model for this study, identifying all the terms in the model.
b. Run an analysis of variance for the data collected in this study. Test for a significant batch effect using α = 0.05.
c. Estimate the variance components associated with batches (σ2a) and containers within batches (σ2e). What proportion of the total variation in percentage of the fade prevention ingredient is due to the batch-to-batch variation?

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Batch Batch 2 Batch 3 Batch 4 Batch 5 Batch 6 Batch 7 Batch 8 Batch9 Batch 10 4.18 5.60 7.59 425 2.18 5.68 4.8.72 467 2.29 4.74 7.46 539 588 76 7.55 714 693 785 140 1.86 5.79 4.8 7 346 2.30 4.6 525 221 8.69 6.29 5.09 7.75 5.25 6.57215 523 879.57 1.01 2.25 5.47 6.10 3.50 6.35 892 3.56 4.34 485