Construct a normal probability plot from the residuals -

Question

Objectives: At the end of this assignment, you should be able to

• conduct a two-factor mixed-effects ANOVA analysis when K 1 and sample sizes are equal.

• Calculate and plot main and interaction effects.

• Design a simple ANOVA experiment.

1. A study was carried out to compare the writing lifetimes of four premium brands of pens. It was thought that the Writing surface might affect lifetime, so three different surfaces were randomly selected.

A writing machine was used to ensure that conditions were otherwise homoge-neous (e.g., constant pressure and a fixed angle).

The 'accompanying table shows the two lifetimes (min) obtained for each brand-surface combination.

		1	Writing Surface 2                              3		xi.
Brand	1 2	709, 659 668, 685	713. 726 722. 740	660, 645 692, 720	4112 4227
of Pen	3	659. 685	666. 684	678, 750	4122
	4	698. 650	704.666	686, 733	4137
	xj	5413	5621	5564	16,598

Carry out an appropriate ANOVA, and state your conclusions.

In this problem you conducted the analysis as if the pen and the paper were both fixed effects. This time, treat the pen type as fixed, but the paper as random. You can get most of the values you need from your work on PS 3 or the notes for PS 3, but will use them a bit differently. Also, use a = 0.10.

2. This time, treat the speed as a random effect. Also, determine a range for p.

The accompanying data resulted from an experiment to particular investigate whether yield from a certain chemical process depended either on the formulation of a input or on mixer speed.

	Speed
	60	70	80
	189.7	185.1	189
1	188.6	179.4	193
	190.1	177.3	191.1
Formulation
	165.1	161.7	163.3
2	165.9	159.8	166.6
	167.6	161.6	170.3

A statistical computer package gave SS (Form) = 2253.44, SS(Speed) = 230.81, SS(Form*Speed) = 18.58, and SSE = 71.87.

a. Does there appear to be interaction between the factors?

b. Does yield appear to depend on either formulation or speed?

c. Calculate estimates of the main effects.

d. The fitted values are x^{^}_ijk = µ^{^} + α^{^}_i + β^{^}_j + y^{^}_ij, and the residuals are x_ijk - x^{^}_ijk. Verify that the residuals are .23, -.87, .63, 4.50, -1.20, -3.30, -2.03, 1.97, .07, -1.10, -.30, 1.40, .67, -1.23, .57, -3.43, -.13, and 3.57.

e. Construct a normal probability plot from the residuals given in part (d). Do the ∈_ijk's appear to be normally distributed?

3. Plot the main and interaction effect for the data in given problem.

The accompanying data was obtained in an experiment to investigate whether compressive strength of concrete cylinders depends on the type of capping material used or variability in different batches ("The Effect of Type of Capping Material on the Compressive Strength of Concrete Cylinders," Proceedings ASTM, 1958: 1166-1186). Each number is a cell total (x_ij.) based on K = 3 observations.

		Batch
		1	2	3	4	5
	1	1847	1942	1935	1891	1795
Capping Material	2	1779	1850	1795	1785	1626
	3	1806	1892	1889	1891	1756

In addition ∑∑∑x²_ijk = 16,815,853 and ∑∑x²_ij = 50,443,409. Obtain the ANOVA table and then test at level .01 the hypotheses H_OG versus H_OA, Him versus Ham and H_OB versus Hap assuming that capping material is a fixed effects factor and batch is a random effects factor.

Use the more efficient way discussed in given problem.

a. Show that E(X^-_i - X^-..) = α_i, so that X^-_i - X^-.. is an unbiased estimator for α, (in the fixed effects model).

b. With Y_ij = X^-_ij, - X^-_i  - X^-_j + X^-. Show that to is an unbiased estimator for y^{^}_ij, (in the fixed effects model).