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In an experiment to assess the effects of curing time (factor A ) and type of mix (factor B ) on the compressive strength of hardened cement cubes, three different curing times were used in combination with four different mixes, with three observations obtained for each of the 12 curing time-mix combinations. The resulting sums of squares were computed to be \(SSA = 30,763.0,SSB = 34,185.6,SSE = 97,436.8,\;and SST\; = 205,966.6\)

a. Construct an ANOVA table.

b. Test at level .05 the null hypothesis \({H_{0AB}}:\;all\;{\gamma _{ij}}\;'s\; = 0\) (no interaction of factors) against \({H_{0AB}}\)at least one

c. Test at level .05 the null hypothesis \({H_{0A}}:{\alpha _1} = {\alpha _2} = {\alpha _3} = 0\) (factor A main effects are absent) against \({H_{0A}}\)at least one

d. Test\({H_{0B}}:{\beta _1} = {\beta _2} = {\beta _3} = {\beta _4} = 0\;versus\;{H_{aB}}:\) at least one using a level .05 test.

e. The values of the\({\bar x_{i = \;'s }},{\bar x_{1L}} = 4010.88,{\bar x_{2L}} = 4029.10,\;and\;{\bar x_{3..}} = 3960.02\). Use Turkey鈥檚 procedure to investigate significant differences among the three curing times.

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

A statistical computer package gave \(SS(\;Form\;) = 2253.44SS(\;Speed\;) = 230.81,\quad SS(\;Form*Speed\;) = 18.58, andSSE = 71.87\;\)

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\({\hat x_{ijk}} = \hat \mu + {\hat \alpha _i} + {\hat \beta _j} + {\hat \gamma _{ij}}\), and the residuals are \({x_{ijk}} - {\hat x_{ij{k^*}}}\)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 they \({ \in _{ijk}}\;'s\;\)appear to be normally distributed?

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: 11661186). Each number is a cell total based on K=3 observations.

Four different coatings are being considered for corrosion protection of metal pipe. The pipe will be buried in three different types of soil. To investigate whether the amount of corrosion depends either on the coating or on the type of soil, 12 pieces of pipe are selected. Each piece is coated with one of the four coatings and buried in one of the three types of soil for a fixed time, after which the amount of corrosion (depth of maximum pits, in .0001 in.) is determined. The data appears in the table.

a. Assuming the validity of the additive model, carry out the ANOVA analysis using an ANOVA table to see whether the amount of corrosion depends on either the type of coating used or the type of soil. Use=.05.

b. Compute\(\hat \mu ,{\hat \alpha _1},{\hat \alpha _2},{\hat \alpha _3},{\hat \alpha _4},{\hat \beta _1},{\hat \beta _2},\;and\;{\hat \beta _3}\)

Because of potential variability in aging due to different castings and segments on the castings, a Latin square design with N 5 7 was used to investigate the effect of heat treatment on aging. With A 5 castings, B 5 segments, C 5 heat treatments, summary statistics include x??? 5 3815.8, oxi 2 ?? 5 297,216.90, ox?j? 2 5 297,200.64, ox??k 2 5 297,155.01, and ooxijskd 2 5 297,317.65. Obtain the ANOVA table and test at level .05 the hypothesis that heat treatment has no effect on aging.

The article 鈥淭he Responsiveness of Food Sales to Shelf Space Requirements鈥� (J. Marketing Research, 1964: 63鈥�67) reports the use of a Latin square design to investigate the effect of shelf space on food sales. The experiment was carried out over a 6-week period using six different stores, resulting in the following data on sales of powdered coffee cream (with shelf space index in parentheses):

\({X_{ij(k)}} = \mu + {\alpha _i} + {\beta _j} + {\partial _k} + {\`o _{ij(k)}},\quad i,j,k = 1,2, \ldots ,N\)

Construct the ANOVA table, and state and test at level .01 the hypothesis that shelf space does not affect sales against the appropriate alternative.

An investigation of the machinability of beryllium-copper alloy using two different dielectric mediums and four different working currents resulted in the following data on material removal rate (this is a subset of the data that appeared in the article 鈥淪tatistical Analysis and Optimization Study on the Machinability of Beryllium Copper Alloy in Electro Discharge Machining,鈥� J. of Engr. Manufacture, 2012: 1847鈥�1861).

a. After constructing an ANOVA table, test at level .05 both the hypothesis of no medium effect against the appropriate alternative and the hypothesis of no working current effect against the appropriate alternative.

b. Use Tukey鈥檚 procedure to investigate differences in expected material removal rate due to different working currents (Q_.05,4,3 = 6.825).

In an experiment to see whether the amount of coverage of light-blue interior latex paint depends either on the brand of paint or on the brand of roller used, one gallon of each of four brands of paint was applied using each of three brands of roller, resulting in the following data (number of square feet covered)

1. Construct the ANOVA table. (Hint: The computations can be expedited by subtracting 400 (or any other convenient number) from each observation. This does not affect the final results.)
2. State and test hypotheses appropriate for deciding whether paint brand has any effect on coverage. Use=.05.
3. Repeat part (b) for brand of roller.
4. Use Tukey鈥檚 method to identify significant differences among brands. Is there one brand that seems clearly preferable to the others?