An experiment to determine the effect of mold temperature on tensile strength involved three different alloys and five different mold temperatures. Four specimens of each alloy were cast at each mold temperature. With mold temperature as the row factor and alloy as the column factor, the sums of squares were: SSA = 69,738, SSB = 8958, SSAB = 7275, and SST = 201,816. a. Construct an ANOVA table. You may give ranges for the P-values. b. Is the additive model plausible? Explain. c. Is it plausible that the main effects of mold temperature are all equal to 0? Provide the value of the test statistic and the P-value. d. Is it plausible that the main effects of alloy are all equal to 0? Provide the value of the test statistic and the P-value.

Archaeologists can determine the diets of ancient civilizations by measuring the ratio of carbon-13 to carbon-12 in bones found at burial sites. Large amounts of carbon-13 suggest a diet rich in grasses such as maize, while small amounts suggest a diet based on herbaceous plants. The article “Climate and Diet in Fremont Prehistory: Economic Variability and Abandonment of Maize Agriculture in the Great Salt Lake Basin” reports ratios, as a difference from a standard in units of parts per thousand, for bones from individuals in several age groups. The data are presented in the following table.

$$\begin{matrix} \text{Age Group (years)} & \text{Ratio}\\ \text{0–11} & \text{17.2 18.4 17.9 16.6 19.0 18.3 13.6 13.5 18.5 19.1 19.1 13.4}\\ \text{12–24} & \text{14.8 17.6 18.3 17.2 10.0 11.3 10.2 17.0 18.9 19.2}\\ \text{25–45} & \text{18.4 13.0 14.8 18.4 12.8 17.6 18.8 17.9 18.5 17.5 18.3 15.2 10.8 19.8 17.3 19.2 15.4 13.2}\\ \text{46+} & \text{15.5 18.2 12.7 15.1 18.2 18.0 14.4 10.2 16.7}\\ \end{matrix}$$

a. Construct an ANOVA table. You may give a range for the P-value. b. Can you conclude that the concentration ratios differ among the age groups?

To assess the effect of piston ring type and oil type on piston ring wear, three types of piston ring and four types of oil were studied. Three replications of an experiment, in which the number of milligrams of material lost from the ring in four hours of running was measured, were carried out for each of the 12 combinations of oil type and piston ring type. With oil type as the row effect and piston ring type as the column effect, the following sums of squares were observed: SSA = 1.0926, SSB = 0.9340, SSAB = 0.2485, SSE = 1.7034. a. How many degrees of freedom are there for the effect of oil type? b. How many degrees of freedom are there for the effect of piston ring type? c. How many degrees of freedom are there for interactions? d. How many degrees of freedom are there for error? e. Construct an ANOVA table. You may give ranges for the P-values. f. Is the additive model plausible? Provide the value of the test statistic and the P-value. g. Is it plausible that the main effects of oil type are all equal to 0? Provide the value of the test statistic and the P-value. h. Is it plausible that the main effects of piston ring type are all equal to 0? Provide the value of the test statistic and the P-value.

Four lighting methods were used in each of three rooms. For each method and each room, the illuminance (in lux) was measured in three separate occasions, resulting in three replications of the experiment. The only effect of interest is the lighting type; the room is a blocking factor. The following sums of squares were calculated: sum of squares for blocks = 11432, sum of squares for treatments = 9943, sum of squares for interactions = 6135, total sum of squares = 51376. a. Construct an ANOVA table. You may give ranges for the P-values. b. Are the assumptions for a randomized complete block design satisfied? Explain. c. Does the ANOVA table provide evidence that lighting type affects illuminance? Explain.

While Simpson’s Rule is generally more accurate than trapezoidal approximation, show that this is not always the case by considering the function $f(x)=9 x^{2}-5 x^{4}$ on the interval [-1, 1] as follows: a. Find the exact area under the curve by integration. b. Use trapezoidal approximation with two trapezoids to approximate the area. c. Use Simpson’s Rule with n = 2 to approximate the area. d. Which approximation method gave greater accuracy?

A study was done to see which of four machines is fastest in performing a certain task. There are three operators; each performs the task twice on each machine. A randomized block design is employed. The MINITAB output follows.

$$\begin{matrix} \text{Source} & \text{DF } & \text{SS} & \text{MS} & \text{F} & \text{P}\\ \text{Machine} & \text{(i)} & \text{257.678} & \text{(ii)} & \text{(iii)} & \text{0.021}\\ \text{Block} & \text{(iv)} & \text{592.428} & \text{(v)} & \text{(vi)} & \text{0.000}\\ \text{Interaction} & \text{(vii)} & \text{(viii)} & \text{(ix)} & \text{(x)} & \text{0.933}\\ \text{Error} & \text{(xi)} & \text{215.836} & \text{17.986}\\ \text{Total} & \text{(xii)} & \text{1096.646}\\ \end{matrix}$$

a. Fill in the missing numbers (i) through (xii) in the output. b. Does the output indicate that the assumptions for the randomized block design are satisfied? Explain. c. Can you conclude that there are differences among the machines? Explain.

Suppose that you want to perform a hypothesis test for a population mean. Assume that the variable under consideration is normally distributed and that the population standard deviation is unknown. a. Can the t-test be used to perform the hypothesis test? Explain your answer. b. Can the Wilcoxon signed-rank test be used to perform the hypothesis test? Explain your answer. c. Which procedure is preferable, the t-test or the Wilcoxon signed-rank test? Explain your answer.

Under what circumstances is using a normal probability plot to assess the normality of a variable usually better than using a histogram, stem-and-leaf diagram, or dotplot?

The article “Multi-objective Scheduling Problems: Determination of Pruned Pareto Sets" presents examples in a discussion of optimization methods for industrial scheduling and production planning. In one example, seven different jobs were performed on each of five machines. The cost of each job on each machine is presented in the following table. Assume that it is of interest to determine whether costs differ between machines, but that it is not of interest whether costs differ between jobs.

$$\begin{matrix} & \text{Job}\\ \text{Machine} & \text{1} & \text{2} & \text{3} & \text{4} & \text{5} & \text{6} & \text{7}\\ \text{A} & \text{16} & \text{24} & \text{18} & \text{22} & \text{26} & \text{20} & \text{22}\\ \text{B} & \text{22} & \text{18} & \text{30} & \text{20} & \text{28} & \text{18} & \text{26}\\ \text{C} & \text{12} & \text{12} & \text{15} & \text{18} & \text{22} & \text{12} & \text{16}\\ \text{D} & \text{18} & \text{28} & \text{16} & \text{26} & \text{34} & \text{18} & \text{18}\\ \text{E} & \text{14} & \text{14} & \text{11} & \text{16} & \text{18} & \text{10} & \text{12}\\. \end{matrix}$$

a. Identify the blocking factor and the treatment factor. b. Construct an ANOVA table. You may give ranges for the P-values. c. Can you conclude that there are differences in costs between some pairs of machines? Explain. d. Which pairs of machines, if any, can you conclude, at the 5% level, to have differing mean costs?

The article “Multi-objective Scheduling Problems: Determination of Pruned Pareto Sets” presents examples in a discussion of optimization methods for industrial scheduling and production planning. In one example, seven different jobs were performed on each of five machines. The means and standard deviations of the processing times for each machine are presented in the following table.

$$\begin{matrix} \text{Machine} & \text{Mean} & \text{ SD} & \text{Sample Size}\\ \text{A} & \text{25.43} & \text{10.67} & \text{7}\\ \text{B} & \text{23.71} & \text{13.92} & \text{7}\\ \text{C} & \text{44.57} & \text{15.90} & \text{7}\\ \text{D} & \text{23.14} & \text{12.75} & \text{7}\\ \text{E} & \text{58.00} & \text{19.11} & \text{7}\\ \end{matrix}$$

a. Construct an ANOVA table. You may give a range for the P-value. b. Can you conclude that there are differences among the mean processing times?

Three different corrosion-resistant coatings are being considered for use on iron pipes. Ten pieces of pipe are marked off in six equal segments. For each pipe, two segments received coating A, two received coating B, and the remaining two received coating C. The pipes were placed in a corrosive environment for a period of time; then the depth of the deepest pit (in mm) caused by corrosion was measured for each segment on each pipe. The effect of interest is the coating; the pipe is a blocking factor, and there were two replications on each pipe. The following sums of squares were calculated: sum of squares for blocks = 11.2, sum of squares for treatments = 4.8, sum of squares for interactions = 18.4, total sum of squares = 44.7. a. Construct an ANOVA table. You may give ranges for the P-values. b. Are the assumptions for a randomized complete block design satisfied? Explain. c. Does the ANOVA table provide evidence that mean pit depth differs among coatings? Explain.

Fill in the blank with an appropriate word, phrase, or symbol(s). If a first experiment can be performed in three distinct ways and a second experiment can be performed in eight distinct ways, the two experiments together can be performed in _ distinct ways.

Which normal distribution has a wider spread: the one with mean 1 and standard deviation 2 or the one with mean 2 and standard deviation 1? Explain your answer.

Match the disorder in Column A with its corresponding cause or description in Column B. Turner syndrome