Statistics - ANOVA Projects

3-1. The tensile strength of portland cement is being studied. Four different mixing techniques can be

3-1. The tensile strength of portland cement is being studied. Four different mixing techniques can be used economically. A completely randomized experiment was conducted and the following data collected:

Mixing Technique	Tensile Strength (lb/in)
1	3129	3000	2865	2890
2	3200	3300	2975	3150
3	2800	2900	2985	3050
4	2600	2700	2600	2765

Test the hypothesis that mixing techniques affect the strength of the cement. Use a = 0.05.
Construct a graphical display as described in Section 3-5.3 to compare the mean tensile strengths for the four mixing techniques. What are your conclusions?
Use the Fisher LSD method with a = 0.05 to make comparisons between pairs of means.
Construct a normal probability plot of the residuals. What conclusion would you draw about the validity of the normality assumption?
Plot the residuals versus the predicted tensile strength. Comment on the plot.
Prepare a scatter plot of the results to aid the interpretation of the results of this experiment.

(a) Rework part (b) of Problem 3-1 using Tukey’s test with  = 0.05. Do you get the same conclusions from Tukey’s test that you did from the graphical procedure and/or the Fisher LSD method?
(b) Explain the difference between the Tukey’s and Fisher procedures.
3-6. A pharmaceutical manufacturer wants to investigate the bioactivity of a new drug. A completely randomized single-factor experiment was conducted with three dosage levels, and the following results were obtained.

Dosage		Observations
20 g	24	28	37	30
30 g	37	44	31	35
40 g	42	47	52	38

Is there evidence to indicate that dosage level affects bioactivity? Use a = 0.05.
If it is appropriate to do so, make comparisons between the pairs of means. What conclusions can you draw?
Analyze the residuals from this experiment and comment on model adequacy.

3.16. The response time in milliseconds was determined for three different types of circuits that could be used in an automatic valve shutoff mechanism. The results are shown in the following table.

Circuit Type Response Time

1 9 12 10 8 15

2 20 21 23 17 30

3 6 5 8 16 7

(a) Test the hypothesis that the three circuit types have the same response time. Use $\alpha $ = 0.01.

(b) Use Tukey’s test to compare pairs of treatment means. Use $\alpha $ = 0.01.

(c) Use the graphical procedure in Section 3.5.3 to compare the treatment means. What conclusions can you draw? How do they compare with the conclusions from part (b)?

(d) Construct a set of orthogonal contrasts, assuming that at the outset of the experiment you suspected the response time of circuit type 2 to be different from the other two.

(e) If you were a design engineer and you wished to minimize the response time, which circuit type would you select?

(f) Analyze the residuals from this experiment. Are the basic analysis of variance assumptions satisfied?

3.29. Refer to problem 3-16. If we wished to detect a maximum difference in the mean response times of 10 milliseconds with a probability of at least 90%, what size should be used? How would you obtain a preliminary estimate of ${{\sigma }^{2}}$ ?

Solution: We use MSE to estimate ${{\sigma }^{2}}$, which means that ${{\sigma }^{2}}\approx 16.9$. Hence,

\[{{\Phi }^{2}}=\frac{n{{D}^{2}}}{2a{{\sigma }^{2}}}=\frac{n\times {{10}^{2}}}{2\times 3\times 16.9}=0.986n\]

We set $\alpha =0.05$, $\Pr \left( \text{Accept} \right)=0.1$, and $\nu =3-1=2$, by trial and error we find that $n\ge 6$, and hence, since we have 3 groups, $N\ge 18$.

4.5. Consider the hardness testing experiment described in 4-1. Suppose that the experiment was conducted as described and the following data are obtained

(a) Analyze the data

(b) Use Fisher LSD to make comparisons

4-7 The effect of three different lubricating oils on fuel economy in diesel truck engines is being studied. Fuel economy is measured using brake-specific fuel consumption after the engine has been running for 15 minutes. Five different truck engines are available for the study, and the experimenters conduct the following randomized complete block design.

	Truck
Oil	1	2	3	4	5
1	0.5	0.634	0.487	0.329	0.512
2	0.535	0.675	0.52	0.435	0.54
3	0.513	0.595	0.488	0.4	0.51

(a) Analyze the data from this experiment.

(b) Use the Fisher LSD method to make comparisons among the three lubricating oils to determine specifically which oils differ in brake-specific fuel consumption.

4-18 The effect of five different ingredients (A. B, C. D, E) on the reaction time of a chemical process is being studied. Each batch of new material is only large enough to permit five runs to he made. Furthermore, each run requires approximately 1 hours, so only five runs can be made in one day. The experimenter decides to run the experiment as a Latin square so that day and batch effects may be systematically controlled. She obtains the data that follow. Analyze the data from this experiment (use a = 0.05) and draw conclusions.

4-31. An engineer is studying the mileage performance characteristics of five types of gasoline additives. In the road test he wishes to use cars as blocks: however, because of a time constraint, he must use an incomplete block design. He runs the balanced design with the five blocks that follow. Analyze the data from this experiment (use a = 0.05) and draw conclusions.

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