Question

Provide R code solution.

Explanation and details.

4. Question 5.7. Johnson and Leone (Statistics and Experimental Design in Engineering and the Physical Sciences, Wiley, 1977) describe an experiment to investigate warping of copper plates. The two factors studied were the temperature and the copper content of the plates. The response variable was a measure of the amount of warping. The data were as follows: Copper Content (%) Temperature (C)40 100 17, 20 16,2 24. 22 28, 27 12.9 18, 13 17, 12 27,31 16, 12 18,21 25, 23 30, 23 21, 17 23, 2 23, 22 29,31 60 80 50 75 100 125 (a) Is there any indication that either factor affects the amount of warping? Is there any interaction between the factors? Use (alpha- 0.05). Provide a graphical evedice for your answer (b) Create an anova table (effect and interaction model), comment on the results. (c) Analyze the residuals from this experiment. (d) Plot the average warping at each level of copper content and compare them to an appropriately scaled t distribution. Describe the differences in the effects of the different levels of specify? (e) Suppose that temperature cannot be easily controlled in the environment in which the copper coger conternt on warngwwrigk desirable what level fcopper conmient woud you plates are to be used. Does this change your answer for part (c)?

Answer 1

Solution :-

1.

Here

One will use two way ANOVA with interactions to check effect of temperature and copper content on the response variable. Df Sum Sq Mean Sq F value Pr(>F)   
x1 3 533.6 177.86 22.858 5e-06
x2 3 156.1 52.03 6.687 0.0039
x1*x2 9 262.5 . 29.17 3.749 0.0104
Residuals 16 124.5 7.78

P-value for interaction between temperature and copper content = 0.0104 > 0.01,

so at 1% level of significance, we fail to reject H₀ and we can conlcude that there is no significant effect of interaction between temperature and copper content in the response variable.

P-value for temperature = 0.0039 < 0.01

and P-value for copper content = 0.000005 < 0.01

so at 1% level of significance we can conclude that there is significant effect of temperature and copper content in response variable.