For hints and examples of similar problems, see last year's Assignment #3 and its solutions.
Use R to re-draw Figs. 8-4 (p. 258), 8-8 (p. 262) and 10-4 (p. 357) from the text.
(a) If you took 9 observations and found x_bar = 38.7 and s = 12, what would be a 95% confidence interval for the mean? If you wanted the width of the confidence interval to be +/-2, how many observations would you need?
(b) The acceptable limits for the weight of a pellet you are producing are 96 g to 104 g. The standard deviation of weight is known to be 2.5 g. The process mean is usually 100 g but 10% of the time the process shifts and the mean is 102 g. If four pellets are sampled and three of them found to be unacceptable, what is the probability that the process has shifted? State any assumptions you make.
Carry out appropriate analyses for the following two data sets. Give graphs. State any assumptions you make. As far as possible, test each assumption. State your conclusions.
(a) In a study to select a method for determining chlorine content in sewage effluent, two different methods were tried with eight different samples of effluent. Observations are in mg/l.
Sample: 1 2 3 4 5 6 7 8 MSI method: 0.39 0.84 1.76 3.35 4.69 7.70 10.52 10.92 SIB method: 0.36 1.35 2.56 3.92 5.35 8.33 10.70 10.91(b) Four laboratories were each asked to make three determinations of percent of methyl alcohol in specimens of a compound taken from a single batch. One of the 12 determinations was discarded because of an accident in the lab; the remaining 11 were as shown.
Laboratory: 1 1 1 2 2 2 3 3 3 4 4 4 % alcohol: 85.06 85.25 84.87 84.99 84.28 NA 84.48 84.72 85.10 84.10 84.05 84.55
The following experimental data show the load (in 1000 lb/ft) necessary to obtain a first crack in a pipe specimen as a function of the age (in days) of the pipe specimen.
Age: 20 20 20 25 25 25 31 31 31 Load: 11.45 10.42 11.14 10.84 11.17 10.54 9.47 9.19 9.54(a) Fit a straight line to the data by least squares, with load as the dependent variable. Plot the data and the fitted line on a graph. Can load to first crack be predicted as a linear function of specimen age? Present your analysis in an ANOVA table with F-Tests for non-linearity and for the slope of the regression line. Give a 95% confidence interval for the residual variance. State your assumptions and your conclusions.
(b) Predict the load at first crack for specimens aged 22 and 100 days. How reliable do you think your predictions are?
The following experimental data show the amount of warping in copper plates, as a function of temperature (degrees C) and the copper content (% Cu) of the plate. Give an interaction plot and a two-factor ANOVA table. Give a 95% confidence interval for the residual variance. State your conclusions.
warping 17 20 16 21 24 22 12 9 18 13 17 12 21 17 23 21 23 22 temp 50 50 50 50 50 50 75 75 75 75 75 75 125 125 125 125 125 125 % Cu 40 40 60 60 80 80 40 40 60 60 80 80 40 40 60 60 80 80
14-8 (p. 520). [Note: you did interaction plots for these data in Assignment #1.]