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###### 日期：2022-03-02 07:22

1 The dataset
consists of information about the production of various tableware products. After casting,
each piece of tableware goes through a series of grinding and polishing steps.
The variables (adapted from the original dataset) are:
? Type: the type of product; a categorical variable with categories Bowl, Plate and Tray;
? Diam: the diameter of the product (in inches);
? Time: the total grinding and polishing time of the product (in minutes);
An engineer in the ceramic factory suggests that the total time it takes to grind and polish
the product can be predicted from its diameter using an equation of the form
time = a ? diameterb (1)
for some constants a and b.
(a) [4 marks] Produce a scatterplot of the grinding and polishing time against the diameter
of the product. Use dierent
colours and/or plotting symbols to show the type of the product.
Your plot should be clearly labelled and contain a legend.
(b) [3 marks] Explain how the relationship in equation (1) as suggested by the engineer
can be transformed into a relationship that can be modelled by a simple linear regression.
What assumptions are you making about the errors in the original relationship, that is in the
original scale of the response?
(c) [4 marks] Fit the simple linear regression model in (b), that is a model for the transformed
relationship. Produce a ‘residuals versus fitted values’ plot and a scale-location
plot of the fitted model and discuss whether linearity and homoscedasticity can be assumed
to hold.
(d) [2 marks] Give a quantitative interpretation of the estimated slope parameter for the
model in (c) in the original scale of the response variable Time.
(e) [2 marks] Predict the time (in minutes) that it will take to grind and polish a product
that has a diameter of 15 inches.
(f) [1 mark] Reproduce the plot from part (a) and add a curve that shows how, according to
the model in (c), the predicted time for grinding and polishing changes with diameter.
Question continues on the next page
1The data is adapted from the Nambe dataset available from DASL, the Data and Story library.
2
(g) [3 marks] The engineer sees your plot which shows the relationship between the
predicted time for grinding/polishing and the diameter. They realise that the
predicted grinding and polishing time for a given products looks like it is (approximately)
proportional to the diameter of the product. They wonder whether a
simple linear regression of time on diameter (possibly through the origin) would have suced.
Discuss whether this would have been a suitable alternative supporting your answer with
appropriate evidence.
(h) [2 marks] The engineer then suggests that the constant a in equation (1) may depend on
the type of the product. Explain how to modify the model in (b)-(c) to accommodate this.
(i) [2 marks] Write out the model equations for the new linear model suggested in (h).
(j) [2 marks] Give a description of the jth row of the design matrix for the model suggested
in (h) using indicator variables.
(k) [2 marks] Fit the model in (h). Reproduce the plot from part (a) and add a curve for
each product type that shows how, according to the fitted model, the predicted time for
grinding and polishing changes with diameter.
Hint: If c is an array of observations for the variable C and d is an array of corresponding
observations for the variable D, then to compute the predictions from a model m with
explanatory variables C and D, we use a command of the form
predict(m, list(C=c, D=d)).
(l) [3 marks] Judging from the plots in (f) and (k) do you think that the engineer was right
to suggest that the constant a in equation (1) should depend on the type of the product.