USE R STUDIO library(alr4)

7.8 Jevons's gold coins (Data file: jevons) The data in this example are deduced from a diagram in Jevons (1868) and provided by Stephen M.

Stigler. In a study of coinage, Jevons weighed 274 gold sovereigns that he had collected from circulation in Manchester, England. For each coin, he

recorded the weight after cleaning to the nearest 0.001 g, and the date of issue. The data file includes Age, the age of the coin in decades, n, the

number of coins in the age class, Weight, the average weight of the coins in the age class, SD, the standard deviation of the weights. The minimum

Min and maximum Max of the weights are also given. The standard weight of a gold sovereign was 7.9876 g; the minimum legal weight was

7.9379 g.

7.8.1 Draw a scatterplot of Weight versus Age, and comment on the applicability of the usual assumptions of the linear regression

model. Also draw a scatterplot of SD versus Age, and summarize the information in this plot.

7.8.2 To fit a simple linear regression model with Weight as the response, wls should be used with variance function Var(Weight|Age)

= n2/SD2. Sample sizes are large enough to assume the SD are population values. Fit the wls model.

7.8.3 Is the fitted regression consistent with the known standard weight for a new coin?

7.8.4 For previously unsampled coins of Age = 1, 2, 3, 4, 5, estimate the probability that the weight of the coin is less than the legal minimum. (Hints: The standard error of prediction is the square root of the sum of two terms, the assumed known variance of an unsampled coin of known Age, which is different for each age, and the estimated variance of the fitted value for that Age; the latter is computed from the formula for the variance of a fitted value. You

should use the normal distribution rather than a t to get the probabilities.)

7.8.5 Determine the Age at which the predicted weight of coins is equal to the legal minimum, and use the delta method to get a standard

error for the estimated age. This problem is called inverse regression, and is discussed by Brown (1993).