#### Assignment 7

Due Thursday, April 25.

Turn in problems # 1 and 2.

Problem 1:
(From Weisberg, 1985.) The data in the file coins.dat are from a paper written by W. Stanley Jevons (1868). 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 date of issue and the weight after cleaning to the nearest .001 gram. In the file coins.dat, the coins have been grouped into age classes coded 1 to 5, roughly corresponding to the age of the coin in decades. For each age class, the data includes the number of coins measured, their average weight, standard deviation, and minimum and maximum weight. Note that the standard weight of a gold sovereign was supposed to be 7.9876 grams; the minimum legal weight was 7.9379 grams.
• Draw a scatter plot of average weight versus coded age, and a scatter plot of the SD's versus coded age. Summarize the information in these plots, and give a plausible real-world explanation of what you observe in these plots. Comment on the applicability of the usual assumptions of the linear regression model.
• Since the number of coins n in each age class are all fairly large, it is reasonable to pretend that the variance of coin weight for each age class is well approximated by , and hence the variance of the average for that class is given by . Use the values given to find weights, and compute the weighted regression of average weight on age.
• Is the fitted regression consistent with the known standard weight for a new coin? If not, give at least two plausible explanations for the discrepancy.
• For a previously unsampled coin of age 1,2,3,4,5 decades, estimate the probability that the weight of the coin is less than the legal minimum. (Hint: Use the value of SD for coins of a given age to estimate their standard deviation, and use your regression results to estimate their mean. Then pretend that coin weights for a given age are normally distributed.) Also determine the age at which the predicted weight of coins is equal to the legal minimum.

Problem 2:
(From Weisberg, 1985.) The data in the file alfalfa.dat were collected to study the variation in rent paid in 1977 for agricultural land planted to alfalfa. The data include

 AlfRent = average rent per acre planted to alfalfa TilRent = average rent paid for all tillable land CowDens = density of dairy cows (number per square mile) Pasture = proportion of farmland used as pasture Liming = whether liming is required to grow alfalfa

The unit of analysis is a county in Minnesota; the 67 counties with appreciable rented farmland are included.

Alfalfa is a high protein crop that is suitable feed for dairy cows. It is thought that rent for land planted to alfalfa relative to rent for other agricultural purposes would be higher in areas with a high density of dairy cows, and rents would be lower in counties where liming is required, since that would mean additional expense.

Use the techniques learned so far to explore these data with regard to understanding rent structure. What do you conclude concerning the hypotheses stated in the last paragraph?

Rawlings:
# 10.3, 10.4, 10.7, 10.10, 11.3 (a), 11.5, 11.8, 11.9 (you may plot the log-likelihood instead of the residual sums of squares), 11.11, 11.16, 11.17, 11.18, 11.19.