Stat 324 – HW 4

Due by 3pm, Friday May 1

1) Exercises 4.7 and 4.8 (p. 135, 136)

Hints: In (c) and (d), you should use the overall model df, n-p-1 in finding the t critical value.

2) Exercise 4.12 (p. 137)

3) The Federal Trade Commission (FTC) annually ranks varieties of domestic cigarettes according to their tar, nicotine, and carbon monoxide contents. The U.S. surgeon general considers each of these three substances hazardous to a smoker’s health. Past studies have shown that increases in the tar and nicotine contents of a cigarette are accompanied by an increase in the carbon monoxide emitted from the cigarette smoke. FTCcigs.mtw contains data for a sample of 25 (filter) brands tested in a recent year (as in Mendenhall and Sincich).

(a) Fit and interpret (e.g., interpret the regression coefficients and the usefulness of the model) a regression model for predicting carbon monoxide from nicotine. Does the model agree with the past studies?

(b) Compare this model to one that predicts carbon monoxide from nicotine and tar (e.g., again, interpret the individual coefficients including their signs, their significance, and the overall model utility).

4) Evolutionary biologists are interested in the characteristics that enable a species to withstand the selective mechanisms of evolution. An interesting variable in this respect is brain size. One might expect that bigger brains are better, but certain penalties seem to be associated with large brains, such as the need for longer pregnancies and fewer offspring. To explore these issues, it is helpful to determine which characteristics are associated with large brains, after getting the effect of body size out of the way. The data in brains.mtw are the average values of brain weight, body weight, gestation length (length of pregnancy), and litter size for 96 species of mammals. (Data from Sascher and Staffeldt (1974) as cited in Ramsey and Schafer (2002).)

(a) Produce a matrix plot of these four variables. Summarize what this series of plots reveals.

(b) Produce a matrix plot of ln(brain weight) against the other variables. Does this suggest transforming some of the explanatory variables as well?

(c) Produce a matrix plot after taking the natural log for all four variables. Which variable appears to have the strongest relationship with ln(brain weight)? Which variables appear strongly associated with each other.

(d) Does the matrix plot in (c) reveal any unusual observations in predicting ln(brain weight)? Circle the observation you feel stands out the most and identify the mammal by name.

(e) Fit the regression model for predicting ln(brain weight) from ln(body weight), ln(gestation length), and ln(litter size). Examine and comment on the residual plots.

(f) Identify one species in the residuals plots revealed to have an unusually large positive residual. Discuss what it means that this mammal has a large positive residual in this context.

(g) The coefficient of ln(gestation) in this model is .4179. The three-toed sloth has a gestation period of 165 days. The Indian fruit bat has a gestation period of 145 days. Does this imply that an estimate of the mean log brain weight for the sloth is (.4179)(.1292) larger than the mean log brain weight for the bat (i.e., that the median if 5.5% higher)? Why or why not? (Being clear where this reasoning (and these numbers) is coming from and whether it is sound or not.)

(h) From your output in (e), is there evidence that brain weight is associated with either gestation period and/or litter size after accounting for the effects of body weight? (Show all your work.)

(i) Is there convincing evidence that litter size is associated with brain weight after accounting for body weight and gestation? (Show your work and being clear how this question differs from that in (g)).

(j) Is there convincing evidence that gestation period is associated with brain weight after accounting for body weight and litter size?

(k) Is it reasonable to draw any causal conclusions from this study? Explain.