In Chapter 3, Exercise 7, we were looking at the relationship between log(mass)

and log(length) for a sample of 100 New Zealand slugs of the species Limax maximus from a study conducted by Barker and McGhie (1984.) These data are in the Minitab worksheet slug.mtw. We identified observation 90 that did not appear to fit the pattern It is likely that this observation is an outlier that was recorded incorrectly, so remove it from the data set. The summary statistics for the 99 remaining observations are. Note: x is log(length), and y is log(weight)

x = 352.399

y = −33.6547
x2 = 1292.94

xy = −18.0147
y2 = 289.598

(a) Calculate the least squares line for the regression of y on x from the formulas.

(b) Using Minitab, calculate the least squares line. Plot a scatterplot of log weight on log length. Include the least squares line on your scatterplot.

(c) Using Minitab, calculate the residuals from the least squares line, and plot the residuals versus x. From this plot, does it appear the linear regression assumptions are satisfied?

(d) Using Minitab, calculate the estimate of the standard deviation of the residuals.

(e) Suppose we use a normal (3, .52) prior for β, the regression slope coefficient. Calculate the posterior distribution of β|data. (Use the standard deviation you calculated from the residuals as if it is the true observation standard deviation.)

(f) Find a 95% credible interval for the true regression slope β.
(g) If the slug stay the same shape as they grow (allotropic growth) the height and width would both be proportional to the length, so the weight would be proportional to the cube of the length. In that case the coefficient of log(weight) on log(length) would equal 3. Test the hypothesis H0 : β = 3 versus H1 : β = 3 at the 5% level of significance. Can you conclude this slug species shows allotropic growth?