The cerebellum is a highly convoluted brain structure It is

The cerebellum is a highly convoluted brain structure. It is thought to facilitate acquisition and use of sensory data by the rest of the brain, particularly motor areas. Studies suggest that the cerebellum may scale up with the size of animals’ bodies and brains. Here are data for 15 mammal species showing weights (in grams) of their bodies and cerebellums.

2. Using the analysis of the log transformed data, construct a 95% confidence interval for ?1.  

3. Using the analysis of the log transformed data, predict the weight of cerebellum when body weight is 250000 grams. (Hint: don’t forget transform back to the original cerebellum from the log transformed data.)

Solution

FALSE. When regressing Cerebellum weight on Body weight, the regression assumptions have not been met.

because from the scatterplot and also from the R-square value(12.1%) it is clear that the prediction formula is very poor.

1. TRUE. the regression assumptions have been met.

because from the analysis of variance table it is found that the p value for the test for existance of regression is 0.000

so the null hypothesis H0: there is no regression is definitely rejected.

so the conclusion is that there exists a significant regression of log Cerebellum weight on log Body weight.

hence the assumptions of regressions are valid.

2. we have to construct a 95% confidence interval for beta1 the slope parameter.

now from the table the estimate of beta1 is 0.78385

ans the standard error is SE coef=0.07183

now here total number of observations is n=15

as here 95% confidence interval is to be constructed

alpha=1-0.95=0.05

so alpha/2=0.025

so the critical value is t_alpha/2;n-2 where t_alpha/2;n-2 is the upper alpha/2 point of a t distribution with df n-2

here alpha/2=0.025 and n-2=13

so t_0.025;13=2.16037 [using minitab]

so the 95% confidence interval is

[estimate of beta1-critical value*standard error,estimate of beta1+critical value*standard error]

=[0.78385-2.16037*0.07183,0.78385+2.16037*0.07183]

=[0.62867,0.93903] [answer]

3.Using the analysis of the log transformed data, the predicted weight of cerebellum when body weight is 250000 grams

log(cerebellum)=-4.97+0.784*log(250000)=4.7745

so cerebellum weight is e^4.7745=118.4517 (grams)