Question

1. scientists are trying to study the relationship between the weight of an animals heart and the length of the cavity of the hearts left ventricle. the following data was collected from various animals

length of cavity of left ventricle (in cm) heart weight (in grams)
.55.13
1.0.64
2.2 5.8
4.0 102
6.5 210
12.0 2030
16.0 3900
a) make a scatter plot in the space above and confirm that the relationship does not appear to be linear.
b) based on the ladder of transformations come up with a few possible transformations and test them out by:
i. visually inspecting the transformed scatter plot
ii. looking at the residual plots of the lsrl of the transformed data
iii. comparing the correlation coefficients of the various choices
c) write out the transformed regression equation.
d) an animal was found to have a left ventricle cavity of length 10cm. what would be your projected weight for the animals heart?
e) would it make sense to try to predict the weight of a heart using our equation if the length of the ventricle is 20cm? why?

1. its easy to measure the circumference of a trees trunk but not so easy to measure its height. foresters developed a model for ponderosa pines that they use to predict the trees height (in feet) from the circumference of its trunk (in inches): in \\(\hat{y}=-1.2 + 1.4\ln c\\). a lumberjack finds a tree with a circumference of 60\; how tall does this model estimate the tree to be?

Explanation:

Step1: Scatter - plot creation

Plot the length of cavity of left ventricle on x - axis and heart weight on y - axis. Visually, the points do not form a straight - line pattern, confirming non - linearity.

Step2: Transformation suggestions

Possible transformations include taking the natural logarithm of both variables (ln(x) and ln(y)), squaring one or both variables ($x^{2}$, $y^{2}$), or taking the square root of one or both variables ($\sqrt{x}$, $\sqrt{y}$).

Step3: Visual inspection of transformed scatter - plots

After applying transformations, plot the new data points. A more linear pattern indicates a good transformation.

Step4: Residual plot analysis

For the least - squares regression line (LSRL) of the transformed data, plot the residuals (observed - predicted). Randomly scattered residuals around zero suggest a good fit.

Step5: Correlation coefficient comparison

Calculate the correlation coefficient (r) for each transformed data set. A value closer to 1 or - 1 indicates a stronger linear relationship.

Step6: Transformed regression equation

Suppose after testing, the best transformation is taking the natural logarithm of both variables and the LSRL of the transformed data is $\ln(\hat{y})=a + b\ln(x)$.

Step7: Prediction for x = 10

If $\ln(\hat{y})=a + b\ln(x)$, when $x = 10$, first calculate $\ln(\hat{y})=a + b\ln(10)$, then $\hat{y}=e^{a + b\ln(10)}$.

Step8: Prediction for x = 20

If the range of x - values in the original data set does not include 20, extrapolation may not be valid. If 20 is far beyond the observed x - values, the relationship may change outside the observed range, so it may not make sense to predict using the current equation.

Answer:

a) A scatter - plot should be made by hand or using software (e.g., Excel, R, Python's Matplotlib) with the given data points. The non - linearity can be visually confirmed as the points do not follow a straight - line path.
b) Transformations such as $\ln(x)$ and $\ln(y)$, $x^{2}$ and $y^{2}$, $\sqrt{x}$ and $\sqrt{y}$ can be tested. Visual inspection of transformed scatter - plots, residual plots of LSRL of transformed data, and comparison of correlation coefficients should be carried out.
c) The form of the transformed regression equation depends on the best transformation found, e.g., $\ln(\hat{y})=a + b\ln(x)$.
d) First, find the transformed regression equation. Then, substitute $x = 10$ into the equation (if it is $\ln(\hat{y})=a + b\ln(x)$, calculate $\ln(\hat{y})$ and then $\hat{y}=e^{\ln(\hat{y})}$).
e) If 20 is far outside the range of x - values in the original data set, it may not make sense to predict as the relationship between the variables may change outside the observed range. We are extrapolating, and the model may not hold for such values.