Question

Heart weights of mammals Here are some data on the hearts of various mammals:

a. Make an appropriate scatterplot for predicting heart weight from length. Describe what you see.

b. Use transformations to linearize the relationship. Does the relationship between heart weight and length seem to follow an exponential model or a power model? Justify your answer.

c. Perform least-squares regression on the transformed data. Give the equation of your regression line. Define any variables you use.

d. Use your model from part (c) to predict the heart weight of a human who has a left ventricle$6.8$ cm long.

Short answer

(a) The scatter plot slope is sloping upwards, hence the direction is positive.

(b) Power model

(c) $\ln y=-0.314+3.139\ln x$

(d)$299.825gms$

Step-by-step solution

Part (a) Step 1: Given information

The given data is

Part (a) Step 2: Explanation

From the given data the graph is

The reason for the curve is that the points are not on a straight line.

The scatter plot slope is sloping upwards, hence the direction is positive.

The reason for this is that all of the points are fairly close together in the same pattern.

Part (b) Step 1: Given information

The given data is

Part (b) Step 2: Explanation

The graph is

The model with the most linear pattern in the associated scatter plot is the model for the relationship between the variables." Power Model

Part (c) Step 1: Given information

The given data is

Part (c) Step 2: Explanation

The logarithm of length is $X$, and the logarithm of heart weight is $Y$.

Find the values using the table

$$\begin{gathered} a=\frac{\sum y·\sum x^2-\sum x\sum xy}{n\sum x^2-{\left(\sum x\right)}^2} \\ a=-0.314 \end{gathered}$$

$$\begin{gathered} b=\frac{n\sum xy-\sum x·\sum y}{n\sum x^2-{\left(\sum x\right)}^2} \\ b=3.139 \end{gathered}$$

then the value of y is

$y=a+b.x$

$y=-0.314+3.139x$

The least-square regression equation is

$\ln y=-0.314+3.139\ln x$.

Part (d) Step 1: Given information

The given data is

Part (d) Step 2: Explanation

From part (c)

$\ln y=-0.314+3.139\ln x$

Putting the value of x

$\ln y=-0.314+3.139\ln (6.8)$

$=5.7032$

Therefore taking the exponential

$$\begin{gathered} y=e^{\ln y} \\ y=e^{5.7032} \end{gathered}$$

$=299.825$

The expected heart weight is$299.825gms$