Q. 4.102 - Elementary Statistics

Question

Crown-Rump Length. Following ate the data on age and crown-rump length for fetuses from Exercise $4.62$.

a. Compute $SST,SSR,SSE$

b. Compute the coefficient of determination 

c. Determine the percentage of variation in the observed values of the response variable explained by the regression, and interpret your answer.

d. State how useful the regression equation appears to be for making predictions.

Step-by-Step Solution

Answer

(a) $$\begin{gathered} SST=48262.1 \\ SSR=48018.5078 \\ SSE=243.5922 \end{gathered}$$

(b) $0.9950$

(c) $99.5\%$

(d) Employing the regression equation to create predictions is quite useful, and the regression can explain more than $99\%$ of the variation.

Part (a) Step 1: Given information

The given data is 

Part (a) Step 2: Explanation

From the values of the given table

The formulas to calculate the sum of squares is  

$SST=\sum y_i^2-{\left(\sum y_i\right)}^2/n$

$SSR=\frac{{\left[\sum x_i{y}_i-\left(\sum x_i\right)\left(\sum y_i\right)/n\right]}^2}{\sum x_i^2-{\left(\sum x_i\right)}^2/n}$

$SSE=SST-SSR$

As shown in the table below, the relevant sums can be determined.  

$$\begin{gathered} SST=302027-\frac{{1593}^2}{10} \\ SST=48262.1 \end{gathered}$$

$$\begin{gathered} SSR=\frac{\mid 32476-178\times 1593÷10{]}^2}{3522-{178}^2÷10} \\ SSR=48018.5078 \end{gathered}$$

$$\begin{gathered} SSE=48262.1-48018.5078 \\ SSE=243.5922 \end{gathered}$$

Part (b) Step 1: Given information

The given data is 

Part (b) Step 2: Explanation

The coefficient of determination is 

$r^2=\frac{SSR}{SST}$

     $$\begin{gathered} =48018.5078/48262.1 \\ =0.9950 \end{gathered}$$

Part (c) Step 1: Given information

The given data is 

Part (c) Step 2: Explanation

The coefficient of determination restated as a percentage is the percentage of variation:    

$0.9950=99.50\%$

Part (d) Step 1: Given information

The given data is 

Part (d) Step 2: Explanation

The regression equation can be used to generate predictions if the estimated $r^2$ is near to 1.

The computed $r^2=99.50$, which is extremely near to $1$.

As a result, employing the regression equation to create predictions is quite useful, and the regression can explain more than $99\%$ of the variation.