Q. 4.102 - Elementary Statistics

Question

For Exercises 4.98 - 4.103,

(a) Compute SST, SSR, SSE using formula 4.2 on page 179.

(b) Compute the coefficient of determination, $r^2$.

(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.

Following are the data on age and crown-lump length for the fetuses from Exercise 4.62.

Step-by-Step Solution

Answer

(a) The values are $SST=48262.1; SSR=48018.51; SSE=243.59$

(b) The coefiicient of variation is $0.995$.

(c) The percentage of variation is $99.5\%$.

(d) The regression line appears to be extremely useful. 

Part (a) Step 1. Given Information.

A table of values.

Part (a) Step 2. Construct the table.

Construct the table.

Part (a) Step 3. Compute SST, SSR, SSE.

Find SST, SSR, SSE.  

$$\begin{gathered} SST=\sum _{}{y}_i^2-\frac{{\left({\displaystyle \sum _{}}{y}_i\right)}^2}{n} \\ =302027-\frac{{\left(1593\right)}^2}{9} \\ =48262.1 \end{gathered}$$

$$\begin{gathered} SSR=\frac{[{\displaystyle \sum _{}}{x}_i{y}_i-{\left({\displaystyle \frac{{\displaystyle \sum _{}{x}_i\sum _{}{y}_i}}{n}}\right)}^2}{{\displaystyle \sum _{}{x_i}^2-\frac{{\left({\displaystyle \sum _{}}{x}_i\right)}^2}{n}}} \\ =\frac{{[32476-{\displaystyle \frac{\left(178\right)\left(1593\right)}{10}}]}^2}{3522-{\displaystyle \frac{{\left(178\right)}^2}{10}}} \\ =48018.51 \end{gathered}$$

$$\begin{gathered} SSE=SST-SSR \\ =48262.1-48018.51 \\ =243.59 \end{gathered}$$

Part (b) Step 1. FInd the coefficient of determination.

$$\begin{gathered} r^2=\frac{SSR}{SST} \\ =\frac{48018.51}{48262.1} \\ =0.995 \end{gathered}$$

Part (c) Step 1. Find the percentage of variation.

Since the coefficient of variation is 0.995, then 99.5% of the variation in the observed value is explained by the regression.

Part (d) Step 1. Useful for making predictions.

The regression line appears to be extremely useful.