Q.4.148 - Elementary Statistics

Question

Following are the data on age and crown-rump length for fetuses.

a. Obtain the linear correlation coefficient. 

b. Interpret the value of $r$in terms of the linear relationship between the two variables in question.

c. Discuss the graphical interpretation of the value of $r$and verify that it is consistent with the graph you obtained in the corresponding exercise in Section 4.2.

d. Square $r$and compare the result with the value of the coefficient of determination you obtained in the corresponding exercise in Section 4.3.

Step-by-Step Solution

Answer

(a) The linear correlation coefficient is $0.9975$

(b) The variables are positively linearly correlated.

(c) 

(d) The coefficient of determination is$0.9950$

Part (a) Step 1: Given Information

The given table is 

We have to obtain the linear correlation coefficient  

Part (a) Step 2: Explanation

Table of computation

The formula of  linear correlation coefficient. is 

$$\begin{gathered} r=\frac{\sum x_i{y}_i-\left(\sum x_i\right){\left(\sum y_i\right)}^{\text{'}n}}{\sqrt{\left[\sum x_i^2-{\left(\sum x_i\right)}^2/n\right]\left[\sum y_i^2-{\left(\sum y_i\right)}^2/n\right.}} \\ =\frac{32476-\left(178\right)\left(1593\right)/10}{\sqrt{\left[3522-(178{)}^2/10\right]\left[302027-(1593{)}^2/10\right]}} \\ =0.9975 \end{gathered}$$

Part (b) Step 1: Given Information

The given table is

We have to interpret the value of $r$ in terms of the linear relationship between the two variables in question.   

Part(b) Step 2: Explanation

The variables are strongly correlated if the estimated $r$is near to $\pm 1$

Close to$1$ is the computed correlation coefficient. As a result, the variables are positively linearly correlated. therefore, age of fetuses increases then crown-rump length for fetuses also increases.

Part (c) Step 1: Given Information

The given table is 

We have to discuss the graphical interpretation of the value of $r$and verify that it is consistent with the graph you obtained in the corresponding exercise in Section 4.2.  

Part(c) Step 2: Explanation

If $r$ is near to $0$, the data points are essentially scattered along a horizontal line. If $r$is the father of $\pm 1$, the data points are more widely dispersed around the regression line. If $r$is close to$\pm 1$, the data points cluster closely around the regression line.

 

The graph in shown below

$r$ is close to $1$ when calculated. As a result, the data points are closely clustered around the regression line. . As a result, the calculated correlation coefficient matches the graph. 

Part (d) Step 1: Given Information

Given table is 

We have to square $r$ and compare the result with the value of the coefficient of determination you obtained in the corresponding exercise in Section 4.3.   

Part (d) Step 2: explanation

The square of  $r$ is

${\left(0.9975\right)}^2=0.9950$