Q. 4.148 - Elementary Statistics

Question

In below exercise, we repeat data from exercises in Section 4.2. For given exercise here.

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.

Crown-Rump Length: Following are the data on age and crown-rump length for fetuses from Exercises 4.62 and 4.102.

x	10	10	13	13	18	19	19	23	25	28
y	66	66	108	106	161	166	177	228	235	280

Step-by-Step Solution

Answer

From the given data the calculated coefficient of determination $r^2=0.995$

From the two values we observe that both the values are equal.

Step 1. Given

x	10	10	13	13	18	19	19	23	25	28
y	66	66	108	106	161	166	177	228	235	280

Step 2. Table for obtaining the linear correlation coefficient.

x	y	xy	x2	y2
10	66	660	100	4356
10	66	660	100	4356
13	108	1404	169	11664
13	106	1378	169	11236
18	161	2898	324	25921
19	166	3154	361	27556
19	177	3363	361	31329
23	228	5244	529	51984
25	235	5875	625	55225
28	280	7840	784	78400
178	1593	32476	3522	302027

The number of observation is n=10

$$\begin{gathered} r=\frac{\underset{}{\overset{}{\sum x_i{y}_{i }- {\displaystyle \frac{\left(\underset{}{\overset{}{\sum x_i}}\right) \left(\underset{}{\overset{}{\sum y_i}}\right)}{n}}}} }{\sqrt{\left[\sum {x^2}_i-{\displaystyle \frac{{\left(\sum x_i\right)}^2}{n}}\right]\left[\sum {y^2}_i-\frac{{\left(\sum y_i\right)}^2}{n}\right]}} \\ =\frac{32476-(178*1593)/10}{\sqrt{\left[ 3522-{\displaystyle \frac{{\left(178\right)}^2}{10}}\right]\left[302027-{\displaystyle \frac{{\left(1593\right)}^2}{10}}\right]}} \\ = 0.9975 \end{gathered}$$

Therefore the linear correlation coefficient value is 0.9975. 

Step 3. Solution b)

From the fraction (a) of the coefficient of connection, 0.9975

Here, it can be noted that the coefficient of integration is positive and close to 1. Therefore, there is a strong positive relationship between age and the length of the crown-rump in the fetus.

Step 4. Solution c

From part (b) there is a strong positive relationship between the age and the length of the crown-rump

fetuses.

Using Excel, follow the steps below to get the scatter data structure provided

1) Enter the data in the excel sheet.

2) Select two columns, insert, disassemble the structure.

The graph is as shown below.

From the above scattering structure, it can be seen that as the years go by the relative length of the rump increases. Therefore, there is a strong positive relationship.

Therefore, the interpretation of the correlation coefficient value obtained in subsection (a) and the definition of the dispersion structure of both are similar.

Step 5. Solution

The square of r =0.9975

$$\begin{gathered} r^2={(0.9975)}^2=0.995 \\ The total sum of squares is \\ SST=\sum _{}{y}^{i^2}-\frac{(\sum _{}{\left(y^{i2}\right)}^2}{n} \\ =30207-\frac{{\left(1593\right)}^2}{10} \\ =48262.1 \\ {r}^2=\frac{SSR}{SST} \\ =\frac{48018.51}{48262.1} \\ =0.995 \end{gathered}$$