Question

researchers wondered whether the size of a persons brain was related to the individuals mental capacity. they selected a sample of 5 females and 5 males and measured their mri image pixel counts and iq scores. the data is reported to the right. complete parts (a) through (d) below. click the icon to view the critical values table. (b) compute the linear correlation coefficient between mri count and iq. are mri count and iq linearly related? select the correct choice below and if necessary, fill in the answer box to complete your choice. (round to three decimal places as needed.) a. no, mri count and iq are not linearly related since the linear correlation coefficient is b. yes, mri count and iq are linearly related since the linear correlation coefficient is

Explanation:

Step1: Recall correlation coefficient formula

The formula for the linear correlation coefficient $r$ is $r=\frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$. Let $x$ be the MRI - count and $y$ be the IQ score. First, we need to calculate the sums: $\sum x$, $\sum y$, $\sum x^{2}$, $\sum y^{2}$, and $\sum xy$ for all 10 data - points (5 females and 5 males).

Step2: Calculate sums for females

For females:
Let $x_1$ be the MRI - count and $y_1$ be the IQ score of females.
$n_1 = 5$
$\sum_{i = 1}^{5}x_{1i}=852244 + 866662+856472+951545+991305=4418228$
$\sum_{i = 1}^{5}y_{1i}=132 + 130+140+137+138=677$
$\sum_{i = 1}^{5}x_{1i}^{2}=852244^{2}+866662^{2}+856472^{2}+951545^{2}+991305^{2}$
$=726314763536+751003986244+733645779584+905448197025+982686493025$
$=4109100219414$
$\sum_{i = 1}^{5}y_{1i}^{2}=132^{2}+130^{2}+140^{2}+137^{2}+138^{2}$
$=17424+16900+19600+18769+19044$
$=91737$
$\sum_{i = 1}^{5}x_{1i}y_{1i}=852244\times132 + 866662\times130+856472\times140+951545\times137+991305\times138$
$=112596208+112666060+119906080+130361665+136890090$
$=612420103$

Step3: Calculate sums for males

For males:
Let $x_2$ be the MRI - count and $y_2$ be the IQ score of males.
$n_2 = 5$
$\sum_{i = 1}^{5}x_{2i}=924059+955003+1079550+949395+1001121=4909128$
$\sum_{i = 1}^{5}y_{2i}=135 + 139+141+140+140=695$
$\sum_{i = 1}^{5}x_{2i}^{2}=924059^{2}+955003^{2}+1079550^{2}+949395^{2}+1001121^{2}$
$=853885965481+912020300009+1165329002500+901241865025+1002243466561$
$=4834720599576$
$\sum_{i = 1}^{5}y_{2i}^{2}=135^{2}+139^{2}+141^{2}+140^{2}+140^{2}$
$=18225+19321+19881+19600+19600$
$=96627$
$\sum_{i = 1}^{5}x_{2i}y_{2i}=924059\times135+955003\times139+1079550\times141+949395\times140+1001121\times140$
$=124747965+132745417+152216550+132915300+140156940$
$=682782172$

Step4: Calculate overall sums

$n=n_1 + n_2=10$
$\sum x=\sum_{i = 1}^{5}x_{1i}+\sum_{i = 1}^{5}x_{2i}=4418228+4909128=9327356$
$\sum y=\sum_{i = 1}^{5}y_{1i}+\sum_{i = 1}^{5}y_{2i}=677+695=1372$
$\sum x^{2}=\sum_{i = 1}^{5}x_{1i}^{2}+\sum_{i = 1}^{5}x_{2i}^{2}=4109100219414+4834720599576=8943820818990$
$\sum y^{2}=\sum_{i = 1}^{5}y_{1i}^{2}+\sum_{i = 1}^{5}y_{2i}^{2}=91737+96627=188364$
$\sum xy=\sum_{i = 1}^{5}x_{1i}y_{1i}+\sum_{i = 1}^{5}x_{2i}y_{2i}=612420103+682782172=1295202275$

Step5: Compute the correlation coefficient

$n\sum xy-(\sum x)(\sum y)=10\times1295202275 - 9327356\times1372$
$=12952022750-12797122432$
$=154900318$
$n\sum x^{2}-(\sum x)^{2}=10\times8943820818990-9327356^{2}$
$=89438208189900 - 86909749452736$
$=2528458737164$
$n\sum y^{2}-(\sum y)^{2}=10\times188364-1372^{2}$
$=1883640 - 1882384$
$=1256$
$r=\frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$
$r=\frac{154900318}{\sqrt{2528458737164\times1256}}$
$r=\frac{154900318}{\sqrt{3175644173877984}}$
$r=\frac{154900318}{5635285.977}$
$r\approx0.027$

Since $|r|\approx0.027$ is very close to 0, we conclude that MRI count and IQ are not linearly related.

Answer:

A. No, MRI count and IQ are not linearly related since the linear correlation coefficient is $0.027$