Q34.7-2ITD. - Campbell Biology

Question

As described in the Scientific Skills Exercise for Chapter 32, the formula for a correlation coefficient is  $r=\frac{\frac{1}{n-1}\sum (x_i-\overline{x)}(y_i-\overline{y})}{s_x{s}_y}$

 

Fill in the column in the data table for the product ( $x_i - \overline{x}$)×( $y_i - \overline{y}$). Use these values and the standard deviations calculated in question 1 to calculate the correlation coefficient r between the brain volume of hominin species (y) and the ages of those species (x).

 

Step-by-Step Solution

Answer

 

Hominin Species	Mean age (millions of years; xi)	$x_i - \overline{x}$	$(x_i - \overline{x)}{ }^2$	Mean Brain Volume  (cm2; yi)	$y_i - \overline{y}$	$(y_i - \overline{y)}{ }^2$	($x_i - \overline{x}$ ) × ( $y_i - \overline{y}$)
Ardipithecus ramidus	-4.4	-2.77	7.67	325	-556.25	309414.06	1540.81
Australopithecus afarensis	-3.4	-1.77	3.13	375	-506.25	256289.06	896.06
Homo habilis	-1.9	-0.27	0.072	550	-331.25	109725.56	89.43
Homo ergaster	-1.6	0.03	0.0009	850	-31.25	976.56	-0.937
Homo erectus	-1.2	0.43	0.184	1,000	118.75	14101.56	51.06
Homo heidelbergensis	-0.5	1.13	1.276	1,200	318.75	101601.56	360.187
Homo neanderthalensis	-0.1	1.53	2.340	1,400	518.75	269101.56	793.68
Homo sapiens	0.0	1.63	2.65	1,350	468.75	219726.56	764.06
	$$\begin{gathered} \overline{x}=-1.63 \\ {s}_x=1.57 \end{gathered}$$			$$\begin{gathered} \overline{y}= 881.25 \\ {s}_y=427.77 \end{gathered}$$			$\sum =1917.92$

 

The correlation coefficient (r) between the ages of the hominin species (x) and the brain volume of those species (y) is 0.407.

 

Step 1: Brain volume and hominin species

The average brain volume is thought to have altered over time and between different hominin species. Therefore, the correlation coefficient is calculated to deduce the relation between hominin species and their brain volume.

Step 2: Calculation of ( x i   -   x ¯ )×( y i   -   y ¯ )

Each ($x_i - \overline{x}$ ) value of one species is multiplied with the ($y_i - \overline{y}$ ) value of that species to obtain ( $x_i - \overline{x}$)×( $y_i - \overline{y}$). The values are then filled in the table.

Step 3: Calculation of correlation coefficient (r)

The correlation coefficient (r) is calculated using the following formula:

 

 $r=\frac{\frac{1}{n-1}\sum (x_i-\overline{x)}(y_i-\overline{y})}{s_x{s}_y}$

 

Substituting the values from the table in the equation,

 

 $\begin{array}{l}r=\frac{\frac{1}{8-1}\sum [1540.81+896.06+89.43+(-0.937)+51.06+360.187+793.68+764.06)]}{1.57\times 427.77}\\ r=\frac{\frac{1}{7}\times 1917.92}{1.57\times 427.77}\\ r=0.407\end{array}$