Q34.7-1ITD. - Campbell Biology

Question

Calculate the means ( $\overline{x}$and $\overline{y}$ ) from the n= 8 data points in the table. Next, fill in the ( $x_i - \overline{x}$) and ($y_i - \overline{y}$ ) columns in the data table, and use those results to calculate the standard deviations s_x and s_y.

 

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
Australopithecus afarensis	-3.4	-1.77	3.13	375	-506.25	256289.06
Homo habilis	-1.9	-0.27	0.072	550	-331.25	109725.56
Homo ergaster	-1.6	0.03	0.0009	850	-31.25	976.56
Homo erectus	-1.2	0.43	0.184	1,000	118.75	14101.56
Homo heidelbergensis	-0.5	1.13	1.276	1,200	318.75	101601.56
Homo neanderthalensis	-0.1	1.53	2.340	1,400	518.75	269101.56
Homo sapiens	0.0	1.63	2.65	1,350	468.75	219726.56
	$\overline{x}=-1.63$			$\overline{y}= 881.25$

 

The standard deviations are $s_x=1.57$ and  $s_y=427.77$.

 

Step 1: Calculation of mean x ¯   and y ¯

The mean age is calculated by the formula: $\overline{x} = \frac{\sum x_i}{n}$

 $\begin{array}{l}\overline{x}=\sum \frac{\left[\right(-4.4)+(-3.4)+(-1.9)+(-1.6)+(-1.2)+(-0.5)+(0.1)+0.0]}{8}\\ =-1.63\end{array}$

 

The mean brain volume is calculated by the formula:  $\overline{y} = \frac{\sum y_i}{n}$

 $\begin{array}{l}\overline{y} = \frac{\sum (325+375+550+850+1000+1200+1400+1350)}{8}\\ = 881.25\end{array}$

 

 

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

( $x_i - \overline{x}$) and ( $y_i - \overline{y}$) are calculated for each data point by subtracting each ith value from the mean value.

 

The values are then squared to obtain $(x_i - \overline{x)}{ }^2$ and  $(y_i - \overline{y)}{ }^2$.

Step 3: Calculation of the standard deviations s x and s y

The standard deviation (s_x) is calculated by the formula:

 

 $s_x =\sqrt{\frac{1}{n-1}}\sum (x_i- \overline{x}{)}^2$

 

Substituting the values from the table into the equation,

 $\begin{array}{l}{s}_x =\sqrt{\frac{1}{8-1}}\sum (7.67+3.13+0.072+0.0009+0.184+1.276+2.34+2.65)\\ = \sqrt{\frac{17.32}{7}}\\ =1.57\end{array}$

 

The standard deviation (s_y) is calculated by the formula:

 $s_y =\sqrt{\frac{1}{n-1}}\sum (y_i- \overline{y}{)}^2$

 

 

Substituting the values from the table into the equation,

 $\begin{array}{l}{s}_y =\sqrt{\frac{1}{8-1}}\sum (309414.06+256289.06+109725.56+976.56+14101.56+101601.56+269101.56+219726.56)\\ = \sqrt{\frac{1280936.48}{7}}\\ =427.77\end{array}$