Q34.7-4ITD. - Campbell Biology

Question

(a) Use your calculated value of r to calculate the slope (m) and the y-intercept (b) of a regression line for this data set. (b) Graph the regression line for the mean brain volume of hominin species versus the mean age of the species. Be careful to select and label your axes correctly. (c) Plot the data from the table on the same graph that shows the regression line. Does the regression line appear to provide a reasonable fit to the data?

Step-by-Step Solution

Answer

(a) $m=110.89$ and  $b=1062.00$

(b) 

(c) 

Step 1: Calculation of slope (m) and y-intercept ( b )

The slope (m) is calculated by:  $m=r\frac{s_y}{s_x}$, where r is the coefficient correlation and s_x and s_y are the standard deviations.

Given:  $r=0.407$,  $s_x=1.57$ and  $s_y=427.77$.

 

Substituting the values in the equation,

 $\begin{array}{l}m=0.407\frac{427.77}{1.57}\\ m=110.89\end{array}$

 

 

The y-intercept (b) is calculated as  $b=\overline{y}-m\overline{x}$.

Given,  $\overline{x}=-1.63$ and  $\overline{y}= 881.25$

Substituting values in the equation,

 $\begin{array}{l}b=881.25-[110.89\times (-1.63\left)\right]\\ b=881.25-(-180.75)\\ b=1062.00\end{array}$

Step 2: Graphing the regression line

The equation for a straight line between x and y variables is , where m is the slope of the line and b is the y-intercept.

The following data is used to graph the regression line.

Hominin Species	Mean age (millions of years; xi)	Mean Brain Volume  (cm2; yi)	$y=mx+b$
Ardipithecus ramidus	-4.4	325	574.08
Australopithecusafarensis	-3.4	375	684.94
Homo habilis	-1.9	550	851.31
Homo ergaster	-1.6	850	884.6
Homo erectus	-1.2	1,000	928.93
Homo heidelbergensis	-0.5	1,200	1006.6
Homo neanderthalensis	-0.1	1,400	1050.92
Homo sapiens	0.0	1,350	1062

 

The mean age of the species is an independent variable, so it would be plotted on the x-axis, whereas the mean brain volume is a dependent variable and would be plotted on the y-axis.

Step 3: Regression line fits the data

The straight line suggests the linear relationship between the two variables analyzed: mean brain volume and the mean age of the hominin species. Thus, the regression line best fits the data because the value of y increases as the value of x increases.