PART A:
Given the table below showing the radius y, in inches, created by growing algae in x days.
Time (x) (days): 1 4 7 10
Radius (y) (inches): 1 8 11 12
We can find the find the correlation coeficient of the data using the table below:
[tex]\begin{center}
\begin{tabular}
{|c|c|c|c|c|}
x & y & x^2 & y^2 & xy \\ [1ex]
1 & 1 & 1 & 1 & 1\\
4 & 8 & 16 & 64 & 32\\
7 & 11 & 49 & 121 & 77\\
10 & 12 & 100 & 144 & 120\\ [1ex]
\Sigma x=22 & \Sigma y=32 & \Sigma x^2=166 & \Sigma y^2=330 & \Sigma xy=230
\end{tabular}
\end{center}[/tex]
Recall that the correlation coefitient is given by the equation:
[tex]r= \frac{n(\Sigma xy)-(\Sigma x)(\Sigma y)}{ \sqrt{(n\Sigma x^2-(\Sigma x)^2)(n\Sigma y^2-(\Sigma y)^2)} } \\ \\ = \frac{4(230)-(22)(32)}{ \sqrt{(4(166)-(22)^2)(4(330)-(32)^2)} } = \frac{920-704}{ \sqrt{(664-484)(1,320-1,024)}} \\ \\ = \frac{216}{ \sqrt{180(296)}} = \frac{216}{ \sqrt{53,280} } = \frac{216}{230.8} =0.94[/tex]
[Notice: even without using the formular to find the value of the correlation coeficient, it can be seen that the value of y strictly increases as the valu of x increases. This means that the value of the correlation coeficient is closer to +1 and from the options the value that is closest to +1 is 0.94]
From the value of the correlation coeffeicient, it can be deduced that the radius of the algae has a strong positive relationship with the time.
Recall the for the value of the correlation coeficient closer to +1, the relationship is strong positive, for the value closer to -1, the value is strong negative and for the values closer to zero, either way of zero is a weak positive if it is positive and weak negative if it is negative.
PART B:
Recall that the slope of a straight line passing through two points
[tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]
is given by
[tex]m= \frac{y_2-y_1}{x_2-x_1} [/tex]
Thus the slope of the graph of radius versus time between 4 and 7 days, [i.e. the line passes through points (4, 8) and (7, 11)] is given by
[tex]m= \frac{11-8}{7-4}= \frac{3}{3} =1[/tex]
The value of the slope means that the radius of the algea grows by 1 inch evry day between day 4 and day 7.
PART C:
We can say that the data above represent both correlation and causationg.
Recall that correlation expresses the relationship between two variables while causation expresses that an event is as a result of another event.
From the information above, we have seen that there is a relationship (correlation) between the passing of days and the growth in the radius of the algae.
Also we can conclude that the growth in the radius of algae is a function of the passing of days, i.e. the growth in the radius of algae is as a result of the passing of days.
Therefore, the data in the table represent both correlation and causation.
