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:
![\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}](https://tex.z-dn.net/?f=%5Cbegin%7Bcenter%7D%0A%5Cbegin%7Btabular%7D%0A%7B%7Cc%7Cc%7Cc%7Cc%7Cc%7C%7D%0Ax%20%26%20y%20%26%20x%5E2%20%26%20y%5E2%20%26%20xy%20%5C%5C%20%5B1ex%5D%0A1%20%26%201%20%26%201%20%26%201%20%26%201%5C%5C%0A4%20%26%208%20%26%2016%20%26%2064%20%26%2032%5C%5C%0A7%20%26%2011%20%26%2049%20%26%20121%20%26%2077%5C%5C%0A10%20%26%2012%20%26%20100%20%26%20144%20%26%20120%5C%5C%20%5B1ex%5D%0A%5CSigma%20x%3D22%20%26%20%5CSigma%20y%3D32%20%26%20%5CSigma%20x%5E2%3D166%20%26%20%5CSigma%20y%5E2%3D330%20%26%20%5CSigma%20xy%3D230%0A%5Cend%7Btabular%7D%0A%5Cend%7Bcenter%7D)
Recall that the correlation coefitient is given by the equation:
[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
and
is given by
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
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, <span>the data in the table represent both correlation and causation.</span>
