PART A:
Given the table below showing the surface area y, in square inches, of a shrinking puddle in x hours.
Time (x) (hours): 1 4 7 10
Surface area (y) (square inches): 100 85 70 55
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 & 100 & 1 & 10,000 & 100\\
4 & 85 & 16 & 7,225 & 340\\
7 & 70 & 49 & 4,900 & 490\\
10 & 55 & 100 & 3,025 & 550\\ [1ex]
\Sigma x=22 & \Sigma y=310 & \Sigma x^2=166 & \Sigma y^2=25,150 & \Sigma xy=1,480
\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(1,480)-(22)(310)}{ \sqrt{(4(166)-(22)^2)(4(25,150)-(310)^2)} } \\ \\ = \frac{5,920-6,820}{ \sqrt{(664-484)(100,600-96,100)}} = \frac{-900}{ \sqrt{180(4,500)}} \\ \\ = \frac{-900}{ \sqrt{810,000} } = \frac{-900}{900} =-1[/tex]
From the value of the correlation
coeffeicient, it can be deduced that the radius of the algae has a
strong negative 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 surface area versus time between 1 and 4 hours,
[i.e. the line passes through points (1, 100) and (4, 85)] is given by
[tex]m= \frac{85-100}{4-1}= \frac{-15}{3} =-5[/tex]
The value of the slope means that the surface area y, in square inches, of a shrinking puddle, shrinks by 5 square inches every hour between the first hour and the fourth hour.
PART C:
We can say that the data above represent both correlation and causation.
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 hours and the shrink in the surface area
of the puddle.
Also we can conclude that the shrink in the surface area
of the puddle is a function of the passing of hourss, i.e. the shrink in the surface area
of the puddle is as a result of the passing of hours.
Therefore, the data in the table represent both correlation and causation.
