the table below shows 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
part a: what is the most likely value of the correlation coefficient of the data in the table? based on the correlation coefficient, describe the relationship between time and surface area of the puddle. [choose the value of the correlation coefficient from −1, −0.99, −0.5, −0.02.] (4 points)
part b: what is the value of the slope of the graph of surface area versus time between 1 and 4 hours, and what does the slope represent? (3 points)
part c: does the data in the table represent correlation or causation? explain your answer. (3 points)

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:


Recall that the correlation coefitient is given by the equation:


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
and
is given by


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


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.