It is possible for an observation to be correlated to its cause, however,
causation is not implied by the existence of correlation
The linear equation that models the data is; [tex]\underline {\hat y \approx 7.08575 + 0.2571\cdot \overline x}[/tex]
The correct option to whether the presence of a causal relationship is
option A. There is a positive correlation and no causal relationship
Reasons:
The given data is presented as follows;
[tex]\displaystyle {\begin{tabular}{|c|c|c|c|c|c|c|} Visits&3&4&5&6&7&8\\Books&12&5&6&8&9&11\end{array}\right] }[/tex]
The least squares regression formula is presented as follows;
[tex]\hat y = a + b\cdot \overline x[/tex]
Where;
[tex]b = \dfrac{n\sum x\cdot y - \left (\sum x \right ) \cdot \left (\sum y \right )}{n\sum x^{2} - \left (\sum x \right )^{2}}[/tex]
Using MS Excel, we have;
∑x·y = 285
∑x = 33
∑y = 51
[tex]\overline {y }[/tex] = 8.5
[tex]\overline {x }[/tex] = 5.5
∑x² = 199
∑x·∑y = 33×51
(∑x)² = 1,089
(∑y)² = 51
Therefore;
a = 8.5 - 0.2575 × 5.5 = 7.08575
The linear equation that model's Nadeem's equation is therefore;
The correlation coefficient, r, is given as follows;
[tex]r = \dfrac{n \cdot \sum X \cdot Y - \left (\sum X \right )\left \cdot (\sum Y \right )}{\sqrt{n \cdot \sum X^{2} - \left (\sum X \right )^{2}\times n \cdot \sum Y^{2} - \left (\sum Y \right )^{2}}}[/tex]
Therefore;
[tex]r = \dfrac{6 \times 285 - 33 \times 51 }{\sqrt{6 \times 199 - 1089\times 6 \times 471 - 51^{2}}} \approx 0.17566[/tex]
Therefore, the there is a positive correlation between the visits and
number of books Nadeem checks out
However, from the weak correlation, given by the low value, of the
correlation coefficient there is no causal relationship between the
Nadeem's visit to the library and the number of books he checks out
The correct option is therefore; There is a positive correlation and no causal relationship
Learn more here:
https://brainly.com/question/24768947
