Part A: Interpretation of the coefficient \(-0.02396\)
The number \(-0.02396\) in the equation \( \log y = -0.02396 x + 4.8153 \) indicates that there is a negative linear relationship between \(x\) and \(\log y\). Specifically, as \(x\) increases, \(\log y\) decreases at a rate of approximately \(0.02396\) per unit increase in \(x\).
In the context of the original data set, this means that as the value of \(x\) increases, the value of \(y\) is decreasing. Since the relationship is in logarithmic form, this suggests that \(y\) is decreasing exponentially with respect to \(x\), meaning for each unit increase in \(x\), the value of \(y\) decreases by a factor related to the exponential function of the logarithmic change.
Part B: Finding the value of \(y\) when \(x = 0\)
To find the value of \(y\) when \(x = 0\), we can substitute \(x = 0\) into the equation:
\[ \log y = -0.02396(0) + 4.8153 \]
This simplifies to:
\[ \log y = 4.8153 \]
To solve for \(y\), we need to exponentiate both sides to eliminate the logarithm. Assuming the base of the logarithm is 10 (common logarithm):
\[ y = 10^{4.8153} \]
Now, we calculate \(10^{4.8153}\):
Calculating this using a calculator or logarithm tables:
\[ y \approx 65061.33 \]
Thus, when \(x = 0\), the value of \(y\) is approximately \(65061.33\).