Part b: Analysis of the Plot

The graph shows the relationship between $\log y$ and $\log x$, with the red line representing the line of best fit. This linear relationship suggests that the data follows a power law, consistent with the equation $y = ax^n$, as stated in the question.

Part c: Estimating $a$ and $n$

The equation of the line of best fit from the $\log$-$\log$ plot is of the form: $\log y = \log a + n \log x$ From this, we can extract the slope $n$ and intercept $\log a$ by fitting the line.

I will now calculate the slope $n$ and intercept $\log a$ from the linear regression model.From the linear regression model:

• The slope $n \approx 1.40$
• The intercept $\log a \approx 0.54$, which means $a \approx 3.46$

Thus, the estimated values are:

• $a \approx 3.5$
• $n \approx 1.4$

These values correspond to the equation $y = ax^n$, with $y = 3.5x^{1.4}$.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the line change if the data followed a different power law, say $y = ax^{2}$?
2. Why is it useful to take the logarithm of both $x$ and $y$ in this type of analysis?
3. What does the value of $n$ represent in terms of the relationship between $x$ and $y$?
4. How can you confirm the accuracy of the estimated $a$ and $n$ values using the original data?
5. How would you interpret negative values of $n$ in the equation $y = ax^n$?

Tip: When dealing with power laws, plotting logarithms often transforms non-linear relationships into straight lines, making it easier to estimate parameters like $a$ and $n$.