Question

Please No graphs. I need help solving the problem with the computations

9. The following table shows the peak wind gust (X) and the number of damaged homes (Y) (in thousands) in seven cities in a typhoon's path. Show that a function of the form y = ax provides a good model for the data. City A А тв B с D E F G X 1 0.98 0.74 1.12 1.34 0.87 0. 65 1 39 Y 25.000 0.950 200.000 150.000 0.940 0 .090 260.000

Answer 1

I am going to provide you a method to estimate a and b of the given model.

Given model is y = ax^{​​​​​b}​ .

Taking logarithm base e on both sides we get

$log_e y = log_e(ax^b)\\log_ey =log_ea + log_ex^b\\ log_ey =log_ea + b\cdot log_ex \\ z= c + b\cdot u$

Where z=log_ey, c = log_{​​​​e}​​​a, u = log_{​​​​e} x

Calculate all the values of corresponding z and u.

Now this gives an equation with two unknowns and 7 data points (z,u).

We can solve this equation with the given data, to find the value of c and b.

estimate of c is given by mean of the z values. Then by taking antilog of c we get a, i.e. a = e^{​​​​​​c}

The estimate of b is given by $\frac{\sum_{i=1}^{7}z_i \cdot u_i}{\sum_{i=1}^{7} (u_i)^2}$ .

To show that it is a good fit, compute the MSE, i.e. $\frac{\sum_{i=1}^{7} (z_i - \hat{z_i})^2}{7}$

Where the $\hat{z_i}$ are obtained from the estimated equation putting in the values of known u values.

Thank you. Hope this helps you with the computation.