Problem 6

	In this problem we explore the use of the bootstrap 
technique to determine the error in a regression problem.  
Consider the attached set of measurements of the heat 
transfer coefficient for turbulent flow through a pipe.  The 
dimensionless heat transfer coefficient is called the Nusselt 
number Nu, and it is expected to obey a power law 
relationship with the Reynolds number Re (a dimensionless 
measure of the fluid velocity) and the Prandtl number Pr (the 
dimensionless ratio of the rate of energy diffusion to fluid 
viscosity).  Thus we expect:

Nu = c * Re^a * Pr^b

where a, b, and c are all constants.

	Calculate the constants and the matrix of covariance of 
the constants using the linear regression formulae described 
in class.  Next, implement the bootstrap technique to estimate 
the matrix of covariance as well, and compare your results 
with that produced by the exact formulae.  You may need to 
use a number of samples which is higher than the number of 
actual observations!

The data is as follows:

       Re           Pr           Nu
   2.0000e+04   8.1600e+00   1.4318e+02
   2.0500e+04   8.3900e+00   1.6686e+02
   2.1000e+04   8.5600e+00   1.5335e+02
   2.1500e+04   7.1900e+00   1.5671e+02
   2.2000e+04   8.0500e+00   1.5221e+02
   2.2500e+04   7.6900e+00   1.5299e+02
   2.3000e+04   7.0500e+00   1.6264e+02
   2.3500e+04   7.4100e+00   1.7429e+02
   2.4000e+04   7.2400e+00   1.5987e+02
   2.4500e+04   7.0100e+00   1.5884e+02
   2.5000e+04   8.3700e+00   1.6875e+02
   2.5500e+04   7.1900e+00   1.7029e+02
   2.6000e+04   7.9500e+00   1.8549e+02
   2.6500e+04   8.3400e+00   1.8124e+02
   2.7000e+04   7.2400e+00   1.6957e+02
   2.7500e+04   7.4500e+00   1.8350e+02
   2.8000e+04   8.2800e+00   1.7738e+02
   2.8500e+04   8.5700e+00   2.0250e+02
   2.9000e+04   8.9400e+00   1.9098e+02
   2.9500e+04   7.8600e+00   1.8808e+02
   3.0000e+04   7.2600e+00   2.0125e+02
   3.0500e+04   7.2000e+00   1.9637e+02
   3.1000e+04   7.5800e+00   2.0122e+02
   3.1500e+04   7.0500e+00   1.9261e+02
   3.2000e+04   8.6600e+00   2.0215e+02
   3.2500e+04   8.8100e+00   2.1756e+02
   3.3000e+04   7.0200e+00   2.1018e+02
   3.3500e+04   8.1000e+00   2.2368e+02
   3.4000e+04   7.2300e+00   2.0038e+02
   3.4500e+04   8.2500e+00   2.1413e+02
   3.5000e+04   8.0700e+00   2.2018e+02
   3.5500e+04   8.9800e+00   2.3234e+02
   3.6000e+04   8.6200e+00   2.4733e+02
   3.6500e+04   7.9700e+00   2.3147e+02
   3.7000e+04   7.6100e+00   2.3418e+02
   3.7500e+04   7.6000e+00   2.3480e+02
   3.8000e+04   8.3700e+00   2.5935e+02
   3.8500e+04   8.9000e+00   2.4134e+02
   3.9000e+04   7.5100e+00   2.2648e+02
   3.9500e+04   7.7800e+00   2.4771e+02
   4.0000e+04   7.7700e+00   2.4293e+02