Chapter 4, Problem 4.31P

For Problems 4.30 through 4.36. itwould be helpful to use a least-squares software package, such as Polynomial Jit, Matlab, or spreadsheet software.

1. The following relations exist between material flexural strength and grain size:

Flexural strength 65 63 67 64 68 62 70 66 68 67 69 71 jc (MPa)
Grain size,
68 66 68 65 69 66 68 65 71 67 68 70
y (nm)
Plot the linear relationship and estimate the standard error of the fit.

Experimental measurements are taken from a physical system that can be modeled as y = bx!". The experimental data is (1.020, 3.270), (1.990, 12.35), (3.000, 28.03), (4.010. 50.04), (4.990, 77.47). Find the coefficients to fit the model using the method of least squares and calculate the standard error of the fit.

Experimental measurements are taken from a physical system that can be modeled as y = tnx + b. The experimental data is (1.00, 2.10), (2.00, 2.91), (3.00, 3.89), (4.00, 5.12), (5.00, 6.09). Find the coefficients to fit the model using the method of least squares, calculating the standard error of the fit and the 95% confidence band about the curve fit (Scheffe band).

In an engineering lab exercise, a student makes 10 independent measurements of a voltage drop across a circuit board resistor:

0.86,	0.83,	0.82,	0.84,	0.95,
0.87,	0.88,	0.89,	0.83,	0.85

Could any value be rejected using Chauvenet’s criterion?

Find the first-order line (y = no + <*!•*) that fits the four data pairs (1, 12), (2, 13), (3, 18), (4, 19). Estimate the standard error of the fit. How many degrees of freedom are there in the curve fit? Show the uncertainty bands about the fit.

A student lab wants to determine the rate of radioactive decay of an isotope by measuring the rate of decay, R, once per hour over a three-hour period. The rate of decay is modeled as R = Roe'¹^ where t is the time constant of decay. The measurements are (0, 13.8), (1, 7.9), (2, 6.1), (3, 2.9). Apply least- squares regression to the data to obtain an estimate for the constant of decay t.

A fan performance test yields the following data pairs for (Q, h): (2000, 5.56), (6000, 5.87), (10000, 4.95), (14000, 4.95), (18000, 3.52), (22000, 1.08). Q is flow rate in m³/s and h is static head in cm-H₂0. Find the lowest-dcgree polynomial that best fits the data as h = f(Q). Note: Physically, a second-order polynomial is a reasonable fit. Explain your choice.