A researcher in the petroleum industry is interested in the sizes of wax crystals produced when wax dissolved in a supercritical fluid is sprayed through a capillary nozzle. In the first experiment, a pre-expansion temperature of 80°C is employed and the diameters of n = 35 crystals are measured with an electron microscope. A sample mean of  = 22.73 pm and a sample standard deviation of sx = 5.20 pm are obtained. In the second experiment, a pre-expansion temperature of 150 C is employed and the diameters of m = 35 crystals are measured, which have a sample mean of  = 12.66 μm and a sample standard deviation of sy = 3.06 μm. The researcher decides that it is not appropriate to assume that the variances of the crystal diameters are the same under both sets of experimental conditions.
(a) Write down an expression for the p-value of the statement that the average crystal size does not depend upon the pre-expansion temperature. Do you think that this is a plausible statement?
(b) Construct a 99% two-sided confidence interval for the difference between the average crystal diameters at the two pre-expansion temperatures.
(c) If a 99% two-sided confidence interval for the difference between the average crystal diameters at the two pre-expansion temperatures is required with a length no larger than L0 = 4.0 μm, how much additional sampling would you recommend?