Experiments conducted at all combinations of the levels of two or more factors are called factorial experiments. Factorial experiments have been shown to be more efficient in exploring the effects of external factors on a response variable than non-factorial arrangements of factor levels. This paper presents a methodology for the analysis of test results obtained at all combinations of the levels of two two-level factors. The response variable is assumed to follow the two parameter Weibull distribution with a shape parameter that, although unknown, does not vary with the factor levels. The scale parameter on the other hand may vary in various ways with the levels of the factors. These assumptions well reflect the behavior of the fracture strength of glassy polymers as it is influenced by factors such as water sorption and silanation of filler. The purpose of the analysis is (1) to compute interval estimates of the common shape parameter and (2) to assess whether either factor singly or in combination with the other affects the Weibull scale parameter. The procedure is illustrated with an example using simulated data for which only one of the factors had a real effect. A further example is given using shear strength measurements made in a 2(2) factorial experiment conducted on a glassy polymer material used in dental restorations.