Comparison of Two-Powder Size Distributions

For quality control purposes, ceramists are often required to determine if the particle size distribution of one batch of powder is the same or different from another. This determination is difficult when the two batches of powder have similar mean sizes. A statistical method [19] must be used to make this distinction. To determine if two particle size distributions are the same or different. Student s t-test is used by applying the null hypothesis to the two sample means. For normal distributions the f-statistic is defined as tl rati of the difference between the two sample arithmetic means (A and A2) to the standard deviation of the difference in the means [20]  

Using the definitions of the normal size distributions, the t-statistic can be formulated as follows [20]  

When two samples are veiy similar, t approaches zero when they are different, t approaches infinity. The value of f is used to calculate the P value using Student s f-test tables, given in the appendix of this book. The P value is tte probability that the two distribution means are the same that is, Aj = Ag. When the P value is greater than a critical accepted value (typically 5% [21] or the experimental error due to both sampling and size determination if it is lai ger) then the null hypothesis (Ho Aj = A2) is accepted (i.e., the two populations are considered to be the same). Ceramic powder size distributions are often represented by log-normal distributions and not by normal distributions. For this reason the t statistic must be augmented for use with lognormal distributions. Equation (2.59) can be modified for this purpose to 

Using the previous equations, we find that the number of degrees of fieedom is 8 and the t value is 0.49. Using the standard t-test, a P value of 64% is obtained. This means that the probability that the two populations are the same is 64%. Because the P value is greater than 5%, we can say that these two powders are essentially the same. For most applications, powder 1 may be substituted for powder 2. For more exacting applications, a higher P value may be necessary for successfiil powder substitution. As a result, for each ceramic powder application a critical P value for powder substitution should be determined. 

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