Power Analysis for Single-Source Variation

A.1 Population of Data

Consider an original sample size with more than 100 data points for each material property to be examined, with preference for one year of data. All samples (including composite samples) are to be collected at the rate mandated by ASTM C311. For reduction in sampling rate, statistical significance of this reduction needs to be justified by using the sample mean and sample standard deviation of the 100 samples (sample set) for analysis.

A.2. Z-Test (two-sided)

The statistical tool, which is based on a two-sided hypothesis test on the mean, is used to examine the expected difference from the mean of the reduced sample size. The methodology is as follows:

1. A Normal population is considered, with variance known.
2. Using the z-score table, or statistical software, determine the expected effect on the difference from the mean of the reduced sample size (n) at a confidence interval of 95%, limiting Type I (α) and Type II (β) errors to roughly 5%.

$n={\left(z_{1-\text{α}/2}+z_{1-\text{β}}\right)}^2{\left(\frac{\text{σ}}{\text{δ}}\right)}^2\left(\text{two}-\text{sided}\text{test}\right)$

$\text{δ}=\left(\text{μ}-{\text{μ}}_o\right)=\sqrt{\frac{{\left(z_{1-\frac{\text{α}}{2}}+z_{1-\text{β}}\right)}^2{\sigma }^2}{n}}$

3. The z-statistics lower and upper bound can be compared with the sample mean of the data set to examine whether variation due to reduced sampling rate is within ASTM Standard limit.

Selection of a reduced sample size shall be determined from the professional judgement of the engineer or testing manager at the plant based on the current trend of the tested data. In the case when no information about the behavior of

a population is given or known, which may be the case for unconventional or reclaimed ashes obtained from landfills or impoundments, the following reduced sample size/sampling rate may be conservatively estimated as follows:

1. No information given about the behavior of the population is known, except the population size N.
2. Determine a reduced sample size/sampling rate (n) at a precision e of 95%, limiting error α to roughly 5%.

$n=\frac{N}{\left(1+N{e}^2\right)}\left(\text{Solvin}\text{Equation}\right)$

3. Proceed to A.2. Z-test to determine the expected effect on the difference of the mean of the reduced sample size.

CSA A3004-A1 Statistical Analysis for Single-Source Variation

B.1. Population of Data

Consider a population with a minimum of 30 samples, with preference for one full year of data if possible (note that any substantial change in the process or change in a set point could generate a new population). The same data population used to certify products should be used for evaluating the adequacy of testing frequency. For less frequent testing, 30 samples (sample set) could represent a period exceeding one year provided there is no significant change in the process.

B.2. T-Test

The statistical evaluation, which is based on a Student’s t-test, is often used to measure process capacities. The methodology is as follows:

1. A Normal population is considered.
2. Using a Student’s t distribution table, or statistical software, determine the t-value corresponding to the number of samples considered and at a confidence level of 99.5% (number of samples – 1 = degrees of freedom).
3. Calculate a t-statistics lower and/or upper value using the average (AVG) and standard deviation (STD) of the population, and the limit of Tables 1 to 8 of CSA A3001.

$\text{t}-\text{statistics}\text{lower}=\frac{\text{AVG}-\text{Low}\text{Spec}\text{Limit}}{\text{STD}}$

or

$\text{t}-\text{statistics}\text{upper}=\frac{\text{High}\text{Spec}\text{Limit}-\text{AV}}{\text{STD}}$

4. The t-statistics lower and/or upper value can be compared to the t-value. If t_upper and t_lower are greater, the process is under control given the current conditions. If lower, then a review of the process might be required.

There should be an annual review of the statistical evaluation.

Levene F-Distribution Test for Homogeneity of Variance

C.1. Population of Data

Consider two independent populations each with k samples, with preference for one full year of data if possible. For testing, 30 samples (sample set) could represent a period exceeding one year provided there is no significant change in the process.

C.2. Levene Test for Homogeneity of Variance (F-test)

The statistical evaluation, which is based on a Fα, k − 1, n − k test, is often used to verify the assumptions that the variances are equal across groups or samples. The methodology is as follows:

1. A null hypothesis that the variances between two groups or samples are equal (i.e., H_o = σ₁ = σ₂ . . . = σ_k).
2. Using the Fα, k − 1, n − k distribution table, or statistical software capable of performing SPSS, determine whether the p-value corresponding to the pair of samples at a confidence level of 95% (α = 0.05) is smaller than the Levene test statistic W.

$\text{W}=\frac{\left(n-k\right)}{\left(k-1\right)}\frac{{\displaystyle {\sum }_{i=1}^k{n}_i}{\left(Z_{i.}-Z_{\mathrm{..}}\right)}^2}{{\displaystyle {\sum }_{i=1}^k{\displaystyle {\sum }_{j=1}^{N_i}{\left(Z_{ij}-Z_i\right)}^2}}}$

where n = sample size, k = groups, n_i = sample size of ith group, Z_i is the group mean, and Z_ij is the overall mean.

3. Given α = 0.05, the Levene test rejects the null hypothesis that the variances are equal if:

$W>\text{F}\alpha ,\text{k}-1,n-\text{k}$

Or if the upper critical value of the F distribution with k − 1 and n − k degrees of freedom at a significance level of α = 0.05 has a p-value ≥ 0.05, cannot reject the null hypothesis.