Appendix B to Subpart F of Part 430 - Sampling Plan For Enforcement Testing

Double Sampling

Step 1. The first sample size (n1) must be four or more units.

Step 2. Compute the mean (x1) of the measured energy performance or water performance (in the case of faucets, showerheads, water closets, and urinals) of the n1 units in the first sample as follows:

ER18MR98.010 where (x1) is the measured energy efficiency, energy or water (in the case of faucets, showerheads, water closets, and urinals) consumption of unit I.

Step 3. Compute the standard deviation (s1) of the measured energy or water performance of the (n1) units in the first sample as follows:

ER18MR98.011

Step 4. Compute the standard error (SX1) of the measured energy or water performance of the n1 units in the first sample as follows:

ER18MR98.012

Step 5. Compute the upper control limit (UCL1) and lower control limit (LCL1) for the mean of the first sample using the applicable DOE energy or water performance standard (EPS) as the desired mean and a probability level of 95 percent (two-tailed test) as follows:

ER18MR98.013 ER18MR98.014 where t is a statistic based on a 95 percent two-tailed probability level and a sample size of n1.

Step 6(a). For an Energy Efficiency Standard, compare the mean of the first sample (x1) with the upper and lower control limits (UCL1 and LCL1) to determine one of the following:

(1) If the mean of the first sample is below the lower control limit, then the basic model is in noncompliance and testing is at an end. (Do not go on to any of the steps below.)

(2) If the mean of the first sample is equal to or greater than the upper control limit, then the basic model is in compliance and testing is at an end. (Do not go on to any of the steps below.)

(3) If the sample mean is equal to or greater than the lower control limit but less than the upper control limit, then no determination of compliance or noncompliance can be made and a second sample size is determined by Step 7(a).

Step 6(b). For an Energy or Water Consumption Standard, compare the mean of the first sample (x1) with the upper and lower control limits (UCL1 and LCL1) to determine one of the following:

(1) If the mean of the first sample is above the upper control limit, then the basic model is in noncompliance and testing is at an end. (Do not go on to any of the steps below.)

(2) If the mean of the first sample is equal to or less than the lower control limit, then the basic model is in compliance and testing is at an end. (Do not go on to any of the steps below.)

(3) If the sample mean is equal to or less than the upper control limit but greater than the lower control limit, then no determination of compliance or noncompliance can be made and a second sample size is determined by Step 7(b).

Step 7(a). For an Energy Efficiency Standard, determine the second sample size (n2) as follows:

ER18MR98.015

where s1 and t have the values used in Steps 4 and 5, respectively. The term “0.05 EPS” is the difference between the applicable energy efficiency standard and 95 percent of the standard, where 95 percent of the standard is taken as the lower control limit. This procedure yields a sufficient combined sample size (n1+n2) to give an estimated 97.5 percent probability of obtaining a determination of compliance when the true mean efficiency is equal to the applicable standard. Given the solution value of n2, determine one of the following:

(1) If the value of n2 is less than or equal to zero and if the mean energy efficiency of the first sample (x1) is either equal to or greater than the lower control limit (LCL1) or equal to or greater than 95 percent of the applicable energy efficiency standard (EES), whichever is greater, i.e., if n2 ≤ 0 and x1 ≥ max (LCL1, 0.95 EES), the basic model is in compliance and testing is at an end.

(2) If the value of n2 is less than or equal to zero and the mean energy efficiency of the first sample (x1) is less than the lower control limit (LCL1) or less than 95 percent of the applicable energy efficiency standard (EES), whichever is greater, i.e., if n2 ≤ 0 and x1 ≥ max (LCL1, 0.95 EES), the basic model is in noncompliance and testing is at an end.

(3) If the value of n2 is greater than zero, then value of the second sample size is determined to be the smallest integer equal to or greater than the solution value of n2 for equation (6a). If the value of n2 so calculated is greater than 20−n1, set n2 equal to 20−n1.

Step 7(b). For an Energy or Water Consumption Standard, determine the second sample size (n2) as follows:

ER18MR98.016 where s1 and t have the values used in Steps 4 and 5, respectively. The term “0.05 EPS” is the difference between the applicable energy or water consumption standard and 105 percent of the standard, where 105 percent of the standard is taken as the upper control limit. This procedure yields a sufficient combined sample size (n1+n2) to give an estimated 97.5 percent probability of obtaining a determination of compliance when the true mean consumption is equal to the applicable standard. Given the solution value of n2, determine one of the following:

(1) If the value of n2 is less than or equal to zero and if the mean energy or water consumption of the first sample (x1) is either equal to or less than the upper control limit (UCL1) or equal to or less than 105 percent of the applicable energy or water performance standard (EPS), whichever is less, i.e., if n2 ≤ 0 and x1 ≤ min (UCL1, 1.05 EPS), the basic model is in compliance and testing is at an end.

(2) If the value of n2 is less than or equal to zero and the mean energy or water consumption of the first sample (x1) is greater than the upper control limit (UCL1) or more than 105 percent of the applicable energy or water performance standard (EPS), whichever is less, i.e., if n2 ≤ 0 and x1 > min (UCL1, 1.05 EPS), the basic model is in noncompliance and testing is at an end.

(3) If the value of n2 is greater than zero, then the value of the second sample size is determined to be the smallest integer equal to or greater than the solution value of n2 for equation (6b). If the value of n2 so calculated is greater than 20−n1, set n2 equal to 20−n1.

Step 8. Compute the combined mean (x2) of the measured energy or water performance of the n1 and n2 units of the combined first and second samples as follows:

ER18MR98.017

Step 9. Compute the standard error (Sx1) of the measured energy or water performance of the n1 and n2 units in the combined first and second samples as follows:

ER18MR98.018 Note:

s1 is the value obtained in Step 3.

Step 10(a). For an Energy Efficiency Standard, compute the lower control limit (LCL2) for the mean of the combined first and second samples using the DOE energy efficiency standard (EES) as the desired mean and a one-tailed probability level of 97.5 percent (equivalent to the two-tailed probability level of 95 percent used in Step 5) as follows:

ER18MR98.019 where the t-statistic has the value obtained in Step 5.

Step 10(b). For an Energy or Water Consumption Standard, compute the upper control limit (UCL2) for the mean of the combined first and second samples using the DOE energy or water performance standard (EPS) as the desired mean and a one-tailed probability level of 102.5 percent (equivalent to the two-tailed probability level of 95 percent used in Step 5) as follows:

ER18MR98.020 where the t-statistic has the value obtained in Step 5.

Step 11(a). For an Energy Efficiency Standard, compare the combined sample mean (x2) to the lower control limit (LCL2) to find one of the following:

(1) If the mean of the combined sample (x2) is less than the lower control limit (LCL2) or 95 percent of the applicable energy efficiency standard (EES), whichever is greater, i.e., if x2 < max (LCL2, 0.95 EES), the basic model is in noncompliance and testing is at an end.

(2) If the mean of the combined sample (x2) is equal to or greater than the lower control limit (LCL2) or 95 percent of the applicable energy efficiency standard (EES), whichever is greater, i.e., if x2 ≥ max (LCL2, 0.95 EES), the basic model is in compliance and testing is at an end.

Step 11(b). For an Energy or Water Consumption Standard, compare the combined sample mean (x2) to the upper control limit (UCL2) to find one of the following:

(1) If the mean of the combined sample (x2) is greater than the upper control limit (UCL2) or 105 percent of the applicable energy or water performance standard (EPS), whichever is less, i.e., if x2 > min (UCL2, 1.05 EPS), the basic model is in noncompliance and testing is at an end.

(2) If the mean of the combined sample (x2) is equal to or less than the upper control limit (UCL2) or 105 percent of the applicable energy or water performance standard (EPS), whichever is less, i.e., if x2 ≤ min (UCL2, 1.05 EPS), the basic model is in compliance and testing is at an end.

Manufacturer-Option Testing

If a determination of non-compliance is made in Steps 6, 7 or 11, the manufacturer may request that additional testing be conducted, in accordance with the following procedures.

Step A. The manufacturer requests that an additional number, n3, of units be tested, with n3 chosen such that n1+n2+n3 does not exceed 20.

Step B. Compute the mean energy or water performance, standard error, and lower or upper control limit of the new combined sample in accordance with the procedures prescribed in Steps 8, 9, and 10, above.

Step C. Compare the mean performance of the new combined sample to the revised lower or upper control limit to determine one of the following:

a.1. For an Energy Efficiency Standard, if the new combined sample mean is equal to or greater than the lower control limit or 95 percent of the applicable energy efficiency standard, whichever is greater, the basic model is in compliance and testing is at an end.

a.2. For an Energy or Water Consumption Standard, if the new combined sample mean is equal to or less than the upper control limit or 105 percent of the applicable energy or water consumption standard, whichever is less, the basic model is in compliance and testing is at an end.

b.1. For an Energy Efficiency Standard, if the new combined sample mean is less than the lower control limit or 95 percent of the applicable energy efficiency standard, whichever, is greater, and the value of n1+n2+n3 is less than 20, the manufacturer may request that additional units be tested. The total of all units tested may not exceed 20. Steps A, B, and C are then repeated.

b.2. For an Energy or Water Consumption Standard, if the new combined sample mean is greater than the upper control limit or 105 percent of the applicable energy or water consumption standard, whichever is less, and the value of n1+n2+n3 is less than 20, the manufacturer may request that additional units be tested. The total of all units tested may not exceed 20. Steps A, B, and C are then repeated.

c. Otherwise, the basic model is determined to be in noncompliance.