Appendix A to Subpart C of Part 429 - Sampling Plan for Enforcement Testing of Covered Consumer Products and Certain High-Volume Commercial Equipment

Appendix A to Subpart C of Part 429 - Sampling Plan for Enforcement Testing of Covered Consumer Products and Certain High-Volume Commercial Equipment

(a) The first sample size (n₁) for enforcement testing must be four or more units, except as provided by § 429.57(e)(1)(i).

(b) Compute the mean of the measured energy performance (x₁) for all tests as follows:

where x_i is the measured energy or water efficiency or consumption from test i, and n₁ is the total number of tests.

(c) Compute the standard deviation (s₁) of the measured energy performance from the n₁ tests as follows:

(d) Compute the standard error (s_x1) of the measured energy performance from the n₁ tests as follows:

(e)

(1) Compute the upper control limit (UCL₁) and lower control limit (LCL₁) for the mean of the first sample using the applicable DOE energy efficiency standard (EES) as the desired mean and a probability level of 95 percent (two-tailed test) as follows:

LCL₁ EES - ts_X1X

where t is the statistic based on a 95 percent two-tailed probability level with degrees of freedom (n₁−1).

(2) For an energy efficiency or water 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:

(i) 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.)

(ii) 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.)

(iii) 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 (e)(3).

(3) For an energy efficiency or water efficiency standard, determine the second sample size (n₂) as follows:

where s₁ and t have the values used in equations 2 and 4, respectively. The term “0.05 EES” is the difference between the applicable energy efficiency or water 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:

(i) If the value of n₂ is less than or equal to zero and if the mean energy or water 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 or water 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.

(ii) 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 or water efficiency standard (EES), whichever is greater, i.e., if n₂≤0 and x₁≤max (LCL₁, 0.95 EES), the basic model is not in compliance and testing is at an end.

(iii) 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 (6). If the value of n₂ so calculated is greater than 21− n₁, set n₂ equal to 21− n₁.

(4) Compute the combined mean (x₂) of the measured energy or water efficiency of the n₁ and n₂ units of the combined first and second samples as follows:

(5) Compute the standard error (S_x2) of the measured energy or water performance of the n₁ and n₂ units in the combined first and second samples as follows:

Note:

s₁ is the value obtained in Step (c).

(6) For an energy efficiency standard (EES), compute the lower control limit (LCL₂) for the mean of the combined first and second samples using the DOE 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 (e)(1)) as follows:

where the t-statistic has the value obtained in Step (e)(1) and s_x2 is the value obtained in Step (e)(5).

(7) For an energy efficiency standard (EES), compare the combined sample mean (x₂) to the lower control limit (LCL₂) to determine one of the following:

(i) 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 not compliant and testing is at an end.

(iii) 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.

(f)

(1) Compute the upper control limit (UCL₁) and lower control limit (LCL₁) for the mean of the first sample using the applicable DOE energy consumption standard (ECS) as the desired mean and a probability level of 95 percent (two-tailed test) as follows:

where t is the statistic based on a 95 percent two-tailed probability level with degrees of freedom (n₁ − 1).

(2) 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:

(i) 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.)

(ii) 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.)

(iii) 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 (f)(3).

(3) For an Energy or Water Consumption Standard, determine the second sample size (n₂) as follows:

where s₁and t have the values used in equations (2) and (10), respectively. The term “0.05 ECS” 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:

(i) 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 consumption standard (ECS), whichever is less, i.e., if n₂ ≤0 and x₁ ≤min (UCL₁, 1.05 ECS), the basic model is in compliance and testing is at an end.

(ii) 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 consumption standard (ECS), whichever is less, i.e., if n₂ ≤0 and x₁ >min (UCL₁, 1.05 EPS), the basic model is not compliant and testing is at an end.

(iii) 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 (11). If the value of n₂ so calculated is greater than 21−n₁, set n₂ equal to 21−n₁.

(4) Compute the combined mean (x₂) of the measured energy or water consumption of the n₁ and n₂ units of the combined first and second samples as follows:

(5) Compute the standard error (S_x2) of the measured energy or water consumption of the n₁ and n₂ units in the combined first and second samples as follows:

Note:

s₁ is the value obtained in Step (c).

(6) For an energy or water consumption standard (ECS), compute the upper control limit (UCL₂) for the mean of the combined first and second samples using the DOE ECS 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 (f)(1)) as follows:

where the t-statistic has the value obtained in (f)(1).

(7) For an energy or water consumption standard (ECS), compare the combined sample mean (x₂) to the upper control limit (UCL₂) to determine one of the following:

(i) If the mean of the combined sample (x₂) is greater than the upper control limit (UCL₂) or 105 percent of the ECS whichever is less, i.e., if x₂ >min (UCL₂, 1.05 ECS), the basic model is not compliant and testing is at an end.

(ii) 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 (ECS), whichever is less, i.e., if x₂≤min (UCL₂, 1.05 ECS), the basic model is in compliance and testing is at an end.