Sampling Plans and Confidence Intervals

Abstract

Before purchasing a large number of devices, a customer will likely ask the supplier about the defect level for the product being offered. The customer’s reliability inquiry is often expressed as: what is the defect level for the population of such devices in terms of number of defective devices per hundred, number of defective devices per thousand, number of defective devices per million (dpm), etc.? To determine the fraction defective, a sample of the devices is randomly selected from the population and this sample is tested/stressed to determine the fraction defective. After the fraction defective is determined for the sample, then it is only natural to ask: based on the sample size used, what is the confidence interval for the population fraction defective? To answer this critically important question, we must understand the basics of sampling theory.

Problems

• 1. To find the fraction of defective medical devices, from a large population of such devices, a sample size of 100 devices was randomly selected from the population and tested. One defective device was found. Using the Chi square distribution, what is the 90 % confidence interval for the fraction defective of the population of such devices?

Answer: 0.00105 ≤ F ≤ 0.0389

• 2. To find the fraction defective of a large population of stainless steel pipes, a sample size of 50 pipes was randomly selected from the population of such pipes and the sample was pressurized to 1,000 psi. Zero defects were found. Using the Chi square distribution, what is the 90 % confidence interval for the fraction defective of the population of such pipes?

Answer: 0 ≤ F ≤ 0.0461

• 3. An electronics store is worried about the defect level for a large population of TVs being sold. Using the Chi square distribution, what sample size should be randomly selected from the population, turned on and tested, to ensure (at 90 % confidence level) that the fraction defective is ≤ 0.5 %? Assume that we accept only on finding zero defects in the sample.

Answer: SS = 47

• 4. Valves, going into the cooling system of a nuclear reactor, must have a very low level fraction defective ≤ 0.1 %. Using the Chi square distribution, at the 90 % confidence level, what sample size should be randomly drawn from a large population of such valves and then tested/stressed with accepting only on zero defective valves being found in the sample?

Answer: SS = 2,303

• 5. Balloons are being tested for defectivity. Using the Chi square distribution, at the 90 % confidence level, how many balloons should be randomly drawn from a large population of such balloons and pressurized/tested to claim that the fraction of defective balloons in the population is ≤1 %?

  • (a) Assume that you accept only when finding zero defects in the sample of size SS.

  • (b) Assume that you accept only when finding one or fewer defects in the sample of size SS.

  • (c) Assume that you accept only when finding two or fewer defects in the sample of size SS.

Answers: (a) SS = 231, (b) SS = 389, (c) SS = 533

• 6. A sample size of 25 O-rings was randomly selected from a large population of such O-rings and the results for the sample followed closely a normal distribution with a mean/median value of (x₅₀)_s = 181.6 mm and with a standard deviation of σ_s = 3.2 mm. What are the 90 % confidence intervals for the population x₅₀ and the population σ?

Answers: (180.5 mm ≤ x₅₀ ≤ 182.7 mm) and (2.8 mm ≤ σ ≤ 4.0 mm)

• 7. In an electromigration test, 30 metal conductors were randomly selected from a large population of such conductors and stressed at a constant current density until the time-to-failure for each conductor was recorded. The time-to-failure data for the sample was described well by a lognormal distribution with a median time-to-failure of (t₅₀)_s = 412 h and with a logarithmic standard deviation of σ_s = 0.52. What are the 90 % confidence intervals for the population t₅₀ and σ?

Answers: (350 h ≤ TF₅₀ ≤ 485 h) and (0.46 ≤ σ ≤ 0.64)

• 8. In a corrosion test, 26 devices were randomly selected from a large population of such devices and stressed at a constant humidity and temperature until the time- to-failure for each device was recorded. The time-to-failure data for the sample was described well by a lognormal distribution with a median time-to-failure of (t₅₀)_s = 212 h and with a logarithmic standard deviation of σ_s = 0.42. What are the 90 % confidence intervals for the population t₅₀ and σ?

Answers: (184 h ≤ TF₅₀ ≤ 245 h) and (0.37 ≤ σ ≤ 0.53)

• 9. In a thermal cycling fatigue test, a sample of 29 devices was randomly selected from a large population of such devices and the sample was cycled from -65 to +150 °C until failure occurred. The cycles-to-failure data for the sample was described well by a lognormal distribution with a median cycle- to-failure of (CTF₅₀)_s = 812 cycles and a logarithmic standard deviation of σ_s = 0.64. What are the 90 % confidence intervals for the population CTF₅₀ and σ?

Answers: (661 cyc ≤ CTF₅₀ ≤ 997 cyc) and (0.56 ≤ σ ≤ 0.79)

• 10. In a time-dependent dielectric breakdown (TDDB) test, a sample of 25 capacitors was randomly selected from a large population of such capacitors and the sample was TDDB stressed at a constant voltage and temperature until the time-to-failure for each device was recorded. The time-to-failure data for the sample was described well by a Weibull distribution with a characteristic time-to-failure of (t₆₃)_s = 505 h and a Weibull slope of β_s = 1.82. What are the 90 % confidence intervals for the population t₆₃ and β?

Answers: (388 h ≤ TF₆₃ ≤ 658 h) and (2.10 ≥ β ≥ 1.47)