III. Reliability in Design and Development
II. B. 3. d. Hypothesis Testing – Comparisons
II. Probability and Statistics for Reliability
B. Statistical inference
3. Hypothesis testing (parametric and non-parametric) (Evaluate)
Apply hypothesis testing for parameters such as means, variance, proportions, and distribution parameters. Interpret significance levels and Type I and Type II errors for accepting/rejecting the null hypothesis.
A good use of hypothesis testing is to make comparisons.
Additional References
Equal Variance Hypothesis (article)
Hypothesis un-equal variance (article)
Paired-Comparison Hypothesis Tests (article)
Two Proportions Hypothesis Testing (article)
Two samples variance hypothesis test (article)
Quick Quiz
II. B. 3. c. Hypothesis Testing – Variance
II. Probability and Statistics for Reliability
B. Statistical inference
3. Hypothesis testing (parametric and non-parametric) (Evaluate)
Apply hypothesis testing for parameters such as means, variance, proportions, and distribution parameters. Interpret significance levels and Type I and Type II errors for accepting/rejecting the null hypothesis.
The testing of measures of dispersion is as simple as comparing means.
Additional References
Hypothesis Tests for Variance Case I (article)
Hypothesis Tests for Variance Case II (article)
Quick Quiz
II. B. 3. b. Hypothesis Testing – Means
II. Probability and Statistics for Reliability
B. Statistical inference
3. Hypothesis testing (parametric and non-parametric) (Evaluate)
Apply hypothesis testing for parameters such as means, variance, proportions, and distribution parameters. Interpret significance levels and Type I and Type II errors for accepting/rejecting the null hypothesis.
The first and simplest of hypothesis tests is with a mean compared to a specification or standard.
Additional References
Hypothesis Testing (article)
Hypothesis Tests for Proportion (article)
Degradation Hypothesis (article)
Quick Quiz
1-34. To compare sample means, which statistical distribution should be used?
(A) chi-square
(B) exponential
(C) normal
(D) t test
(D) t test
The key phrase here is “sample means”, thus the t-test is the best of the options. While the t-test is primarily for use with small samples —less then 30— drawn from normal populations it is fairly robust to non-normal populations for the comparison of means.
The exponential distribution is not used for the comparison of means using hypothesis testing. The normal or z-test is used to compare the population (not sample) means. The chi-square distribution is used to compare population variances or to compare the observed and expected frequencies of test outcome.
II. B. 3. a. Hypothesis Testing – The Process
II. Probability and Statistics for Reliability
B. Statistical inference
3. Hypothesis testing (parametric and non-parametric) (Evaluate)
Apply hypothesis testing for parameters such as means, variance, proportions, and distribution parameters. Interpret significance levels and Type I and Type II errors for accepting/rejecting the null hypothesis.
Here is a simple process to use when comparing a datasets to a standard or another dataset.
Additional References
Hypothesis Testing (article)
Run Test for Randomness (article)
Hypothesis Test Selection (article)
Hypothesis Test Selection Flowchart (article)
Hypothesis Test Sample Size (article)
Quick Quiz
1-26. The level of significance is defined as the probability of which of the following?
(A) accepting a null hypothesis when it is true
(B) not accepting a null hypothesis when it is true
(C) rejecting a null hypothesis when it is true
(D) not rejecting a null hypothesis when it is true
(C) rejecting a null hypothesis when it is true
The null hypothesis is rejected if the p-value is less than the significance or α level. The α level is the probability of rejecting the null hypothesis given that it is true (type I error).
The null hypothesis is a statement about a belief. We may doubt that the null hypothesis is true, which might be why we are “testing” it. The alternative hypothesis might, in fact, be what we believe to be true. The test procedure is constructed so that the risk of rejecting the null hypothesis, when it is in fact true, is small. This risk, α is often referred to as the significance level of the test. By having a test with a small value of α, we feel that we have actually “proved” something when we reject the null hypothesis. (NIST Engineering Statistical Handbook)
1-37. Which term is commonly used as the probability of rejecting material produced at an unacceptable quality level?
(A) α
(B) β
(C) 1 − α
(D) 1 − β
(D) 1 − β
β is the probability of accepting the lot or batch when we shouldn’t. It is the chance of a false acceptance given the sample data when the population would not be acceptable. Therefore the meaning of 1 − β is the probability of rejecting (properly so) an unacceptable lot of material. The sample indicates the population is bad and it is bad.
1 − β is also known as the power of the test which means the ability to sample to detect a unacceptable population. β is also called the Type II error or consumer’s risk. In contract, α is the Type I error or producer’s risk.
II. B. 2. b. Statistical Intervals – MTBF Estimates
II. Probability and Statistics for Reliability
B. Statistical inference
2. Statistical interval estimates (Evaluate)
Compute confidence intervals, tolerance intervals, etc., and draw conclusions from the results.
The common metric, MTBF, has confidence intervals.
Additional References
Confidence Intervals for MTBF (article)
Perils of MTBF (NoMTBF.com article)
Quick Quiz
1-21. A randomly failing component is tested to X failures and a total of N observations are made. How many degrees of freedom are used in calculating confidence limits on the mean?
(A) 2X + 2
(B) 2X
(C) N
(D) N − 1
(B) 2X
There are a few clues in the question that help narrow down the problem to the correct formula to examine. First, “randomly” implies the exponential distribution. The testing terminated when it reached “X failures” suggesting a failure terminated test design. It calls× for the calculation of the “mean” “confidence interval”.
MTBF is the mean of the exponential distribution and the lower one-sided confidence interval is
$$ \frac{2T}{\chi _{\left( \alpha ,\text{ }2r \right)}^{2}}\le \theta $$
where 2r (or in terms of this problem 2X) is the degrees of freedom.
II. B. 2. a. Statistical Intervals – Point Estimates
II. Probability and Statistics for Reliability
B. Statistical inference
2. Statistical interval estimates (Evaluate)
Compute confidence intervals, tolerance intervals, etc., and draw conclusions from the results.
When using a sample to estimate a parameter, we can calculate where the true (unknown) value may reside.
Additional References
Point and Interval Estimates (article)
Confidence Interval for a Proportion – Normal Approximation (article)
Confidence Interval for Variance (article)
Statistical Confidence (article)
Quick Quiz
1-39. Identify which of the following statements concerning statistical inference is false.
(A) The confidence interval is a range of values that may include the true value of a population parameter.
(B) The confidence interval normally encompasses the statistical tolerance limits of the population parameter.
(C) Estimation is the process of analyzing a sample result to predict the value of the population parameter.
(D) The point estimate is a single value used to estimate the population parameter.
(B) The confidence interval normally encompasses the statistical tolerance limits of the population parameter.
Confidence intervals apply to the limits of the statistical parameter being estimated such as the mean or variance. Tolerance intervals provide a set of limits for the future individual values, not the mean or variance. Tolerance intervals do not apply to estimates of population parameters.
1-42. What sort of mathematical models are used for statistical inference?
(A) exponential
(B) inferential
(C) deterministic
(D) probabilistic
(D) probabilistic
We are dealing with variation and using a sample to estimate parameters for a population. We base our ability to estimate confidence intervals and hypothesis testing on the tenets of probability.
II. B. 1. Point Estimates of Parameters
II. Probability and Statistics for Reliability
B. Statistical inference
1. Point estimates of parameters (Evaluate)
Obtain point estimates of model parameters using probability plots, maximum likelihood methods, etc. Analyze the efficiency and bias of the estimators.
Additional References
Quick Quiz
1-17. In a life test of 4 power cells, failures were observed after 12, 22, 30, and 37 hours. A fifth cell was tested for 75 hours without failure, at which time the test was terminated. Calculate the estimated mean time to failure and the failure rate.
(A) MTTF = 35.2; failure rate = 0.0284
(B) MTTF = 44; failure rate = 0.0227
(C) MTTF = 35.2; failure rate = 0.0227
(D) MTTF = 44; failure rate =0.0284
(B) MTTF = 44; failure rate = 0.0227
The formula for MTTF is the total test time divided by the number of failures. In this case add the time to failure times of the four units that failed and the total time for the one unit that did not fail. That is 12 + 22 + 30 + 37 + 75 = 176 hours of total test time. Then divide by the number of failure, 176 / 4 = 44 hours MTTF. The inverse of MTTF is an estimate of the failure rate, 1 / 44 = 0.0227.
The question did not mention the units being replace or quickly repaired after failures, that the test time ends for each unit upon failure. Plus it is looking for the MTTF both of which imply non-repairable units. If you divided by 5 or didn’t check failure rate calculation is correct, there is an answer listed.
1-31. The failure rate for a flash drive is 0.00023 per hour of operation. Calculate the MTBF assuming that the failure rate is constant.
(A) 435 hours
(B) 3125 hours
(C) 4,348 hours
(D) 43,478 hours
(C) 4,348 hours
For the exponential distribution (assuming constant failure rate) the MTBF is the inverse of the failure rate. Thus
$$ \theta =\frac{1}{\lambda }=\frac{1}{0.00023}=4,348\text{ hours}$$
Statistical Inference Introduction
II. B. Statistical Inference
II. A. 7. j. Pre-Control Charts
II. Probability and Statistics for Reliability
A. Basic concepts
7. Statistical process control (SPC) and process capability (Understand)
Define and describe SPC and process capability studies (Cp, Cpk, etc.), their control charts, and how they are all related to reliability.
An alternative control chart process that is especially useful for line start-up.
Additional References
Pre-Control Charts (article)
Quick Quiz
II. A. 7. i. Capability and Charts
II. Probability and Statistics for Reliability
A. Basic concepts
7. Statistical process control (SPC) and process capability (Understand)
Define and describe SPC and process capability studies (Cp, Cpk, etc.), their control charts, and how they are all related to reliability.
Process capability and control charts are related – and lets talk about how.
Additional References
Quick Quiz
II. A. 7. h. Standard Normal and z-values
II. Probability and Statistics for Reliability
A. Basic concepts
7. Statistical process control (SPC) and process capability (Understand)
Define and describe SPC and process capability studies (Cp, Cpk, etc.), their control charts, and how they are all related to reliability.
We use the normal distribution and the z-table for many applications here it is useful, too.
Additional References
z value (article)
Reading a Standard Normal Table (article)
How to read a standard normal table (article)
Interpolation within Distribution Tables (article)
Quick Quiz
II. A. 7. g. Process Capability
II. Probability and Statistics for Reliability
A. Basic concepts
7. Statistical process control (SPC) and process capability (Understand)
Define and describe SPC and process capability studies (Cp, Cpk, etc.), their control charts, and how they are all related to reliability.
Once a process is stable, the next step is to compare the specification to the variation produced.
Additional References
Process Capability (article)
Supply Chain Process Control and Capability (article)
SOR 013 The Design and the Supplier’s Capability (podcast)
Basic Approach to Achieve Process Stability (article)
Reliability and Statistical Process Control (recorded webinar)
Quick Quiz
