The correct answer is C. 0.67. The CSSBB source defines the Cpm index as the Taguchi-based capability index that includes both process variation and the distance of the process mean from the target. It is given by:

Cpm = (USL − LSL) / [6 × √(σ² + (μ − T)²)]

From the question:

USL = 510, LSL = 490, so USL − LSL = 20

σ = 4

μ = 503

T = 500

Now compute:

σ ² + ( μ − T)² = 4² + (503 − 500)² = 16 + 9 = 25

√25 = 5

Denominator = 6 × 5 = 30

So:

Cpm = 20 / 30 = 0.667

Rounded to two decimal places, this is 0.67, which matches option C. The CSSBB material notes that Cpm places more emphasis on how well the process is centered on the target, not just on spread within specifications. Since this process is off target by 3 units, the Cpm is reduced accordingly. Therefore, the verified answer is C.

The correct answer is A. It takes the same amount of time to get to work by route X as by route Y . In CSSBB hypothesis testing, the null hypothesis (H0) is typically stated as no difference or equality , while the alternative hypothesis states the suspected difference or effect. The CSSBB examples for two-sample mean tests show the standard null form as H0: μ1 = μ2 and the alternative as H1: μ1 ≠ μ2 or directional inequalities depending on the claim.

Since the commuter suspects that route X may take longer, that suspicion belongs in the alternative hypothesis , not the null. The null should begin from the assumption that the average travel times are the same until evidence suggests otherwise. Option C describes a general alternative statement, not a null. Option D is about variation, not average travel time. Therefore, the proper null hypothesis is that the two routes take the same amount of time on average , which corresponds to option A .

The correct answer is C. 2.50 . The CSSBB financial-measures material gives the widely used formula:

ROI = Net Income / Investment

To apply that formula correctly, the net income or project gain is the savings minus the investment :

Net income = 700,000 − 200,000 = 500,000

Then divide by the investment:

ROI = 500,000 / 200,000 = 2.50

So the project returns 2.5 times the invested amount , which matches option C . This is consistent with Six Sigma project financial evaluation, where costs and benefits are compared to determine whether the project is worthwhile and how strongly it contributes to profitability. The CSSBB source also emphasizes that financial benefits are used to evaluate, prioritize, select, and track Six Sigma projects.

Option D would result only if someone incorrectly divided total savings by investment without subtracting the investment itself. Under the CSSBB ROI definition using net income, the verified answer is 2.50 .

In DOE, factors, blocks, and design resolution are determined during experiment planning, before data are collected. Factors are the independent variables selected for study, blocks are chosen to account for nuisance variation, and resolution is part of the experimental design selection. The CSSBB supplemental material explains that factors and blocks are design elements established in advance, while resolution reflects the level of confounding and is also chosen as part of the design setup. By contrast, the estimate of variance depends on the observed experimental results. The CSSBB materials describe the need for “an estimate of the inherent error variation” and place this in the context of replication, power, and post-data understanding of noise in the system. That variance estimate comes from the collected response data and is not fully known until the experiment has been run. This is fundamental in Analyze Phase thinking, because the team uses the experimental results to quantify error, assess significance, and separate signal from noise. Therefore, among the listed choices, the only item that can be determined only after data collection is the estimate of variance.

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In DOE terminology, a factor is an input variable that the experimenter deliberately controls or changes to study its effect on a response. For a pie-baking process, cooking time is an input setting and therefore qualifies as a factor. By contrast, flavor, texture, and crust color are typically outputs or response characteristics that result from the chosen factor settings. The CSSBB and related Lean Six Sigma experimental-design materials describe factors as variables adjusted at selected levels, while responses are the measured outcomes used to judge the effect of those settings. For example, time, temperature, ingredient quantity, or oven setting are common factors in a food-processing experiment, whereas quality attributes such as doneness, appearance, and texture are responses. This distinction is fundamental in the Improve Phase because DOE is used to determine which controllable inputs drive desired outputs. A Black Belt must clearly separate Xs from Ys: factors are the Xs and results are the Ys. In this question, cooking time is the only controllable input variable listed, so it is the correct factor in the experiment.

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PPM means parts per million and is calculated as defectives divided by total units, multiplied by 1,000,000. Since there is one opportunity per unit, the PPM calculation can be applied directly from the table values.

For January:

Facility X = 152 / 15,200 × 1,000,000 = 10,000 PPM

Facility Y = 123 / 12,500 × 1,000,000 = 9,840 PPM

Lower PPM indicates better quality performance, so Facility Y performed better in January.

For February:

Facility X = 161 / 17,000 × 1,000,000 ≈ 9,471 PPM

Facility Y = 141 / 14,000 × 1,000,000 ≈ 10,071 PPM

Again, lower PPM is better, so Facility X performed better in February.

This is a standard Measure Phase and process-performance comparison exercise because Black Belts often normalize defect data into PPM to compare sites with different production volumes. Raw defect counts alone can mislead, but PPM creates a fair basis for comparison. Therefore, the correct answer is D: January, Facility Y; February, Facility X.

The correct answer is B. critical-to-quality factors . In the CSSBB materials, the voice of the customer (VOC) may come from both internal and external customers , and this input is used to identify what is critical to the process, product, or service. The CTQ process specifically says to capture the VOC, then translate VOC to CTQs , prioritize them, and set requirements. The stakeholder and VOC material also states that customer input is used to “translate the needs into product, service, and process characteristics and metrics.”

That makes critical-to-quality (CTQ) the correct term for the characteristics derived from customer input and used to shape design and improvement efforts. SIPOC elements are suppliers, inputs, process, outputs, and customers; they are process-map categories, not customer-derived design characteristics. Key performance indicators are measures used to track performance, and initial-design concepts are not the standard CSSBB term here. Therefore, the verified answer is B. critical-to-quality factors .

This problem uses normal distribution logic and the Central Limit Theorem to determine control limits for the sample mean. If 99.73% of individual parts meet specification under a normal distribution, that corresponds to approximately ±3 standard deviations from the process mean. Since the specification is 25 ± 1 mm, the individual-process standard deviation is 1/3 = 0.333 mm. For subgroup means with sample size n = 36, the standard deviation of the sampling distribution is σx̄ = σ/√n. The CSSBB material explicitly states that the standard deviation of sample means equals the population standard deviation divided by the square root of sample size. Thus, σx̄ = 0.333/6 = 0.0556. Control limits for an x-bar chart are typically centered at the process mean and placed at ±3σx̄, giving ±3(0.0556) = ±0.1667, or about ±0.167. Therefore, the correct control limits for the sample means are 25 ± 0.167. This is a classic Control Phase application of statistical process control using the sampling distribution of the mean.

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The correct answer is A. One or more of the constraints might move . In the CSSBB control-phase material, the project control phase is described as necessary to sustain the project gains , and the control plan must remain a living document so the process can continue to be monitored and updated over time. The supplemental Lean Six Sigma material also states that the Control phase ensures that the new performance of the Y is sustained over time and that a monitoring system should continually track leading X’s and lagging Y indicators.

This ongoing monitoring is especially important because, in Theory of Constraints, once one constraint is improved or broken, the system must return to step 1 and look for new constraints . The CSSBB source explicitly states, “After the constraint has been broken, look for new constraints.” That means the location of the bottleneck or limiting factor can shift after improvements are made. Option B is too narrow, option C is not a valid control-phase principle, and option D incorrectly assumes another pilot run is the reason for monitoring. Therefore, the best verified answer is A .

The correct answer is C. identify what specific areas of knowledge need to be improved through training. The CSSBB source describes training needs assessment as a critical step used to uncover the gaps between adequate and inadequate job performance. It explains that the assessment reveals what is being done now and what should be done, and at the job/performer level it helps identify the areas the performer needs to address.

That is exactly what a training skills gap analysis is for: finding the specific knowledge or skill deficiencies that training should close. A later training plan may identify participants, budget, organization, content, and timetable, but those are downstream planning tasks. The first analytical purpose is to diagnose the gap. Another CSSBB source states that training is intended to close the gap between current and desired abilities, which reinforces that the focus is on identifying what needs improvement. Therefore, among the given options, the best verified answer is C, because a training skills gap analysis is performed to identify the specific knowledge areas that require improvement through training.