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Question 1Technical Mathematics

A sample contains the five values 4, 8, 6, 10 and 12. What is the arithmetic mean?

  1. 8 Correct answer.
  2. 10
  3. 7
  4. 9

Explanation

The mean is the sum of the observations divided by the number of observations: (4 + 8 + 6 + 10 + 12) / 5 = 40 / 5 = 8. It uses every value in the data set.

Question 2Technical Mathematics

What is the median of the data set 3, 7, 5, 9, 11, 13?

  1. 11
  2. 7
  3. 8 Correct answer.
  4. 9

Explanation

Ordering the values gives 3, 5, 7, 9, 11, 13. With an even number of observations the median is the average of the two middle values, (7 + 9) / 2 = 8.

Question 3Technical Mathematics

For the sample 12, 15, 9, 20 and 14, what is the range?

  1. 9
  2. 20
  3. 11 Correct answer.
  4. 8

Explanation

The range is the largest value minus the smallest value: 20 − 9 = 11. It depends on only two observations.

Question 4Technical Mathematics

Which measure of dispersion is expressed in the same units as the original data and uses every observation?

  1. Standard deviation Correct answer.
  2. Coefficient of variation
  3. Range
  4. Variance

Explanation

The standard deviation is the square root of the variance, so it returns to the original units of measurement and it is computed from all observations. Variance is in squared units, the coefficient of variation is a dimensionless ratio, and the range uses only two values.

Question 5Technical Mathematics

A process has specification limits of 10 and 20 and a process standard deviation of 1.0. What is Cp?

  1. 1.67 Correct answer.
  2. 0.83
  3. 5.00
  4. 1.00

Explanation

Cp = (USL − LSL) / (6σ) = (20 − 10) / (6 × 1.0) = 10 / 6 = 1.67. It compares the specification width with the natural 6-sigma process width.

Question 6Technical Mathematics

What is the mode of the data set 2, 4, 4, 6, 7, 4, 9?

  1. 5.14
  2. 4.5
  3. 4 Correct answer.
  4. 6

Explanation

The mode is the value that occurs most often. The value 4 appears three times, more than any other value in the set.

Question 7Technical Mathematics

In a normal distribution, approximately what percentage of observations lie within one standard deviation of the mean?

  1. About 99.7%
  2. About 95.5%
  3. About 68.3% Correct answer.
  4. About 50.0%

Explanation

For a normal distribution the empirical rule gives roughly 68.3% of the area within ±1σ of the mean. The corresponding figures are about 95.5% for ±2σ and about 99.7% for ±3σ.

Question 8Technical Mathematics

Why is the sample standard deviation calculated with n − 1 rather than n in the denominator?

  1. It corrects the bias caused by estimating the mean from the same sample Correct answer.
  2. It is required whenever the data are normally distributed
  3. It converts the variance into a range
  4. It makes the calculated value larger and therefore safer

Explanation

Dividing by n − 1 applies Bessel's correction: because deviations are measured from the sample mean estimated from the same data, the sum of squared deviations tends to be too small. Using n − 1 gives an unbiased estimator of the population variance regardless of the distribution shape.

Question 9Technical Mathematics

A characteristic has USL = 60, LSL = 40, process mean = 53 and process standard deviation = 2. What is Cpk?

  1. 0.58
  2. 2.17
  3. 1.67
  4. 1.17 Correct answer.

Explanation

Cpk = min[(USL − mean), (mean − LSL)] / (3σ) = min[(60 − 53), (53 − 40)] / 6 = min(7, 13) / 6 = 1.17. The nearer specification limit governs the index.

Question 10Technical Mathematics

An X-bar chart is based on samples of n = 9 from a process with known standard deviation σ = 3 and a grand mean of 50. What is the upper control limit?

  1. 51
  2. 53 Correct answer.
  3. 59
  4. 56

Explanation

The control limits for an X-bar chart with known σ are placed at the grand mean ± 3σ/√n. Here 3 × 3/√9 = 3 × 1 = 3, so UCL = 50 + 3 = 53.

Question 11Technical Mathematics

A process is inspected in subgroups whose size varies from day to day, and each unit is classified as conforming or nonconforming. Which control chart is appropriate?

  1. X-bar and R chart
  2. np-chart
  3. p-chart Correct answer.
  4. c-chart

Explanation

The p-chart plots the proportion nonconforming and its limits are calculated separately for each subgroup size, so varying sample sizes are handled correctly. The np-chart also tracks nonconforming units but requires a constant subgroup size, and c-charts count defects rather than defective units.

Question 12Technical Mathematics

A gauge R&R study reports that the measurement system variation equals 45% of the tolerance width. How should this be interpreted?

  1. Unacceptable; the measurement system should be improved Correct answer.
  2. Excellent; no action is needed
  3. Acceptable for most purposes
  4. Marginal; may be acceptable depending on criticality

Explanation

A common rule of thumb judges %GRR of tolerance below 10% as acceptable, 10–30% as conditionally acceptable, and above 30% as unacceptable. A value of 45% exceeds the upper threshold, so the measurement system is inadequate for that tolerance.

Question 13Technical Mathematics

Two independent events have probabilities 0.5 and 0.4. What is the probability that both occur?

  1. 0.90
  2. 0.45
  3. 0.10
  4. 0.20 Correct answer.

Explanation

For independent events the multiplication rule applies: P(A and B) = P(A) × P(B) = 0.5 × 0.4 = 0.20. Independence is what justifies multiplying the probabilities.

Question 14Technical Mathematics

A sample of n = 25 is drawn from a process with standard deviation σ = 10. What is the standard error of the sample mean?

  1. 5.0
  2. 2.0 Correct answer.
  3. 0.4
  4. 10.0

Explanation

The standard error of the mean is σ/√n = 10/√25 = 10/5 = 2.0. It describes the spread of sample means about the process mean.

Question 15Technical Mathematics

In statistical hypothesis testing, what is a Type I error?

  1. Rejecting a true null hypothesis Correct answer.
  2. Rejecting a false null hypothesis
  3. Failing to reject a true null hypothesis
  4. Failing to reject a false null hypothesis

Explanation

A Type I error is rejecting the null hypothesis when it is actually true; its probability is the significance level α. A Type II error, by contrast, is failing to reject a false null hypothesis.

Question 16Technical Mathematics

The number of surface defects found on each painted panel is recorded, with defects occurring randomly and independently at a constant average rate. Which probability distribution models this count?

  1. Normal
  2. Poisson Correct answer.
  3. Exponential
  4. Binomial

Explanation

The Poisson distribution models counts of rare events occurring in a fixed interval of time, area or unit when the average rate is constant and events are independent. Its mean and variance are both equal to the rate parameter λ.

Question 17Technical Mathematics

A fitted simple linear regression has a slope of 2.5. How is this slope interpreted?

  1. A one-unit increase in x is associated with a 2.5-unit increase in y Correct answer.
  2. y increases by 2.5 units regardless of the value of x
  3. 2.5% of the variation in y is explained by x
  4. x increases by 2.5 units for each unit increase in y

Explanation

In y = a + bx, the slope b is the estimated change in the mean response for a one-unit increase in the predictor. Here b = 2.5, so the predicted y rises by 2.5 units per unit of x.

Question 18Technical Mathematics

A process is centred exactly halfway between its specification limits. Which relationship between Cp and Cpk holds?

  1. Cp and Cpk are unrelated
  2. Cp is greater than Cpk
  3. Cp equals Cpk Correct answer.
  4. Cp is less than Cpk

Explanation

Cpk equals Cp minus an allowance for off-centring; when the process mean coincides with the specification midpoint there is no off-centring penalty. The two indices therefore coincide, and Cp becomes an upper bound on Cpk in all other cases.

Question 19Technical Mathematics

For subgroups of size 5, the average range is 5.0 and the constant d2 is 2.326. What is the estimated process standard deviation R-bar/d2?

  1. 11.63
  2. 2.15 Correct answer.
  3. 2.33
  4. 2.50

Explanation

Within-subgroup variation is estimated as σ̂ = R̄/d2 = 5.0/2.326 = 2.15. The constant d2 converts average range into a standard deviation for a given subgroup size.

Question 20Technical Mathematics

A sample of n = 36 is taken from a process with known σ = 12. What is the half-width of the 95% confidence interval for the mean, using z = 1.96?

  1. 2.00
  2. 1.96
  3. 3.92 Correct answer.
  4. 0.33

Explanation

The half-width is z × σ/√n = 1.96 × 12/6 = 1.96 × 2 = 3.92. The interval is the sample mean plus or minus this amount.

Question 21Technical Mathematics

A process has Cp = 1.5 and its mean is displaced from the specification midpoint by exactly one standard deviation. What is Cpk?

  1. 0.50
  2. 1.50
  3. 1.17 Correct answer.
  4. 1.33

Explanation

With Cp = 1.5 the tolerance half-width is 1.5 × 3σ = 4.5σ. The distance from the mean to the nearer limit is therefore 4.5σ − 1σ = 3.5σ, so Cpk = 3.5σ / 3σ = 1.17.

Question 22Technical Mathematics

Using the common six-sigma convention of a 1.5-sigma long-term shift, what defect rate corresponds to six-sigma performance?

  1. 233 defects per million opportunities
  2. 6,210 defects per million opportunities
  3. 66,807 defects per million opportunities
  4. 3.4 defects per million opportunities Correct answer.

Explanation

The six-sigma benchmark of 3.4 defects per million opportunities incorporates an assumed 1.5σ drift between short-term and long-term performance. Without the shift, six sigma corresponds to about 0.002 defects per million.

Question 23Technical Mathematics

Two independent standard uncertainty contributions are 3 units and 4 units. What is their combined standard uncertainty?

  1. 12.0
  2. 5.0 Correct answer.
  3. 3.5
  4. 7.0

Explanation

Independent uncertainty components are combined in quadrature: uc = √(3² + 4²) = √25 = 5.0. Uncertainties are not simply added because independent errors can partially cancel.

Question 24Technical Mathematics

A component has a constant failure rate with MTBF of 1000 hours. Assuming an exponential time-to-failure distribution, what is the reliability at 1000 hours of operation?

  1. 0.100
  2. 0.632
  3. 0.368 Correct answer.
  4. 0.500

Explanation

Exponential reliability is R(t) = e^(−t/MTBF). At t = MTBF = 1000 hours, R = e^(−1) = 0.368, so about 37% of units survive to the mean life.

Question 25Technical Mathematics

In attribute acceptance sampling with a fixed acceptance number, what is the effect of increasing the sample size on the operating characteristic curve?

  1. The curve becomes flatter, so more lots are accepted
  2. The curve is unchanged because only the acceptance number matters
  3. The curve becomes steeper, discriminating more sharply between good and bad lots Correct answer.
  4. The curve shifts so that the producer's risk increases

Explanation

Larger samples give more information about lot quality, so the OC curve approaches a step function at the chosen quality level. Discrimination between acceptable and unacceptable lots improves, although sampling costs rise.

Question 26Technical Mathematics

A sample of five bearing diameters is 12, 15, 18, 20, and 25 mm. What is the sample mean?

  1. 20 mm
  2. 17 mm
  3. 19 mm
  4. 18 mm Correct answer.

Explanation

The mean is the sum of the observations divided by the number of observations. Here the sum is 90 mm and 90 ÷ 5 = 18 mm, so the sample mean is 18 mm.

Question 27Technical Mathematics

The ordered sample is 3, 4, 5, 8, 9, and 12. What is the median?

  1. 8.5
  2. 6.5 Correct answer.
  3. 5.0
  4. 8.0

Explanation

With an even number of observations, the median is the average of the two middle values. The middle values are 5 and 8, so the median is (5 + 8) ÷ 2 = 6.5.

Question 28Technical Mathematics

For the observations 14, 22, 9, 31, and 17, what is the range?

  1. 22 Correct answer.
  2. 17
  3. 24
  4. 19

Explanation

The range is the largest observation minus the smallest observation. The maximum is 31 and the minimum is 9, so the range is 31 − 9 = 22.

Question 29Technical Mathematics

A quality engineer calculates the average fill volume of 50 bottles selected from one day's production. This average is best described as a:

  1. census value
  2. sample statistic Correct answer.
  3. control limit
  4. population parameter

Explanation

A value computed from a subset of the population is a sample statistic. The 50 bottles form a sample, so their average fill volume estimates the population parameter.

Question 30Technical Mathematics

If P(A) = 0.35, what is P(not A)?

  1. 0.55
  2. 0.75
  3. 0.65 Correct answer.
  4. 0.35

Explanation

The complement rule states that P(not A) = 1 − P(A). Therefore, P(not A) = 1 − 0.35 = 0.65.

Question 31Technical Mathematics

A fair six-sided die is rolled once. What is the probability of rolling an even number?

  1. 1/3
  2. 1/2 Correct answer.
  3. 1/6
  4. 2/3

Explanation

The favorable outcomes are 2, 4, and 6, giving 3 favorable outcomes out of 6 equally likely outcomes. Thus the probability is 3/6 = 1/2.

Question 32Technical Mathematics

Which measure of central tendency is least affected by a single extreme outlier?

  1. Median Correct answer.
  2. Range
  3. Mean
  4. Standard deviation

Explanation

The median depends on the position of the middle value rather than the magnitude of every observation. An extreme outlier can move the mean and inflate the range or standard deviation, but it usually has little effect on the median.

Question 33Technical Mathematics

The number of nonconforming units found in each inspected lot is best classified as which type of variable?

  1. Categorical
  2. Continuous
  3. Ordinal
  4. Discrete Correct answer.

Explanation

A count of nonconforming units can take only whole-number values such as 0, 1, 2, and so on. Such count data are discrete numerical measurements.

Question 34Technical Mathematics

For a binomial process with n = 5 trials and p = 0.20, what is the probability of exactly 2 successes?

  1. 0.2048 Correct answer.
  2. 0.4096
  3. 0.0512
  4. 0.5120

Explanation

The binomial probability is C(5,2) × (0.20)^2 × (0.80)^3. This equals 10 × 0.04 × 0.512 = 0.2048.

Question 35Technical Mathematics

What z-value corresponds most closely to the 95th percentile of the standard normal distribution?

  1. 1.960
  2. 1.645 Correct answer.
  3. 2.326
  4. 1.282

Explanation

The 95th percentile leaves 5% of the area in the upper tail, which corresponds to z ≈ 1.645. This value is commonly used for one-sided 95% normal calculations.

Question 36Technical Mathematics

An R chart uses subgroups of n = 5. The average range is Rbar = 4.0, and for n = 5 the constant D4 = 2.114. What is the upper control limit for the R chart?

  1. 4.000
  2. 0.000
  3. 10.570
  4. 8.456 Correct answer.

Explanation

The upper control limit for an R chart is UCL_R = D4 × Rbar. Substituting D4 = 2.114 and Rbar = 4.0 gives 2.114 × 4.0 = 8.456.

Question 37Technical Mathematics

A c chart monitors the number of defects per unit. The average number of defects per unit is cbar = 9. What is the upper control limit?

  1. 15
  2. 9
  3. 18 Correct answer.
  4. 12

Explanation

For a c chart, the upper control limit is cbar + 3√cbar. With cbar = 9, this is 9 + 3√9 = 9 + 9 = 18.

Question 38Technical Mathematics

A process has specification limits LSL = 50 and USL = 70. The process standard deviation is estimated as 2.5. What is Cp?

  1. 1.00
  2. 1.33 Correct answer.
  3. 1.50
  4. 1.20

Explanation

Cp is the specification width divided by six standard deviations: Cp = (USL − LSL) / (6σ). Here (70 − 50) / (6 × 2.5) = 20 / 15 = 1.33.

Question 39Technical Mathematics

Sixteen repeated measurements give a sample standard deviation of 0.8 mm. What is the standard uncertainty of the mean (Type A) if the readings are treated as independent?

  1. 0.80 mm
  2. 3.20 mm
  3. 0.20 mm Correct answer.
  4. 0.05 mm

Explanation

The Type A standard uncertainty of the mean is s/√n. With s = 0.8 mm and n = 16, this is 0.8/4 = 0.20 mm.

Question 40Technical Mathematics

Two independent components are connected in series. Their reliabilities are 0.90 and 0.80. What is the reliability of the system?

  1. 0.72 Correct answer.
  2. 0.90
  3. 0.98
  4. 0.80

Explanation

For a series system, the system reliability is the product of the component reliabilities because all components must function. Thus 0.90 × 0.80 = 0.72.

Question 41Technical Mathematics

In acceptance sampling, producer's risk is the probability of:

  1. rejecting a lot whose quality is at the AQL Correct answer.
  2. accepting a lot whose quality is worse than the LTPD
  3. accepting a lot whose quality is at the AQL
  4. rejecting a lot whose quality is worse than the LTPD

Explanation

Producer's risk is the chance that a lot of acceptable quality, typically at the AQL, is rejected by the sampling plan. Consumer's risk is the chance of accepting a lot that is worse than the LTPD.

Question 42Technical Mathematics

A simple linear regression has R² = 0.81. Which statement best interprets this value?

  1. The fitted slope equals 0.81
  2. 81% of future observations will be predicted exactly
  3. 81% of response variation is explained by the model Correct answer.
  4. The correlation coefficient equals 0.81

Explanation

R² is the proportion of total variation in the response that is explained by the regression model. An R² of 0.81 therefore means about 81% of the response variation is explained by the fitted relationship.

Question 43Technical Mathematics

A hypothesis test produces a p-value of 0.03. Using a significance level of α = 0.05, what is the appropriate decision?

  1. Accept the null hypothesis as true
  2. Fail to reject the null hypothesis
  3. Reject the null hypothesis Correct answer.
  4. Increase the sample size before deciding

Explanation

When the p-value is less than the chosen significance level, the null hypothesis is rejected. Since 0.03 < 0.05, the decision is to reject the null hypothesis.

Question 44Technical Mathematics

A 95% confidence interval for a process mean is calculated as 48.2 to 51.8. Which is the correct frequentist interpretation?

  1. About 95% of similarly constructed intervals contain the true mean Correct answer.
  2. 95% of individual process observations fall between the limits
  3. There is a 95% chance the true mean is between the limits
  4. The sample mean lies between the limits with 95% probability

Explanation

A confidence interval is interpreted in terms of the long-run performance of the procedure, not the probability of a fixed parameter lying in one interval. About 95% of intervals constructed the same way would contain the true mean.

Question 45Technical Mathematics

On a control chart, seven consecutive points fall on the same side of the center line. This pattern most strongly indicates:

  1. A stratified process
  2. A measurement resolution problem
  3. A process shift Correct answer.
  4. Common-cause variation only

Explanation

A run of seven points on one side of the center line is a standard signal of a sustained process shift. Such a run is unlikely under stable common-cause variation alone.

Question 46Technical Mathematics

A study plans to estimate a proportion with 95% confidence (z = 1.96) and a margin of error of 0.05. Using p = 0.5 as a conservative planning value, what sample size is required?

  1. 400
  2. 385 Correct answer.
  3. 384
  4. 271

Explanation

The planning formula is n = (z/E)² × p(1 − p). Substituting gives (1.96/0.05)² × 0.25 = 384.16, which must be rounded up to 385.

Question 47Technical Mathematics

A one-way ANOVA compares 4 treatment levels with 25 total observations. What is the within-groups (error) degrees of freedom?

  1. 25
  2. 21 Correct answer.
  3. 24
  4. 3

Explanation

In a one-way ANOVA, error degrees of freedom equal N − k, where N is the total number of observations and k is the number of groups. Here 25 − 4 = 21.

Question 48Technical Mathematics

A full factorial experiment has 3 factors, each at 2 levels, and each treatment combination is replicated twice. How many experimental runs are required?

  1. 12
  2. 8
  3. 16 Correct answer.
  4. 32

Explanation

A 2³ full factorial has 2 × 2 × 2 = 8 treatment combinations. With two replicates per combination, the number of runs is 8 × 2 = 16.

Question 49Technical Mathematics

In a Weibull reliability model, a shape parameter β greater than 1 indicates that the failure rate is:

  1. Undefined over time
  2. Decreasing over time
  3. Constant over time
  4. Increasing over time Correct answer.

Explanation

The Weibull shape parameter determines the behavior of the hazard function. When β > 1, the failure rate increases with time; when β = 1 it is constant, and when β < 1 it decreases.

Question 50Technical Mathematics

A control chart has probability 0.20 of detecting a particular shift on any single sample. What is the average run length (ARL) to detection after the shift occurs?

  1. 2
  2. 20
  3. 5 Correct answer.
  4. 10

Explanation

The average run length to detection is the reciprocal of the per-sample detection probability. With detection probability 0.20, ARL = 1/0.20 = 5 samples.