Book summary “Applied statistics and probability for engineers”: summary

Introduction

Engineering problems rarely feature predictable outcomes and complete certainty. You face raw materials that vary, manufacturing processes that drift, and measurements that contain error.

Engineering problem solving requires statistical thinking and rigorous probabilistic modeling to understand and control variability in physical systems. You must move beyond deterministic assumptions to capture real-world uncertainty.

The book provides a systematic framework to collect data, build models, and test hypotheses. It covers probability foundations, random variables, statistical inference, regression, experimental design, and quality control.

• How do you model random phenomena and quantify risk?
• What methods extract reliable parameters from noisy samples?
• How do you optimize processes and isolate sources of variation?

1. The Role of Statistics in Engineering

The engineering method is a structured, iterative approach to formulating and solving problems faced by society. You use this sequence to refine existing products or processes, or design new ones that meet customers' needs.

The engineering method is a structured, iterative approach to formulating and solving problems faced by society.

You develop a solution by following specific steps:

• Develop a clear and concise description of the problem.
• Identify, at least tentatively, the important factors that affect this problem or that may play a role in its solution.
• Propose a model for the problem, using scientific or engineering knowledge of the phenomenon being studied.
• Conduct appropriate experiments and collect data to test or validate the tentative model or conclusions.
• Refine the model on the basis of the observed data.
• Manipulate the model to assist in developing a solution to the problem.
• Conduct an appropriate experiment to confirm that the proposed solution is both effective and efficient.
• Draw conclusions or make recommendations based on the problem solution.

Collecting Engineering Data

Observed engineering measurements always exhibit variability, which can be modeled mathematically as a constant plus a random disturbance. Successive observations do not produce identical results due to small environmental changes, test equipment differences, and part variations.

Observed engineering measurements always exhibit variability, which can be modeled mathematically as a constant plus a random disturbance.

Eight prototype nylon connectors measured for pull-off force yielded varying results such as 12.6, 12.9, 13.4, 12.3, 13.6, 13.5, 12.6, and 13.1 pounds. The initial wall thickness specification for the nylon connector was 3/32 inch, while an alternative specification was 1/8 inch.

You collect data through three primary methods: retrospective studies, observational studies, and designed experiments. Retrospective studies use historical data but often suffer from missing information, unrecorded factors, and narrow operating ranges. Observational studies disturb the process as little as possible over a short duration. Designed experiments actively manipulate controllable variables to determine their impact on system output.

Analyzing historical logs of an acetone-butyl alcohol distillation column is a retrospective study. Actively setting reboil temperatures, condensate temperatures, and reflux rates forms a designed experiment.

Plotting data over time helps judge process stability, and making adjustments based on random disturbances can actually increase process variation. Time series plots reveal shifts in process mean levels that dot diagrams alone might hide. W. Edwards Deming demonstrated that trying to compensate for every random error in a process actually moves the system further from the target.

Deming dropped marbles through a funnel onto a target. Strategy 1 left the funnel fixed, while Strategy 2 adjusted the funnel position after every drop to compensate for previous errors. Strategy 2 resulted in a variability roughly 2 times larger than Strategy 1.

2. Probability Foundations

Probability models quantify risk and uncertainty by defining sample spaces, events, set operations, conditional probabilities, and independence.

A random experiment is any procedure that can result in different outcomes even when repeated identically. The set of all possible outcomes defines the sample space. Sample spaces can be discrete, consisting of finite or countably infinite outcomes, or continuous, containing an interval of real numbers. The appropriate choice of a sample space depends entirely on the specific objectives of your engineering analysis.

Measuring the thickness of a molded plastic connector can use a sample space of the positive real line, an interval between 10 and 11 millimeters, discrete categories like low, medium, and high, or a binary conformance check of yes and no.

Events are subsets of a sample space. You can construct complex events using set operations such as unions, intersections, and complements. Set operations obey specific mathematical laws including commutative, distributive, and DeMorgan's laws. Two events that share no common outcomes are designated as mutually exclusive.

Analyzing 50 polycarbonate plastic samples for scratch and shock resistance yields joint event frequencies where high shock and high scratch resistance intersect in 40 samples.

Every probability model must satisfy three fundamental axioms governing sample spaces and mutually exclusive events. Probability is a number assigned to each event in a random experiment sample space. The three axioms require that the probability of the entire sample space is 1, the probability of any event lies between 0 and 1, and the probability of the union of two mutually exclusive events is the sum of their individual probabilities. If the probability of event E is 0.4, the complement E prime has a probability of 0.6.

Sample Spaces and Events

The general addition rule calculates the probability of the union of two events by summing their individual probabilities and subtracting their intersection. When determining the probability that event A or event B occurs, the intersection of the two events is added twice if you simply sum P(A) and P(B). Therefore, the intersection probability must be subtracted to correct the total sum.

In a study of 940 semiconductor manufacturing wafers, the probability of high contamination or center location is calculated by adding individual probabilities and subtracting the overlapping center-high contamination wafers.

To calculate union probabilities, you can follow these practical steps:

• Identify the individual probabilities P(A) and P(B).
• Find the intersection probability P(A intersection B).
• Apply the formula P(A union B) = P(A) + P(B) - P(A intersection B).

Probability Rules and Bayes Theorem

Conditional probability adjusts the likelihood of an event based on the knowledge that another event has occurred. The conditional probability of event B given event A is defined as the probability of their intersection divided by the probability of A, provided that the probability of A is greater than zero. It reflects a reevaluation of probabilities under a new state of knowledge. For 400 parts classified by surface flaws and functional defects, the probability of a defect given a surface flaw is 10 divided by 40, whereas the unconditional defect probability is 28 divided by 400.

You compute conditional probabilities using this sequence:

• Determine the probability of event A occurring.
• Determine the probability of the intersection of events A and B.
• Divide the intersection probability by P(A) to find P(B given A).

The total probability rule breaks down the probability of an event across a set of mutually exclusive and exhaustive subsets. When an event can occur under several different conditions that partition the sample space, its overall probability is computed as the sum of the joint probabilities of the event with each condition, expressed as a weighted average. A semiconductor product failure probability is calculated using chips exposed to high, medium, and low contamination levels, weighted by their respective proportions of 20%, 30%, and 50%.

To find total probabilities, you apply these steps:

• Identify a collection of mutually exclusive and exhaustive sets that partition the sample space.
• Compute the conditional probability of the target event given each set.
• Multiply each conditional probability by the probability of the corresponding set.
• Sum all products to obtain the total probability.

Bayes' theorem inverts conditional probabilities, allowing the calculation of a cause given an observed outcome. By utilizing the definition of conditional probability and the total probability rule, Bayes' theorem computes the probability of a prior event or condition given that a subsequent event has been observed. In medical screening, despite a test having 99% positive identification accuracy and 95% negative identification accuracy, a positive test result yields only a 0.002 probability of actually having a rare illness due to its low population incidence of 0.0001.

You can apply Bayes' theorem with these steps:

• Identify the prior probabilities of the mutually exclusive conditions.
• Identify the conditional probabilities of the observed outcome given each condition.
• Apply Bayes' formula by dividing the specific joint probability by the total probability of the observed outcome.

The multiplication rule allows the calculation of the probability of the intersection of two events using conditional probabilities. By rewriting the definition of conditional probability, the intersection of event A and event B can be computed as the conditional probability of B given A multiplied by the probability of A. The probability that an automobile battery has low charging current and is subject to high engine compartment temperature is calculated as 0.7 multiplied by 0.05.

You execute the multiplication rule through these steps:

• Identify the unconditional probability of the first event P(A).
• Identify the conditional probability of the second event given the first P(B given A).
• Multiply these values to get the intersection probability P(A intersection B).

Two events are independent when the knowledge that one event has occurred does not affect the probability of the other. Mathematical independence is defined equivalently by statements where the conditional probability equals the unconditional probability, or where the joint probability equals the product of the individual probabilities. Analyzing polycarbonate plastic disks or semiconductor wafers where defect rates do not change based on the presence of secondary characteristics like scratch resistance illustrates this property.

You can verify independence by following these steps:

• Check if P(A given B) equals P(A), or if P(A intersection B) equals P(A) times P(B).
• If true, treat the events as independent for probability calculations.

3. Random Variables and Probability Distributions

Random variables abstract experimental outcomes into mathematical functions, categorized by discrete and continuous probability distributions.

A random variable is a mathematical function that assigns a real number to each outcome in the sample space of a random experiment. Random variables abstract outcomes into numerical values, categorized as discrete when their range is finite or countably infinite, or continuous when their range encompasses an interval of real numbers.

You can work with random variables by following these steps:

• Define the random experiment and its sample space.
• Assign numerical values to outcomes based on the variable of interest.
• Classify the random variable as discrete or continuous depending on whether its range is countable or an interval.

The number of lines in use in a voice communication system with 48 external lines represents a discrete random variable ranging from 0 to 48.

Discrete Random Variables

The mean and variance of a discrete random variable summarize the center and dispersion of its probability distribution.

The mean or expected value E(X) is the weighted average of possible values using probabilities as weights, acting as the balance point of the distribution. The variance V(X) measures dispersion using squared deviations from the mean, and its square root is the standard deviation.

For a digital transmission channel where X is the number of bits in error in four transmissions with probabilities ranging from P(X=0)=0.6561 to P(X=4)=0.0001, the expected value is E(X)=0.4 and the variance is V(X)=0.36.

The binomial distribution models the number of successes in a series of independent Bernoulli trials with a constant probability of success. A Bernoulli trial has two possible outcomes labeled success and failure. The binomial probability mass function is calculated using combinations, and its mean and variance are given by np and np(1-p) respectively.

When a bit transmitted through a channel has a 0.1 probability of error over 4 transmissions, the probability of exactly 2 errors is P(X=2)=0.0486.

The hypergeometric distribution models the number of successes in a sample selected randomly without replacement from a finite population. Unlike the binomial distribution, trials are not independent because sampling is done without replacement. Its variance includes a finite population correction factor.

When four parts are selected without replacement from a batch containing 100 parts from a local supplier and 200 from another, the probability that all four are from the local supplier is P(X=4)=0.0119.

The Poisson distribution expresses the probability of a given number of events occurring in a fixed interval of time or space. It arises from a Poisson process where counts occur randomly, independently, and proportionally to the interval length. For a Poisson random variable, both the mean and the variance are equal to the parameter lambda.

If flaws in a thin copper wire follow a Poisson distribution with a mean of 2.3 flaws per millimeter, the probability of exactly 2 flaws in 1 millimeter is 0.265.

Continuous Random Variables

The mean and variance of a continuous random variable are calculated using integration instead of summation.

The expected value or mean of X is the integral of x multiplied by its probability density function over its range. The variance is the integral of the squared deviation from the mean multiplied by the probability density function.

For the copper current measurement in Example 4-6, the mean of X is 10 milliamperes and the variance is 33.33 milliamperes squared.

The normal distribution is a symmetric bell-shaped probability distribution characterized by its mean and variance. A normal random variable is defined by its parameters mu and sigma squared. Because its probability density function lacks a closed-form integral, probabilities are computed numerically or via standard normal distribution tables using standardization.

In Example 4-10, current measurements follow a normal distribution with a mean of 10 milliamperes and a variance of 4 milliamperes squared. Calculations show a 0.6827 probability within plus or minus 1 standard deviation, a 0.9545 probability within plus or minus 2 standard deviations, and a 0.9973 probability within plus or minus 3 standard deviations.

The exponential distribution models the distance or time between successive counts in a Poisson process and possesses a unique lack of memory property. The exponential random variable is defined by parameter lambda, with a mean of 1 over lambda and a variance of 1 over lambda squared. Its lack of memory property means that the conditional probability of waiting additional time is independent of how much time has already passed.

In Example 4-21, corporate computer network user log-ons follow a Poisson process with a mean of 25 log-ons per hour, modeled by an exponential distribution over a 6 minute time interval.

4. Joint Probability Distributions

Multiple random variables acting simultaneously are analyzed using joint probability mass or density functions, covariance, correlation, and linear combinations.

Joint Distributions and Independence

You define bivariate probability distributions when two or more random variables operate together. The joint probability mass function determines the probability of each simultaneous point in the range of discrete random variables.

For discrete random variables $X$ and $Y$, the joint probability mass function $f_{XY}(x,y)$ satisfies specific conditions:

• it is greater than or equal to zero for all points;
• the sum of probabilities over all points equals one;
• it equals the joint probability that $X$ equals $x$ and $Y$ equals $y$.

In a digital information receiver experiment, bits are classified as acceptable, suspect, or unacceptable with probabilities $0.9$, $0.08$, and $0.02$. The joint probability distribution determines the simultaneous number of acceptable and suspect bits in four transmitted bits.

Two random variables are independent if their joint probability mass function equals the product of their marginal probability mass functions for all values. Knowledge of one variable does not change the probabilities associated with the values of the other variable. If the set of points with positive probability does not form a rectangular range, the variables cannot be independent.

Parts classified by color and length specifications in a plastic molding operation demonstrate independence when joint probabilities equal the products of marginal probabilities. In this large shipment example, $1\%$ of parts do not conform to specifications.

The joint probability density function for continuous random variables determines probabilities by integrating over regions in multidimensional space. The probability that continuous random variables assume values in a region $R$ is equal to the double or multiple integral of the joint probability density function over that region, interpreted geometrically as the volume under a surface.

Computer server connection time $X$ and authorization time $Y$ are modeled with a joint exponential density function using $6 \times 10^6$ as the scaling multiplier. You find the probability of specific wait times by integrating over the appropriate region.

Multiple continuous random variables are defined as independent if and only if their joint probability density function equals the product of their individual marginal probability density functions for all points in their range. If the equality fails for even a single point, the random variables are not independent.

The thicknesses of a substrate, active layer, and coating layer are analyzed for a chemical product. By utilizing the independence of these normally distributed layers with specified means and standard deviations, the proportion of products meeting all thickness specifications is calculated as $0.7778$.

Covariance and Linear Combinations

Correlation scales the covariance by the standard deviations of each variable, creating a dimensionless quantity bounded between $-1$ and $1$ that measures the strength of the linear relationship. The correlation coefficient $\rho_{XY}$ is defined as the covariance divided by the product of the standard deviations of $X$ and $Y$. Values near $1$ or $-1$ indicate that points fall along a straight line. Independent variables always have a correlation of zero, though zero correlation does not strictly imply independence in general.

Calculations for a discrete joint distribution yield an expected product $E(XY)$ of $4.5$, a variance $V(X)$ of $1.36$, a covariance of $1.26$, and a correlation coefficient $\rho_{XY}$ of $0.926$.

The mean of a linear combination of random variables is the linear combination of their individual means, while the variance includes terms for the covariance of every pair unless the variables are independent. Given a linear combination, the expected value is the linear combination of expected values. The variance incorporates all pairwise covariances multiplied by their respective constants, which simplifies to a sum of squared constants times individual variances when the random variables are independent.

The perimeter of a part given by $Y = X_1 + 2X_2$ has a mean of $14$ centimeters and a standard deviation of $0.4$ centimeters, calculated using individual means, standard deviations, and a covariance of $0.005$.

5. Data Summary and Point Estimation

Observational data are summarized using descriptive statistics and graphical displays, while point estimation methods provide parameter values.

Data Summary and Display

Sample statistics such as the mean, variance, and standard deviation summarize the location and dispersion of observational data. The sample mean acts as a balance point for the data. The sample variance divides the sum of squared deviations by n - 1 to account for the degrees of freedom lost by using the sample average instead of the true population mean.

Calculating the sample mean of 13.0 pounds and sample variance of 0.2286 pounds squared from eight observations of prototype engine connector pull-off forces.

You can compute sample statistics using these steps:

• Sum all observations to compute the sample mean by dividing by n.
• Calculate deviations of each observation from the sample mean.
• Square the deviations, sum them, and divide by n - 1 to get the sample variance.
• Take the positive square root of the sample variance to find the sample standard deviation.

Box plots provide a compact graphical display of data center, spread, symmetry, and outliers using quartiles and whiskers. A rectangular box encloses the interquartile range with edges at the first and third quartiles, while a line inside the box marks the median. Whiskers extend from the box to the smallest and largest data points within 1.5 interquartile ranges, and points beyond are classified as outliers or extreme outliers.

Manufacturing quality index measurements on semiconductor devices at three plants revealed excessive variability at plant 2 and the need for plants 2 and 3 to raise performance.

Methods of Point Estimation

An estimator is unbiased if the expected value of its sampling distribution is exactly equal to the true population parameter being estimated.

When an estimator is unbiased, its bias is zero, meaning its distribution is centered precisely at the true parameter value. While unbiasedness is a desirable baseline property, multiple unbiased estimators can exist for the same parameter. Mathematical derivation confirms that both the sample mean and the sample variance are unbiased estimators of the population mean and population variance respectively.

The method of moments finds parameter estimators by equating population moments, defined in terms of expected values, to corresponding sample moments and solving the resulting equations. Population moments are functions of unknown distribution parameters. Equating the first m population moments to the first m sample moments yields moment estimators for the unknown parameters.

Using failure time data where x equals 21.65 hours, the moment estimate for the exponential distribution parameter lambda is 0.0462.

You can derive moment estimators with these steps:

• Define population moments as functions of unknown parameters in terms of expected values.
• Calculate the corresponding sample moments from the random sample.
• Equate the population moments to the sample moments.
• Solve the resulting equations to obtain the parameter estimators.

Maximum likelihood estimation yields parameter estimators by finding the parameter values that maximize the likelihood function of the observed sample. For discrete random variables, the likelihood function is the joint probability of obtaining the observed sample values. For continuous distributions, it is the product of the probability density functions. Maximum likelihood estimators are generally preferred over moment estimators due to superior efficiency and asymptotic properties.

For a normal distribution with unknown mean and known variance, the maximum likelihood estimator of the mean is the sample mean, identical to its moment estimator.

6. Statistical Inference for a Single Sample

A confidence interval provides an interval estimate of plausible values for an unknown population parameter with a specified high confidence coefficient.

For a normal population with a known variance, a 100(1 - alpha)% confidence interval on the mean is constructed using the sample mean, the known variance, the sample size, and percentage points of the standard normal distribution. You can follow these steps to build the interval:

• Obtain a random sample of size n from the normal population.
• Calculate the sample mean x from the sample data.
• Select the appropriate percentage point z of the standard normal distribution corresponding to the confidence coefficient 1 - alpha.
• Compute the lower and upper confidence limits by adding and subtracting the margin of error from the sample mean.

Ten measurements of impact energy on A238 steel specimens with a known standard deviation of 1 J yield a 95% confidence interval for the mean impact energy. The sample mean impact energy is 64.46 J, and the standard normal percentage point z_0.025 corresponding to a 95% confidence interval is 1.96.

Hypothesis Testing Foundations

Testing statistical hypotheses can lead to type I errors (rejecting a true null hypothesis) and type II errors (failing to reject a false null hypothesis), which are inversely related.

A type I error probability is denoted by alpha and represents the significance level or size of the test. A type II error probability is denoted by beta and depends on the specific alternative hypothesis and sample size. Decreasing one error probability generally increases the other unless the sample size is increased.

In testing a propellant burning rate where H0 is 50 centimeters per second, an observed mean outside the critical region 48.5 to 51.5 when the true mean is 50 creates a type I error. A sample mean inside that region when the true mean is 52 creates a type II error. The probability of type I error for n equal to 10 with critical limits at 48.5 and 51.5 is 0.057434, whereas the probability of type II error when the true mean is 52 and n is 10 is 0.2643.

7. Statistical Inference for Two Samples and Regression

Two-sample inference procedures compare means, variances, and proportions, while regression analysis builds empirical models for relationships between variables.

When you compare the means of two independent normal populations with unknown but equal variances, a pooled t-test is used. You apply this test following specific operational steps:

• State the parameters of interest as 1 and 2.
• Formulate the null hypothesis H0: 1 - 2 = 0.
• Choose the alternative hypothesis H1.
• Select the significance level alpha.
• Define the test statistic T0 based on sample means and pooled standard deviation.
• Establish the rejection region using t-distribution critical values with n1 + n2 - 2 degrees of freedom.
• Compute the test statistic from the sample data.
• Reach a conclusion by comparing the test statistic to the critical region or using a P-value.

In Example 10-5, two chemical process catalysts were compared, resulting in a pooled standard deviation of 2.70 and a t-value of 0.35, leading to a failure to reject the null hypothesis.

When observations from two populations are collected in homogeneous pairs, a paired t-test analyzes the differences to eliminate extraneous variability. You execute this procedure through the following sequence:

• Collect data in homogeneous pairs (X1j, X2j).
• Calculate the difference Dj = X1j - X2j for each pair.
• Compute the sample average difference d and sample standard deviation of the differences sD.
• Define the null hypothesis H0: mu_D = 0.
• Calculate the test statistic t0 = d / (sD / sqrt(n)).
• Compare t0 to the critical t-distribution value with n - 1 degrees of freedom.

In Example 10-9, shear strength prediction methods (Karlsruhe versus Lehigh) were compared across nine steel plate girders, yielding a paired t-statistic of 6.05.

Hypothesis tests on the equality of two population variances rely on the F-distribution, defined as the ratio of two independent chi-square variables divided by their degrees of freedom. You follow these steps to conduct the test:

• Assume both populations are normally distributed.
• Formulate the null hypothesis H0: sigma_1^2 = sigma_2^2.
• Calculate sample variances S1^2 and S2^2.
• Compute the test statistic F0 = S1^2 / S2^2.
• Compare F0 against critical values from the F-distribution table with n1 - 1 numerator and n2 - 1 denominator degrees of freedom.

In Example 10-11, oxide layer thicknesses on semiconductor wafers for two gas mixtures were compared using 20 wafers each, resulting in an F test statistic of 0.85.

Simple Linear Regression

Karl Gauss's method of least squares estimates regression parameters by minimizing the sum of the squares of the vertical deviations of observations from the estimated regression line.

You build the fitted regression model by executing these computational tasks:

• Calculate the sample means for x and y.
• Compute the sum of squares of the regressor (Sxx) and the sum of cross-products (Sxy).
• Determine the slope estimator beta_1_hat by dividing Sxy by Sxx.
• Determine the intercept estimator beta_0_hat by subtracting beta_1_hat times x_mean from y_mean.
• Construct the fitted regression line equation y_hat = beta_0_hat + beta_1_hat * x.

Fitting a simple linear regression model to the oxygen purity data yielded a sum of hydrocarbon levels equal to 23.92, a sum of purity values equal to 1,843.21, and a fitted model equation of y_hat = 74.283 + 14.947x.

8. Design of Experiments and Statistical Process Control

Designed experiments isolate sources of variation and optimize processes, while statistical quality control charts monitor operational stability.

Design of Experiments

Designed experiments allow engineers to test a single factor across multiple levels while systematically isolating and controlling experimental variability. By randomizing experimental runs, you approximately balance out the effect of nuisance variables. You execute designed experiments sequentially through conjecture, experimentation, analysis, and conclusion.

• Formulate the original conjecture or hypothesis.
• Perform the test experiment.
• Conduct the statistical analysis of the data.
• Formulate conclusions based on the findings.

A civil engineer investigates five curing methods on concrete strength, and a paper manufacturer studies four hardwood concentrations ranging from 5% to 20% to improve tensile strength.

Statistical Quality Control

Control charts distinguish between natural background variability and assignable causes, serving as an online process-monitoring and problem-solving tool. A typical control chart uses a center line and upper and lower control limits typically set at 3-sigma distances. You plot points outside limits or exhibiting nonrandom patterns to indicate that the process is out of control and requires investigation.

• Collect sample data periodically according to the rational subgroup concept.
• Compute the sample statistic and plot it on the chart against the sample number or time.
• Check for points beyond the control limits or nonrandom patterns.
• Investigate and eliminate assignable causes when an out-of-control signal occurs.

Dr. Walter A. Shewhart of Bell Telephone Laboratories wrote a memorandum in 1924 showing a modern control chart.

10 Key Ideas

1. Observed engineering measurements always exhibit variability, which can be modeled mathematically as a constant plus a random disturbance.
2. The engineering method is a structured, iterative approach to formulating and solving problems faced by society.
3. Probability models quantify risk and uncertainty by defining sample spaces, events, set operations, conditional probabilities, and independence.
4. Random variables abstract experimental outcomes into mathematical functions, categorized by discrete and continuous probability distributions.
5. Multiple random variables acting simultaneously are analyzed using joint probability mass or density functions, covariance, correlation, and linear combinations.
6. An estimator is unbiased if the expected value of its sampling distribution is exactly equal to the true population parameter being estimated.
7. A confidence interval provides an interval estimate of plausible values for an unknown population parameter with a specified high confidence coefficient.
8. Testing statistical hypotheses can lead to type I errors (rejecting a true null hypothesis) and type II errors (failing to reject a false null hypothesis), which are inversely related.
9. Karl Gauss's method of least squares estimates regression parameters by minimizing the sum of the squares of the vertical deviations of observations from the estimated regression line.
10. Designed experiments isolate sources of variation and optimize processes, while statistical quality control charts monitor operational stability.