🧰Engineering Applications of Statistics Unit 1 – Intro to Probability & Statistics

Key Concepts and Definitions

• Probability quantifies the likelihood an event will occur expressed as a number between 0 and 1
• Statistics involves collecting, analyzing, interpreting, and presenting data to make informed decisions
  • Descriptive statistics summarize and describe the main features of a data set (measures of central tendency, variability)
  • Inferential statistics use sample data to make predictions or draw conclusions about a larger population
• Variables can be classified as discrete (distinct, separate values like counts) or continuous (any value within a range)
• Probability distributions describe the likelihood of different outcomes for a random variable
  • Common discrete distributions include binomial, Poisson, and geometric
  • Common continuous distributions include normal (bell curve), exponential, and uniform
• Hypothesis testing evaluates claims or assumptions about a population parameter using sample data
  • Null hypothesis ($H_0$) represents the default or status quo position
  • Alternative hypothesis ($H_a$ or $H_1$) represents the claim being tested

Probability Fundamentals

• Probability of an event (A) is denoted as P(A) and ranges from 0 (impossible) to 1 (certain)
• Complementary events have probabilities that sum to 1 P(A) + P(A') = 1
• Mutually exclusive events cannot occur simultaneously P(A and B) = 0
• Independent events do not influence each other's probability P(A|B) = P(A)
• Conditional probability P(A|B) measures the likelihood of event A occurring given that event B has occurred
• Bayes' theorem relates conditional probabilities P(A|B) = P(B|A) * P(A) / P(B)
• Law of total probability states P(A) = P(A|B) * P(B) + P(A|B') * P(B')
• Expected value (mean) of a discrete random variable X is E(X) = Σ[x * P(X=x)]

Types of Data and Distributions

• Nominal data consists of categories with no inherent order (colors, gender)
• Ordinal data has categories with a meaningful order but no consistent scale (rankings, survey responses)
• Interval data has ordered categories with consistent scale but no true zero (temperature in Celsius or Fahrenheit)
• Ratio data has ordered categories, consistent scale, and true zero (height, weight, temperature in Kelvin)
• Normal distribution is symmetric and bell-shaped described by mean μ and standard deviation σ
  • Empirical rule (68-95-99.7%) states the percentage of values within 1, 2, and 3 standard deviations of the mean
  • Standard normal distribution (z-distribution) has μ=0 and σ=1
• Binomial distribution models the number of successes in a fixed number of independent trials with constant probability
• Poisson distribution models the number of rare events occurring in a fixed interval of time or space

Descriptive Statistics

• Measures of central tendency describe the center or typical value of a dataset
  • Mean (average) is sensitive to extreme values and best for symmetric distributions
  • Median (middle value) is resistant to outliers and best for skewed distributions
  • Mode (most frequent value) is used for categorical or discrete data
• Measures of variability describe the spread or dispersion of a dataset
  • Range is the difference between the maximum and minimum values
  • Variance is the average squared deviation from the mean $s^2 = Σ(x_i - x̄)^2 / (n-1)$
  • Standard deviation is the square root of variance and measures typical distance from the mean
• Skewness measures the asymmetry of a distribution (positive skew has a long right tail, negative skew has a long left tail)
• Kurtosis measures the heaviness of the tails relative to a normal distribution (high kurtosis has heavy tails, low kurtosis has light tails)
• Percentiles and quartiles divide a dataset into equal parts (25th percentile is the first quartile Q1, 50th percentile is the median)

Inferential Statistics

• Population refers to the entire group of interest while a sample is a subset of the population
• Parameter is a numerical summary of a population (μ, σ) while a statistic is a numerical summary of a sample (x̄, s)
• Sampling error is the difference between a sample statistic and the corresponding population parameter
• Sampling distributions describe the variability of a sample statistic over many samples
  • Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as sample size increases, regardless of the population distribution shape
• Confidence intervals estimate a population parameter using sample data and provide a range of plausible values
  • Confidence level (e.g., 95%) indicates the long-run proportion of intervals that will contain the true parameter value
  • Margin of error determines the width of the interval and decreases with larger sample sizes
• Hypothesis tests use sample evidence to make decisions about population parameters
  • p-value measures the strength of evidence against the null hypothesis (smaller p-values provide stronger evidence)
  • Significance level α is the threshold for rejecting the null hypothesis (common levels are 0.01, 0.05, 0.10)

Hypothesis Testing

• Null hypothesis ($H_0$) represents the status quo or default position (often a statement of equality)
• Alternative hypothesis ($H_a$ or $H_1$) represents the claim being tested (often a statement of inequality)
• One-tailed tests have a directional alternative hypothesis (< or >)
• Two-tailed tests have a non-directional alternative hypothesis (≠)
• Type I error (false positive) occurs when rejecting a true null hypothesis
  • Significance level α controls the probability of a Type I error
• Type II error (false negative) occurs when failing to reject a false null hypothesis
  • Power (1-β) is the probability of correctly rejecting a false null hypothesis
• Test statistic (e.g., z, t, F) measures the difference between the sample statistic and the null hypothesis value in standardized units
• Rejection region (critical region) contains the test statistic values that lead to rejecting the null hypothesis
• p-value is the probability of observing a test statistic as extreme or more extreme than the actual result, assuming the null hypothesis is true

Statistical Software and Tools

• Spreadsheet programs (Microsoft Excel, Google Sheets) can perform basic statistical analyses and create charts
• Statistical software packages provide more advanced capabilities
  • R is a free, open-source programming language and environment for statistical computing and graphics
  • Python is a general-purpose programming language with libraries for data analysis (NumPy, SciPy, Pandas)
  • SAS (Statistical Analysis System) is a proprietary software suite for advanced analytics, business intelligence, and predictive modeling
  • SPSS (Statistical Package for the Social Sciences) is a proprietary software package used for interactive statistical analysis
• Online calculators and web applets can perform specific statistical tests and calculations
• Data visualization tools (Tableau, PowerBI, Plotly) create interactive dashboards and explore data graphically

Real-World Engineering Applications

• Quality control uses statistical process control (SPC) charts to monitor manufacturing processes and detect defects
  • Control charts (x̄, R, s, p, np, c, u) track process stability over time and identify out-of-control conditions
  • Process capability indices (Cp, Cpk) measure the ability of a process to meet specifications
• Reliability engineering assesses the probability and consequences of system failures
  • Reliability is the probability a system will perform its intended function under specified conditions for a specified period
  • Failure rate is the frequency with which a system fails, often modeled using the exponential distribution
  • Mean time between failures (MTBF) and mean time to repair (MTTR) are key metrics for maintainability
• Design of experiments (DOE) optimizes product and process designs by systematically varying input factors
  • Factorial designs investigate the effects of multiple factors simultaneously
  • Response surface methodology (RSM) builds empirical models to find optimal factor settings
• Simulation and risk analysis use probability distributions to model uncertain inputs and outcomes
  • Monte Carlo simulation generates random samples from input distributions to estimate the distribution of an output variable
  • Sensitivity analysis determines which inputs have the greatest influence on the output
• Forecasting and time series analysis predict future values based on historical patterns
  • Moving averages and exponential smoothing are used for short-term forecasts
  • Autoregressive integrated moving average (ARIMA) models are used for longer-term forecasts