📊Intro to Business Analytics Unit 3 – Probability Distributions & Sampling

Key Concepts

• Probability distributions describe the likelihood of different outcomes in a random experiment or process
• Random variables can be discrete (distinct values) or continuous (any value within a range)
• Probability density functions (PDFs) and cumulative distribution functions (CDFs) mathematically define distributions
• Expected value represents the average outcome of a random variable over many trials
• Variance and standard deviation measure the spread or dispersion of a distribution
  • Higher variance indicates greater variability in the possible outcomes
• Central Limit Theorem states that the distribution of sample means approximates a normal distribution as the sample size increases, regardless of the shape of the population distribution

Types of Probability Distributions

• Normal (Gaussian) distribution is symmetric and bell-shaped, characterized by its mean and standard deviation
  • Useful for modeling many natural phenomena and averages of large samples
• Binomial distribution describes the number of successes in a fixed number of independent trials with two possible outcomes (success or failure)
• Poisson distribution models the number of events occurring in a fixed interval of time or space, given a known average rate
• Exponential distribution represents the time between events in a Poisson process
• Uniform distribution has constant probability over a defined range and zero probability outside that range
• Other notable distributions include Beta, Gamma, and Chi-square, each with specific applications

Measures of Central Tendency and Dispersion

• Mean is the arithmetic average of a dataset, calculated by summing all values and dividing by the number of observations
• Median is the middle value when a dataset is ordered from lowest to highest
  • Robust to outliers and useful for skewed distributions
• Mode is the most frequently occurring value in a dataset
• Range is the difference between the maximum and minimum values
• Interquartile range (IQR) is the difference between the 75th and 25th percentiles, representing the middle 50% of the data
• Variance measures the average squared deviation from the mean, indicating how far values typically are from the average
• Standard deviation is the square root of the variance, expressed in the same units as the original data

Sampling Techniques

• Simple random sampling selects a subset of individuals from a population such that each individual has an equal probability of being chosen
• Stratified sampling divides the population into subgroups (strata) based on a specific characteristic and then randomly samples from each stratum
  • Ensures representation of key subgroups in the sample
• Cluster sampling divides the population into clusters, randomly selects a subset of clusters, and then samples all individuals within those clusters
  • Useful when a complete list of the population is not available or when clusters naturally occur (geographic regions)
• Systematic sampling selects individuals at regular intervals from a population list
• Convenience sampling selects individuals who are easily accessible or willing to participate, but may introduce bias

Probability Distribution Applications

• Quality control uses normal distribution to set acceptable ranges for product specifications
• Finance employs various distributions to model asset returns, portfolio risk, and option pricing
• Marketing may use binomial distribution to analyze the success of a campaign or product launch
• Operations management can apply Poisson distribution to model the number of customer arrivals or machine failures in a given time period
• Exponential distribution is often used to model waiting times, such as customer service call durations or equipment failure rates

Common Statistical Tests

• Z-test compares a sample mean to a population mean when the population standard deviation is known
• T-test compares means between two groups or against a hypothesized value when the population standard deviation is unknown
• ANOVA (Analysis of Variance) tests for differences among three or more group means
• Chi-square test assesses the association between two categorical variables
• Regression analysis examines the relationship between a dependent variable and one or more independent variables
  • Linear regression assumes a linear relationship between variables
  • Logistic regression predicts binary outcomes based on predictor variables

Data Visualization for Distributions

• Histogram displays the frequency distribution of a continuous variable using bins
  • Shape, center, and spread of the distribution can be easily observed
• Box plot (box-and-whisker plot) summarizes the five-number summary (minimum, Q1, median, Q3, maximum) of a distribution
  • Useful for comparing distributions across groups
• Probability plot (Q-Q plot) assesses if a dataset follows a specific theoretical distribution by plotting the quantiles of the data against the quantiles of the theoretical distribution
  • Points falling along a straight line indicate a good fit
• Cumulative frequency plot shows the cumulative proportion or percentage of observations less than or equal to each value
• Violin plot combines a box plot with a kernel density plot to display the distribution shape and summary statistics

Real-world Business Examples

• A manufacturing company monitors the diameters of ball bearings produced, expecting the diameters to follow a normal distribution with a mean of 10mm and a standard deviation of 0.1mm
  • Quality control limits are set at ±3 standard deviations from the mean
• An e-commerce retailer analyzes the number of daily website visits, which follows a Poisson distribution with an average of 1,000 visits per day
  • This information helps plan server capacity and customer support staffing
• A financial institution assesses the risk of its loan portfolio by modeling the probability of default for each loan using a binomial distribution
  • The institution can then determine the appropriate interest rates and reserve requirements
• A market research firm conducts a survey using stratified sampling based on age groups to ensure adequate representation of each age category in the sample
  • Results are then weighted to reflect the age distribution of the target population
• A hospital manages patient wait times, which are modeled using an exponential distribution with an average wait of 30 minutes
  • This information is used to optimize staffing levels and improve patient satisfaction