⛽️Business Analytics Unit 5 – Probability and Statistical Inference

Key Concepts and Definitions

• Probability measures the likelihood of an event occurring, expressed as a number between 0 and 1
  • 0 indicates an impossible event, while 1 represents a certain event
• Random variable associates a numerical value with each possible outcome of an experiment
  • Discrete random variables have countable values (number of defective items in a batch)
  • Continuous random variables can take on any value within a range (time until a machine fails)
• Population refers to the entire group of individuals, objects, or events of interest
• Sample is a subset of the population used to make inferences about the whole
• Parameter is a numerical summary measure describing a characteristic of a population (mean, standard deviation)
• Statistic is a numerical summary measure calculated from a sample (sample mean, sample standard deviation)
• Sampling distribution describes the distribution of a statistic over many samples
  • Helps determine the likelihood of obtaining a particular sample result

Probability Fundamentals

• Probability of an event (A) is denoted as P(A) and ranges from 0 to 1
• Complementary events have probabilities that sum to 1, P(A) + P(A') = 1
• Mutually exclusive events cannot occur simultaneously, P(A ∩ B) = 0
• Independent events have probabilities unaffected by the occurrence of other events, P(A|B) = P(A)
• Conditional probability P(A|B) measures the likelihood of event A given that event B has occurred
• Bayes' Theorem relates conditional probabilities: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
• Law of Total Probability states that for mutually exclusive events B₁, B₂, ..., Bₙ: $P(A) = P(A|B₁)P(B₁) + P(A|B₂)P(B₂) + ... + P(A|Bₙ)P(Bₙ)$
• Expected value of a random variable X is the average value over many trials: $E(X) = \sum_{i=1}^{n} x_i P(X=x_i)$

Types of Probability Distributions

• Probability distribution describes the likelihood of each possible outcome for a random variable
• Discrete probability distributions are used for random variables with countable outcomes
  • Binomial distribution models the number of successes in a fixed number of independent trials (defective items in a batch)
  • Poisson distribution models the number of events occurring in a fixed interval of time or space (customer arrivals per hour)
• Continuous probability distributions are used for random variables with an infinite number of possible values
  • Normal (Gaussian) distribution is symmetric and bell-shaped, characterized by its mean and standard deviation
  • Exponential distribution models the time between events in a Poisson process (time between customer arrivals)
• Uniform distribution has equal probability for all values within a given range
• Probability density function (PDF) describes the relative likelihood of a continuous random variable taking on a specific value
• Cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a particular value

Sampling Methods and Techniques

• Simple random sampling selects a subset of individuals from a population such that each individual has an equal chance of being chosen
• Stratified sampling divides the population into subgroups (strata) based on a specific characteristic, 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 includes all individuals within those clusters
  • Useful when a complete list of individuals in the population is not available
• Systematic sampling selects individuals from a population at regular intervals (every 10th customer)
• Convenience sampling selects individuals who are easily accessible or readily available (mall intercept surveys)
• Sampling error is the difference between a sample statistic and the corresponding population parameter due to chance
• Non-sampling error arises from sources other than sampling, such as measurement error or non-response bias

Statistical Inference Basics

• Statistical inference uses sample data to make conclusions about a population
• Point estimate is a single value used to estimate a population parameter (sample mean)
• Interval estimate provides a range of values that likely contains the population parameter (confidence interval)
• Sampling distribution of a statistic describes its variability over repeated samples
  • Central Limit Theorem states that the sampling distribution of the mean approaches a normal distribution as sample size increases, regardless of the population distribution
• Standard error measures the variability of a statistic across different samples
  • For the sample mean, standard error is calculated as: $\frac{\sigma}{\sqrt{n}}$, where σ is the population standard deviation and n is the sample size
• Margin of error is the maximum expected difference between a sample statistic and the corresponding population parameter
  • Calculated as the critical value (z-score) multiplied by the standard error

Hypothesis Testing

• Hypothesis testing is a statistical method for making decisions about a population based on sample data
• Null hypothesis (H₀) states that there is no significant difference or effect
• Alternative hypothesis (H₁ or Hₐ) states that there is a significant difference or effect
• Type I error (false positive) occurs when the null hypothesis is rejected when it is actually true
  • Significance level (α) is the probability of making a Type I error, typically set at 0.05
• Type II error (false negative) occurs when the null hypothesis is not rejected when it is actually false
  • Power of a test (1-β) is the probability of correctly rejecting the null hypothesis when the alternative is true
• Test statistic is a value calculated from the sample data used to determine whether to reject the null hypothesis (z-score, t-score)
• P-value is the probability of obtaining a test statistic as extreme as the observed value, assuming the null hypothesis is true
  • If the p-value is less than the significance level (α), the null hypothesis is rejected

Confidence Intervals and Estimation

• Confidence interval is a range of values that is likely to contain the true population parameter with a specified level of confidence
• Confidence level is the probability that the confidence interval contains the true population parameter, typically set at 95%
• Margin of error determines the width of the confidence interval
  • Smaller margin of error results in a narrower confidence interval but requires a larger sample size
• Point estimate is the center of the confidence interval, usually the sample statistic (sample mean)
• For a population mean with known standard deviation, the confidence interval is calculated as: $\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$
• For a population mean with unknown standard deviation, the confidence interval is calculated as: $\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$
  • $\bar{x}$ is the sample mean, $z_{\alpha/2}$ is the critical z-score, $t_{\alpha/2, n-1}$ is the critical t-score, σ is the population standard deviation, s is the sample standard deviation, and n is the sample size

Applications in Business Decision-Making

• A/B testing compares two versions of a product or service to determine which performs better
  • Null hypothesis: no difference between versions; Alternative hypothesis: one version outperforms the other
• Quality control uses sampling and hypothesis testing to ensure products meet specified standards
  • Null hypothesis: product meets standards; Alternative hypothesis: product does not meet standards
• Market research employs sampling techniques to gather data on consumer preferences and behavior
  • Stratified sampling ensures representation of key demographic groups
  • Cluster sampling is useful when a complete customer list is not available
• Forecasting uses historical data and probability distributions to predict future demand or sales
  • Normal distribution is often assumed for long-term forecasts
  • Poisson distribution models rare events (stockouts)
• Risk analysis assesses the likelihood and impact of potential events using probability distributions
  • Monte Carlo simulation generates multiple scenarios based on input probability distributions
• Inventory management balances the costs of holding inventory against the risk of stockouts
  • Economic Order Quantity (EOQ) model determines the optimal order size based on demand, ordering costs, and holding costs
  • Reorder point is set based on lead time demand and a specified service level (probability of not stocking out)