🧮Calculus and Statistics Methods Unit 6 – Inferential Statistics

Key Concepts and Terminology

• Inferential statistics involves drawing conclusions about a population based on a sample
• 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 population
• Parameter is a numerical characteristic of a population (mean, standard deviation)
• Statistic is a numerical characteristic of a sample used to estimate a population parameter
• Sampling distribution is the probability distribution of a statistic obtained from all possible samples of a given size from a population
• Null hypothesis ($H_0$) is a statement of no effect or no difference, assumed to be true unless evidence suggests otherwise
• Alternative hypothesis ($H_a$) is a statement that contradicts the null hypothesis, representing the researcher's claim

Foundations of Probability

• Probability is a measure of the likelihood of an event occurring, expressed as a value between 0 and 1
• Classical probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes (assuming all outcomes are equally likely)
• Empirical probability is based on observed frequencies of events in a large number of trials
• Probability distributions describe the likelihood of different outcomes for a random variable
  • Discrete probability distributions (binomial, Poisson) are used for countable outcomes
  • Continuous probability distributions (normal, exponential) are used for measurable outcomes
• Expected value is the average outcome of a random variable over a large number of trials, calculated by multiplying each possible outcome by its probability and summing the results
• Bayes' theorem describes how to update the probability of an event based on new information or evidence

Sampling Techniques and Distributions

• Simple random sampling ensures each member of the population has an equal chance of being selected
• Stratified sampling divides the population into homogeneous subgroups (strata) and randomly samples from each stratum
• Cluster sampling involves dividing the population into clusters, randomly selecting clusters, and sampling all members within the selected clusters
• Systematic sampling selects every $k$th element from a list of the population, where $k$ is the sampling interval
• Central Limit Theorem states that the sampling distribution of the mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution
• Standard error is the standard deviation of the sampling distribution, measuring the variability of a statistic from sample to sample
• $z$-score standardizes a value by measuring its distance from the mean in standard deviation units, calculated as $z = \frac{x - \mu}{\sigma}$

Hypothesis Testing Fundamentals

• Hypothesis testing is a statistical method for making decisions about a population based on sample data
• Null hypothesis ($H_0$) assumes no effect or difference, while the alternative hypothesis ($H_a$) represents the researcher's claim
• Type I error (false positive) occurs when rejecting a true null hypothesis, with probability $\alpha$ (significance level)
• Type II error (false negative) occurs when failing to reject a false null hypothesis, with probability $\beta$
• Power is the probability of correctly rejecting a false null hypothesis, calculated as $1 - \beta$
• $p$-value is the probability of obtaining a test statistic as extreme as or more extreme than the observed value, assuming the null hypothesis is true
• Rejecting the null hypothesis when the $p$-value is less than the significance level $\alpha$ (typically 0.05)

Confidence Intervals and Estimation

• Confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence
• Point estimate is a single value (statistic) used to estimate a population parameter
• Margin of error is the maximum expected difference between the point estimate and the true population parameter
• Confidence level is the probability that the confidence interval contains the true population parameter (typically 95%)
• Larger sample sizes and lower variability lead to narrower confidence intervals and more precise estimates
• Confidence intervals for means, proportions, and differences can be constructed using the appropriate formulas and critical values

Statistical Tests and Their Applications

• $t$-test compares means between two groups or a sample mean to a hypothesized population mean
  • Independent samples $t$-test compares means of two independent groups
  • Paired samples $t$-test compares means of two related groups or repeated measures
  • One-sample $t$-test compares a sample mean to a hypothesized population mean
• ANOVA (Analysis of Variance) compares means among three or more groups
  • One-way ANOVA compares means of one factor with three or more levels
  • Two-way ANOVA examines the effects of two factors and their interaction on a dependent variable
• Chi-square test assesses the association between two categorical variables in a contingency table
• Correlation measures the strength and direction of the linear relationship between two continuous variables (Pearson's $r$)
• Regression predicts the value of a dependent variable based on one (simple) or more (multiple) independent variables

Interpreting Results and Drawing Conclusions

• Statistical significance indicates that the observed results are unlikely to have occurred by chance alone, but does not necessarily imply practical importance
• Effect size measures the magnitude of the difference or relationship, independent of sample size (Cohen's $d$, $r^2$)
• Confidence intervals provide a range of plausible values for the population parameter, allowing for an assessment of both statistical significance and practical importance
• Limitations of the study design, sample size, and generalizability should be considered when interpreting results
• Correlation does not imply causation; experimental designs with random assignment are needed to establish causal relationships
• Results should be interpreted in the context of previous research, theoretical frameworks, and practical implications

Real-World Applications and Case Studies

• Market research uses inferential statistics to draw conclusions about consumer preferences and behavior based on surveys or focus groups
• Clinical trials employ hypothesis testing to evaluate the effectiveness and safety of new drugs or treatments compared to placebos or existing interventions
• Quality control in manufacturing relies on sampling techniques and statistical process control to monitor and maintain product quality
• Polling organizations use inferential statistics to estimate population opinions or voting intentions based on representative samples
• Psychological research applies statistical tests to compare groups, assess relationships between variables, and evaluate the effectiveness of interventions
• Epidemiological studies use inferential statistics to investigate the prevalence, incidence, and risk factors of diseases in populations
• A/B testing in web design and online marketing uses hypothesis testing to compare the effectiveness of different versions of websites or ads in terms of user engagement or conversion rates