Fun with Numbers: Detective Techniques

Extract base rates, sensitivity, and specificity from an app's promotional data and client complaint records.

Organize screening results into a 2x2 confusion matrix and compute all diagnostic accuracy metrics to assess the app's real-world reliability.

Apply Bayes' theorem to determine the true probability that a client flagged as 'compatible' is genuinely compatible, using both claimed accuracy and observed prevalence.

Simulate how the app's reliability changes as the true compatibility base rate varies, to identify the scenario most consistent with the volume of complaints.

Compute marginal and stratum-specific satisfaction associations from two studios' data to demonstrate how both can truthfully claim superiority.

Identify a confounding variable in the client intake process that, when conditioned on, creates a misleading association between studio choice and satisfaction.

Apply the backdoor adjustment formula to determine the true causal effect of studio choice on satisfaction, isolating it from confounding factors.

Estimate the causal effect through a mediating variable using the front-door adjustment formula when direct adjustment is not possible.

Assess the plausibility of a coach's claimed perfect success rate by computing the binomial probability of the observed outcome under a chance-only hypothesis.

Test whether paired before-after client satisfaction improvements are statistically significant using a non-parametric sign test.

Investigate the relationship between pre- and post-program satisfaction scores using both linear and rank-based correlation measures to uncover potential non-linearity.

Establish a baseline by modeling client satisfaction as a function of a single key predictor variable.

Build a comprehensive multiple linear regression model using all available predictors to identify the primary drivers of satisfaction scores.

Scrutinize the comprehensive model's fit (R-squared), residuals, and individual coefficient significance to find statistical evidence of data manipulation.

Build a logistic regression model to predict the probability of a client returning based on their profile and engagement data.

Evaluate the model's overall performance in explaining the observed pattern of client attrition.

Translate the model's coefficients into odds ratios to pinpoint and interpret the key factors that most significantly increase the odds of a client disappearing.

Decompose total satisfaction variation into between-group and within-group components (ANOVA) to determine if relationship arrangement type genuinely matters.

Compute the F-statistic to formally assess whether the satisfaction differences across groups are statistically significant.

Apply the Markov and Chebyshev inequalities to establish credible bounds on client metrics for which only partial data is available.

Use Jensen's inequality to compute a bound on the expected value of a transformed engagement metric, given only the group's average engagement.

Model the arrival of booking requests using a Poisson process to determine if the observed frequency in a specific time window is anomalous.

Use the exponential distribution to analyze the time between bookings and calculate the probability of suspiciously long or short gaps.

Analyze merged and thinned booking streams from multiple studio channels to detect irregularities hidden in the aggregate data.

Compute the survival and hazard functions for relationship durations to identify if and when the risk of a relationship ending accelerates.

Estimate the true attrition rate from a dataset containing both observed and right-censored relationship durations using Maximum Likelihood Estimation.

Use order statistics to determine the expected value and variance of the highest and lowest scores in a design contest, establishing a benchmark for fairness.

Apply the optimal stopping rule (1/e rule) to determine the ideal point to stop evaluating candidates and make a selection in a sequential process.

Calculate the required sample size to estimate the mean quality of designs with a specified precision and confidence level.

Evaluate and compare two competing estimators for design quality by computing their bias, variance, and Mean Squared Error (MSE).

Standardize a multivariate dataset of reviewer behaviors and compute the covariance matrix.

Perform Principal Component Analysis (PCA) to reduce reviewer profiles to their most informative components.

Project the reviewer data onto the principal components and create visualizations to identify clusters of suspicious activity.

Group the suspects by applying and comparing both single-linkage and complete-linkage hierarchical clustering methods to the reduced data.

Classify a new client by applying the K-Nearest Neighbors (KNN) algorithm to their profile data.

Re-classify the client using a Naive Bayes model, handling both discrete and continuous (Gaussian) features appropriately.

Construct a two-level decision tree profile for the client by identifying the most informative features using Gini impurity or information gain.

Compare the classifications from all three models (KNN, Naive Bayes, Decision Tree) to reach a final conclusion about the mystery client's identity.

Compute the Economic Order Quantity (EOQ) for a product line to establish the optimal inventory policy under normal conditions.

Determine the reorder point for the boutique's stock, and use it to pinpoint when the inventory shrinkage likely began.

Perform a sensitivity analysis on the EOQ model to reveal the scale of inventory manipulation based on discrepancies in demand data.

Use dimensionality reduction (PCA) on a complex behavioral dataset to isolate the most significant patterns of activity.

Apply clustering algorithms to the reduced data to segment individuals into distinct behavioral groups and identify a primary suspect cluster.

Build a multiple regression model focused on the suspect cluster to predict their next move and quantify the factors driving their strategy.