🔬 MemoLearning Inferential Statistics

Make data-driven conclusions and predictions using statistical inference methods

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Inferential Statistics Curriculum

12
Core Units
~85
Statistical Concepts
20+
Statistical Tests
30+
Real Examples
1

Introduction to Inference

Understand the fundamentals of statistical inference and how to make conclusions about populations from samples.

  • Statistical inference concepts
  • Population vs sample inference
  • Sampling distributions
  • Central limit theorem
  • Standard error
  • Estimation vs hypothesis testing
  • Type I and Type II errors
  • Statistical significance
2

Probability Distributions

Master key probability distributions used in statistical inference and hypothesis testing.

  • Normal distribution
  • Student's t-distribution
  • Chi-square distribution
  • F-distribution
  • Binomial and Poisson distributions
  • Distribution properties
  • Choosing appropriate distributions
  • Distribution parameters
3

Confidence Intervals

Construct and interpret confidence intervals for population parameters using sample data.

  • Confidence interval concepts
  • Confidence level interpretation
  • Margin of error
  • CI for population mean
  • CI for population proportion
  • CI for difference of means
  • Sample size determination
  • Bootstrap confidence intervals
4

Hypothesis Testing Framework

Learn the systematic approach to hypothesis testing and statistical decision making.

  • Null and alternative hypotheses
  • Test statistics
  • P-values and interpretation
  • Critical values and regions
  • Decision rules
  • One-tailed vs two-tailed tests
  • Power and effect size
  • Multiple testing corrections
5

One-Sample Tests

Test hypotheses about single population parameters using one-sample statistical tests.

  • One-sample t-test
  • One-sample z-test
  • One-sample proportion test
  • Wilcoxon signed-rank test
  • Goodness-of-fit tests
  • Normality testing
  • Assumptions and conditions
  • Effect size measures
6

Two-Sample Tests

Compare two groups or populations using appropriate two-sample statistical tests.

  • Independent t-test
  • Paired t-test
  • Two-sample proportion test
  • Mann-Whitney U test
  • Wilcoxon signed-rank test
  • F-test for variances
  • Pooled vs unpooled variance
  • Equal variance assumptions
7

Analysis of Variance (ANOVA)

Compare multiple groups simultaneously using ANOVA and understand variance partitioning.

  • One-way ANOVA
  • Two-way ANOVA
  • ANOVA assumptions
  • Post-hoc tests
  • Multiple comparisons
  • Effect size in ANOVA
  • Repeated measures ANOVA
  • Non-parametric alternatives
8

Chi-Square Tests

Analyze categorical data relationships using chi-square tests for independence and goodness-of-fit.

  • Chi-square test of independence
  • Chi-square goodness-of-fit
  • Expected vs observed frequencies
  • Contingency table analysis
  • Degrees of freedom
  • Effect size measures
  • Fisher's exact test
  • McNemar's test
9

Correlation and Regression Inference

Make inferences about relationships between variables using correlation and regression analysis.

  • Correlation significance testing
  • Regression coefficient testing
  • Confidence intervals for slopes
  • Model significance (F-test)
  • Prediction intervals
  • Regression assumptions
  • Residual analysis
  • Multiple regression inference
10

Non-Parametric Tests

Apply distribution-free statistical tests when parametric assumptions are violated.

  • When to use non-parametric tests
  • Mann-Whitney U test
  • Kruskal-Wallis test
  • Friedman test
  • Sign test
  • Runs test
  • Rank correlation tests
  • Permutation tests
11

Bootstrap and Resampling

Use modern resampling methods for statistical inference when traditional methods are inadequate.

  • Bootstrap principle
  • Bootstrap confidence intervals
  • Bootstrap hypothesis testing
  • Jackknife method
  • Permutation tests
  • Cross-validation
  • Resampling applications
  • Advantages and limitations
12

Applied Statistical Inference

Apply statistical inference methods to real-world problems and learn best practices for data analysis.

  • Choosing appropriate tests
  • Sample size and power analysis
  • Multiple testing corrections
  • Effect size reporting
  • Practical significance
  • Publication bias
  • Reproducibility considerations
  • Communicating results

Unit 1: Introduction to Inference

Understand the fundamentals of statistical inference and how to make conclusions about populations from samples.

Statistical Inference Concepts

Learn the fundamental idea of using sample data to make conclusions about populations.

Estimation Hypothesis Testing Prediction
Statistical inference allows us to make educated guesses about populations using limited sample information, quantifying our uncertainty in the process.

Population vs Sample Inference

Understand how sample statistics are used to estimate population parameters and the inherent uncertainty involved.

# Population parameter (unknown)
population_mean = μ # Unknown

# Sample statistic (known)
sample_mean = np.mean(sample_data)
# Use sample_mean to estimate μ

Sampling Distributions

Learn about the distribution of sample statistics and how it forms the foundation of statistical inference.

Sampling Distribution of x̄:
Mean = μ
Standard Error = σ/√n
# Simulate sampling distribution
sample_means = []
for i in range(1000):
  sample = np.random.normal(50, 10, 30)
  sample_means.append(np.mean(sample))

Central Limit Theorem

Understand how the CLT enables inference even when the population distribution is unknown.

For large sample sizes (n ≥ 30), the sampling distribution of the mean approaches a normal distribution, regardless of the population distribution shape.
from scipy import stats
# CLT in action
# Even with skewed population
population = stats.expon.rvs(size=10000)
sample_means = [np.mean(np.random.choice(population, 30)) for _ in range(1000)]

Standard Error

Calculate and interpret the standard error as a measure of sampling variability.

Standard Error of Mean: SE = σ/√n
Standard Error of Proportion: SE = √(p(1-p)/n)
import numpy as np
# Calculate standard error
n = len(sample)
sample_std = np.std(sample, ddof=1)
standard_error = sample_std / np.sqrt(n)

Estimation vs Hypothesis Testing

Distinguish between two main approaches to statistical inference: estimation and hypothesis testing.

Estimation: "What is the likely range of the population mean?"
Hypothesis Testing: "Is the population mean equal to a specific value?"
# Estimation approach
confidence_interval = stats.t.interval(0.95, df, sample_mean, se)

# Hypothesis testing approach
t_stat, p_value = stats.ttest_1samp(sample, hypothesized_mean)

Type I and Type II Errors

Learn about the two types of errors that can occur in hypothesis testing and their implications.

Type I (α) Type II (β) Power (1-β)
Type I Error: Rejecting a true null hypothesis (False Positive)
Type II Error: Failing to reject a false null hypothesis (False Negative)

Statistical Significance

Understand what statistical significance means and how to interpret p-values correctly.

# Interpreting p-values
if p_value < 0.05:
  print("Statistically significant at α = 0.05")
  print("Reject null hypothesis")
else:
  print("Not statistically significant")
  print("Fail to reject null hypothesis")

Unit 2: Probability Distributions

Master key probability distributions used in statistical inference and hypothesis testing.

Normal Distribution

Understand the most important distribution in statistics and its role in inference.

X ~ N(μ, σ²)
Z = (X - μ)/σ ~ N(0, 1)
from scipy import stats
# Normal distribution operations
normal_dist = stats.norm(loc=50, scale=10)
# Probability calculations
prob = normal_dist.cdf(60) - normal_dist.cdf(40)

Student's t-Distribution

Learn when and how to use the t-distribution for inference with small samples.

# t-distribution for small samples
n = 15
df = n - 1 # degrees of freedom
t_dist = stats.t(df)
# Critical value for 95% confidence
t_critical = t_dist.ppf(0.975)