Lesson 5 Topic 2 Solutions

Central Limit Theorem (CLT) & Comprehensive Z-Score Analysis

Data Science & Statistical Inference Report

I. Introduction & Theoretical Background

The Central Limit Theorem (CLT) serves as a foundational principle within probability theory and modern data science. It dictates that given a sufficiently large sample size from a population with a finite variance, the distribution of the sample means will be approximately normally distributed (following a Gaussian "bell curve"), regardless of the underlying population's original distribution shape.

In practical analytics and engineering environments, the CLT is indispensable for several core reasons:

Parametric Hypothesis Testing: It justifies the application of normal-distribution-based statistics- such as Z-tests, t-tests, and Analysis of Variance (ANOVA) -and the construction of confidence intervals even when original datasets exhibit skewness or non-normality.
Experimentation & A/B Testing: It provides the standard mathematical foundation required to evaluate variance across user cohorts, feature tests and metric averages (e.g., conversion rates, click-through metrics).
Asymptotic Error Estimation: It guarantees that the standard error of calculated parameters diminishes predictably at a rate proportional to 1/√n as the sample size increases.

II. Population Parameter Calculations

Consider a discrete population consisting of 28 consecutive integers: X = {1, 2, 3, …, 28}. This finite sequence follows a Discrete Uniform Distribution. Below are the primary population parameters derived mathematically:

Table I: Population Statistical Parameters
Parameter	Mathematical Formula	Calculation Detail	Result
Population Size (N)	Count	28 consecutive integers	28
Population Mean (μ)	μ = (∑ X_i) / N	(1 + 2 + … + 28) / 28 = 406 / 28	14.50
Population Variance (σ²)	σ² = ∑(X_i - μ)² / N	Sum of squared deviations / 28	65.25
Population Standard Deviation (σ)	σ = √σ²	√65.25	8.0777

III. Sampling Distribution & Z-Score Limit Evaluation

According to the Central Limit Theorem, drawing repeated independent random samples of size n from this population produces a sampling distribution of means with the following formal properties:

Expected Sampling Mean: μ_x̄ = μ = 14.50

Standard Error (SE): σ_x̄ = σ / √n

To establish standardised operational boundaries, lower and upper limits are computed at Z = -2.00 and Z = +2.00 standard deviations from the expected mean. Under a standard normal model, this interval (μ_x̄ ± 2σ_x̄) encompasses approximately 95.4% of all sample means.

Table II: Theoretical Sampling Properties and Standardized Limits Across Sample Sizes
Sample Size (n)	Expected Mean (μ_x̄)	Standard Error (σ_x̄)	Lower Limit (Z = -2.00)	Upper Limit (Z = +2.00)	Theoretical Interval Coverage
n = 2	14.50	8.0777 / √2 = 5.7118	3.08	25.92	95.4% Expected Range
n = 5	14.50	8.0777 / √5 = 3.6125	7.28	21.73	95.4% Expected Range
n = 10	14.50	8.0777 / √10 = 2.5544	9.39	19.61	95.4% Expected Range
n = 20	14.50	8.0777 / √20 = 1.8062	10.89	18.11	95.4% Expected Range

IV. Empirical Simulation & Visual Convergence

An empirical Monte Carlo simulation was executed comprising 5,000 independent draws at each specified sample size (n ∈ {2, 5, 10, 20}). The results demonstrate clear visual convergence toward a standard Gaussian density curve centered tightly at μ = 14.50.

Fig. 1. Empirical sampling distributions of sample means across 5,000 iterations for sample sizes n = 2, 5, 10, and 20. Overlaid red lines depict the theoretical standard normal curve.

V. Z-Score Hints & Conclusion

Z-Score Boundary Breakdown:

Lower Limit (Z = -2.00): Calculated via μ_x̄ - 2(σ_x̄). Represents the lower 2.3% threshold of the expected sampling distribution.
Upper Limit (Z = +2.00): Calculated via μ_x̄ + 2(σ_x̄). Represents the upper 2.3% threshold of the expected sampling distribution.
Sample Size Impact: As sample size n increases from 2 to 20, standard error (σ_x̄) drops sharply from 5.7118 down to 1.8062. This contracts the 95.4% boundary from [3.08, 25.92] down to [10.89, 18.11], demonstrating how larger sample sizes systematically constrain variance around the population mean.

Conclusion: The numerical study and simulation confirm the Central Limit Theorem. As the sample size n increases, the sampling distribution of means rapidly approaches a normal distribution N(14.50, σ²/n), narrowing standard error and improving estimation precision.