##### Scenario Assessing statistical knowledge for disease-testing model evaluation ##### Question Explain the Central Limit Theorem, its prerequisites, and when it fails. Describe Bayesian inference and why it is widely used. A disease affects 1/1000 people. A test predicts positives with 95% accuracy and negatives with 98% accuracy. If the test flags someone positive, what’s the probability they are truly infected? Show your reasoning. ##### Hints Discuss i.i.d. assumptions, sampling size, Bayes’ formula; build a confusion matrix and compute posterior probability.

This interview question evaluates statistical assumptions, formulas, estimation strategy, uncertainty, edge cases, and interpretation in a realistic interview setting. A strong answer for Explain Central Limit Theorem and Its Limitations states assumptions, handles edge cases, explains trade-offs, and shows how to validate the result clearly.

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This is a medium difficulty Statistics & Math question, commonly asked during Onsite rounds at Amazon.

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This question is commonly asked for Data Scientist candidates at Amazon during technical interviews.

Explain Central Limit Theorem and Its Limitations

Statistics Concepts and Disease-Test Evaluation

Context

You are assessing core statistical concepts used in evaluating diagnostic tests and in data science decision-making.

Assume:

"Predicts positives with 95% accuracy" refers to sensitivity: P(test+ | infected) = 0.95.
"Predicts negatives with 98% accuracy" refers to specificity: P(test− | not infected) = 0.98.

Questions

Central Limit Theorem (CLT)
- Explain the CLT, its prerequisites/assumptions, and situations where it can fail or be unreliable.
Bayesian Inference
- Describe Bayesian inference and why it is widely used in practice.
Disease Test Posterior Probability
- A disease affects 1 in 1,000 people (prevalence = 0.001). A test has sensitivity 95% and specificity 98%.
- If the test flags someone positive, what is the probability they are truly infected? Show your reasoning. You may use a confusion matrix and Bayes' rule.

Hints

Discuss i.i.d. assumptions and sample size for the CLT.
Use Bayes’ formula for the posterior probability.
Build a confusion matrix and compute the positive predictive value (PPV).

Constraints & Assumptions

Preserve the scope, facts, inputs, and requested outputs from the prompt above.
If the prompt leaves a detail unspecified, state a reasonable assumption before relying on it.
Keep the answer interview-ready: concise enough to present, but concrete enough to implement or evaluate.

Clarifying Questions to Ask

Clarify the random variables, distributional assumptions, independence assumptions, and desired output.
Show enough derivation for the interviewer to follow the reasoning.
Explain how you would validate the result with simulation or sensitivity checks.

What a Strong Answer Covers

A correct setup with definitions, formulas, and boundary conditions.
A step-by-step derivation or estimation plan.
Interpretation of the result, including uncertainty and practical limitations.
Checks for assumptions, edge cases, and numerical stability.

Follow-up Questions

How would the result change if the assumptions were relaxed?
Can you verify the answer with a simulation?
What is the most likely source of estimation error?