Explanation 1
Estimated time: 62 minutes.
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Lesson 6 of 11
Use rule checks and continuity correction, then compare approximate and exact probabilities.
Lesson 7.5
Use rule checks and continuity correction, then compare approximate and exact probabilities. Business question: Approximate P(X>=25).
Explanation 1
Estimated time: 62 minutes.
Explanation 2
Prior knowledge: complements, algebra, probability notation, and careful unit conversion.
Explanation 3
Decide whether a normal approximation is reasonable for a binomial or Poisson count.
Explanation 4
Apply continuity correction and quantify approximation error against an exact result.
Explanation 5
Interpret a normal approximations to binomial and poisson models result in business context with consistent units.
Explanation 6
Use and explain the relevant base R distribution functions without confusing density, probability, quantiles, or simulation.
Core explanation
Approximation: A continuous model used to estimate a discrete probability. Continuity correction: Move a discrete boundary by 0.5 to match bar widths. Error check: Compare approximate and exact results when software permits.
Explanation 1
A probability is an area over an event, not an unexplained curve height.
Explanation 2
State the random variable, support, units, and assumptions before using a formula.
Explanation 3
Connect every numerical answer to a practical decision and limitation.
Terminology
A continuous model used to estimate a discrete probability.
Normal estimate of a binomial tail.
Move a discrete boundary by 0.5 to match bar widths.
X>=18 becomes Y>17.5.
Compare approximate and exact results when software permits.
Absolute probability difference.
Notation and formulas
Binomial: mu=np, sigma=sqrt(np(1-p)); Poisson: mu=lambda, sigma=sqrt(lambda)
Symbols: n is trial count; p is success probability; lambda is a Poisson mean count; mu and sigma are the approximating normal parameters. Counts are unit-free and continuity boundaries move by 0.5. Use: apply this relationship only after defining the model and event. Plain-text equivalent: Binomial: mu=np, sigma=sqrt(np(1-p)); Poisson: mu=lambda, sigma=sqrt(lambda).
Worked example
Scenario
X~Binomial(160,0.12).
R check
pbinom() and ppois() return exact cumulative count probabilities; pnorm() returns the continuity-corrected approximation.Use boundary 24.5 and compare with pbinom.
Guided practice
Scenario
X~Binomial(400,0.52).
R check
pbinom() and ppois() return exact cumulative count probabilities; pnorm() returns the continuity-corrected approximation.Use boundaries 189.5 and 220.5.
Interactive Mission
np
20.00
Interpretation: The rule-of-10 gate passes; apply a 0.5 continuity correction and compare with the exact binomial.
Independent practice
Scenario
X~Binomial(40,0.03).
R check
pbinom() and ppois() return exact cumulative count probabilities; pnorm() returns the continuity-corrected approximation.np=1.2 fails the rule check; use exact binomial.
Independent practice
Scenario
X~Poisson(36).
R check
pbinom() and ppois() return exact cumulative count probabilities; pnorm() returns the continuity-corrected approximation.Use a normal model with boundary 44.5.
Independent practice
Scenario
X~Poisson(9).
R check
pbinom() and ppois() return exact cumulative count probabilities; pnorm() returns the continuity-corrected approximation.The rate is small; exact Poisson is preferable.
Independent practice
Scenario
Exact probability is 0.143 and approximate is 0.151.
R check
pbinom() and ppois() return exact cumulative count probabilities; pnorm() returns the continuity-corrected approximation.The error is 0.008 and should be disclosed.
Independent practice
Scenario
For integer X find P(X>12).
R check
pbinom() and ppois() return exact cumulative count probabilities; pnorm() returns the continuity-corrected approximation.More than 12 begins above 12.5.
R connection
pbinom() and ppois() return exact cumulative count probabilities; pnorm() returns the continuity-corrected approximation. Run the original starter code, inspect every printed intermediate value, then modify one assumption and explain the decision impact. Keep code, formulas, function names, column names, and numerical output left-to-right.
Live R Lab
Compare exact count probabilities with continuity-corrected normal approximations.
Ready to run
Common mistakes and model safeguard
Never present a normal approximation as exact; show the rule check, correction, and error where possible. Also check tail direction, parameter units, support, and whether software returned density, cumulative probability, a quantile, or generated data.
Originality and provenance
Original synthetic data generated for STATLAB Academy. No textbook data used.
Reflection
Name the assumption, the evidence you would seek, and how the recommendation could change.
Lesson summary
Use rule checks and continuity correction, then compare approximate and exact probabilities. The reliable workflow is: define X and units, justify the model, state the event, calculate, verify, interpret, and disclose limitations.
Explanation 1
Approximation: A continuous model used to estimate a discrete probability.
Explanation 2
Continuity correction: Move a discrete boundary by 0.5 to match bar widths.
Explanation 3
Error check: Compare approximate and exact results when software permits.
Assistant
Coach me through Normal Approximations to Binomial and Poisson Models one decision at a time. Before submission, give hints only and do not reveal quiz answers or complete my recommendation.
Resource
Download an original, accessible STATLAB Academy reference and practice sheet.
DownloadExit check
Use the short exit check after instruction and practice. Reach 70% to complete this lesson; explanations appear only after submission.
Checkpoint
Question 1 of 4. Answered 0/4. Passing score: 70%.