توضیح 1
Estimated time: 62 minutes.
Loading
Loading
درس 6 از 11
Use rule checks and continuity correction, then compare approximate and exact probabilities.
وضعیت ترجمه محتوای درس
ترجمه دانشگاهی فارسی در حال بازبینی انسانی است. متن آموزشی تأییدشده انگلیسی در ادامه نمایش داده میشود؛ رابط کاربری فارسی است و فرمولها، کد R، نمادگذاری و نام ستونها چپبهراست باقی میمانند.
Lesson 7.5
Use rule checks and continuity correction, then compare approximate and exact probabilities. Business question: Approximate P(X>=25).
توضیح 1
Estimated time: 62 minutes.
توضیح 2
Prior knowledge: complements, algebra, probability notation, and careful unit conversion.
توضیح 3
Decide whether a normal approximation is reasonable for a binomial or Poisson count.
توضیح 4
Apply continuity correction and quantify approximation error against an exact result.
توضیح 5
Interpret a normal approximations to binomial and poisson models result in business context with consistent units.
توضیح 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.
توضیح 1
A probability is an area over an event, not an unexplained curve height.
توضیح 2
State the random variable, support, units, and assumptions before using a formula.
توضیح 3
Connect every numerical answer to a practical decision and limitation.
اصطلاحات
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.
نمادگذاری و فرمولها
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).
مثال حلشده
سناریو
X~Binomial(160,0.12).
بررسی با R
pbinom() and ppois() return exact cumulative count probabilities; pnorm() returns the continuity-corrected approximation.Use boundary 24.5 and compare with pbinom.
تمرین هدایتشده
سناریو
X~Binomial(400,0.52).
بررسی با R
pbinom() and ppois() return exact cumulative count probabilities; pnorm() returns the continuity-corrected approximation.Use boundaries 189.5 and 220.5.
فعالیت تعاملی
np
20.00
تفسیر: شرط قاعده ۱۰ برقرار است؛ تصحیح پیوستگی ۰٫۵ را اعمال و نتیجه را با احتمال دقیق دوجملهای مقایسه کنید.
تمرین مستقل
سناریو
X~Binomial(40,0.03).
بررسی با R
pbinom() and ppois() return exact cumulative count probabilities; pnorm() returns the continuity-corrected approximation.np=1.2 fails the rule check; use exact binomial.
تمرین مستقل
سناریو
X~Poisson(36).
بررسی با R
pbinom() and ppois() return exact cumulative count probabilities; pnorm() returns the continuity-corrected approximation.Use a normal model with boundary 44.5.
تمرین مستقل
سناریو
X~Poisson(9).
بررسی با R
pbinom() and ppois() return exact cumulative count probabilities; pnorm() returns the continuity-corrected approximation.The rate is small; exact Poisson is preferable.
تمرین مستقل
سناریو
Exact probability is 0.143 and approximate is 0.151.
بررسی با R
pbinom() and ppois() return exact cumulative count probabilities; pnorm() returns the continuity-corrected approximation.The error is 0.008 and should be disclosed.
تمرین مستقل
سناریو
For integer X find P(X>12).
بررسی با R
pbinom() and ppois() return exact cumulative count probabilities; pnorm() returns the continuity-corrected approximation.More than 12 begins above 12.5.
ارتباط با R
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.
بازاندیشی
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.
توضیح 1
Approximation: A continuous model used to estimate a discrete probability.
توضیح 2
Continuity correction: Move a discrete boundary by 0.5 to match bar widths.
توضیح 3
Error check: Compare approximate and exact results when software permits.
دستیار
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.
منبع
Download an original, accessible STATLAB Academy reference and practice sheet.
دریافتارزیابی پایانی