توضیح 1
Estimated time: 62 minutes.
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درس 4 از 11
Translate business thresholds into z scores and compute left, right, and middle areas reliably.
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 7.3
Translate business thresholds into z scores and compute left, right, and middle areas reliably. Business question: Find P(X<20).
توضیح 1
Estimated time: 62 minutes.
توضیح 2
Prior knowledge: complements, algebra, probability notation, and careful unit conversion.
توضیح 3
Standardize a normal value and interpret its signed distance from the mean.
توضیح 4
Calculate and communicate left-tail, right-tail, and interval normal probabilities.
توضیح 5
Interpret a standard normal and normal probabilities 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
z score: Signed distance from the mean in standard-deviation units. Left tail: Cumulative area below a threshold. Right tail: Complementary area above a threshold.
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
Signed distance from the mean in standard-deviation units.
z=1.5 is 1.5 SD above target.
Cumulative area below a threshold.
Deliveries completed by 30 minutes.
Complementary area above a threshold.
Packages above a tolerance limit.
نمادگذاری و فرمولها
z=(x-mu)/sigma; P(a<X<b)=Phi(z_b)-Phi(z_a)
Symbols: a and b are measurement thresholds; mu is the mean; sigma>0 is the standard deviation; Phi is the standard-normal CDF; z_a and z_b are unit-free standardized boundaries. Use: apply this relationship only after defining the model and event. Plain-text equivalent: z=(x-mu)/sigma; P(a<X<b)=Phi(z_b)-Phi(z_a).
مثال حلشده
سناریو
X~N(24,4).
بررسی با R
pnorm() returns cumulative probability. Use lower.tail=FALSE for a right tail and subtract two CDF values for an interval.z=-1 and the left-tail probability is about 0.1587.
تمرین هدایتشده
سناریو
X~N(500,12).
بررسی با R
pnorm() returns cumulative probability. Use lower.tail=FALSE for a right tail and subtract two CDF values for an interval.Use a right-tail probability, about 0.0478.
فعالیت تعاملی
مساحت بازه نرمال
0.6827
تمرین مستقل
سناریو
X~N(30,5).
بررسی با R
pnorm() returns cumulative probability. Use lower.tail=FALSE for a right tail and subtract two CDF values for an interval.Subtract the two cumulative probabilities.
تمرین مستقل
سناریو
X is normal with mean 70.
بررسی با R
pnorm() returns cumulative probability. Use lower.tail=FALSE for a right tail and subtract two CDF values for an interval.Symmetry gives exactly 0.5.
تمرین مستقل
سناریو
A metric is flagged below 40 or above 60 for N(50,5).
بررسی با R
pnorm() returns cumulative probability. Use lower.tail=FALSE for a right tail and subtract two CDF values for an interval.Add the two disjoint tail probabilities.
تمرین مستقل
سناریو
A continuous threshold says at most 12.
بررسی با R
pnorm() returns cumulative probability. Use lower.tail=FALSE for a right tail and subtract two CDF values for an interval.The endpoint has zero probability, so both statements match.
تمرین مستقل
سناریو
Z is standard normal.
بررسی با R
pnorm() returns cumulative probability. Use lower.tail=FALSE for a right tail and subtract two CDF values for an interval.The middle area is approximately 0.95.
تمرین مستقل
سناریو
X~N(100,15).
بررسی با R
pnorm() returns cumulative probability. Use lower.tail=FALSE for a right tail and subtract two CDF values for an interval.z=2, so the upper tail is about 0.0228.
ارتباط با R
pnorm() returns cumulative probability. Use lower.tail=FALSE for a right tail and subtract two CDF values for an interval. 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
Calculate left, right, and interval probabilities without tail errors.
Ready to run
Common mistakes and model safeguard
Draw the requested region; many wrong answers are correct calculations for the wrong tail. 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
Translate business thresholds into z scores and compute left, right, and middle areas reliably. The reliable workflow is: define X and units, justify the model, state the event, calculate, verify, interpret, and disclose limitations.
توضیح 1
z score: Signed distance from the mean in standard-deviation units.
توضیح 2
Left tail: Cumulative area below a threshold.
توضیح 3
Right tail: Complementary area above a threshold.
دستیار
Coach me through Standard Normal and Normal Probabilities 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