Explanation 1
Estimated time: 62 minutes.
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Lesson 4 of 11
Translate business thresholds into z scores and compute left, right, and middle areas reliably.
Lesson 7.3
Translate business thresholds into z scores and compute left, right, and middle areas reliably. Business question: Find P(X<20).
Explanation 1
Estimated time: 62 minutes.
Explanation 2
Prior knowledge: complements, algebra, probability notation, and careful unit conversion.
Explanation 3
Standardize a normal value and interpret its signed distance from the mean.
Explanation 4
Calculate and communicate left-tail, right-tail, and interval normal probabilities.
Explanation 5
Interpret a standard normal and normal probabilities 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
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.
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
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.
Notation and formulas
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).
Worked example
Scenario
X~N(24,4).
R check
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.
Guided practice
Scenario
X~N(500,12).
R check
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.
Interactive Mission
Normal interval area
0.6827
Independent practice
Scenario
X~N(30,5).
R check
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.
Independent practice
Scenario
X is normal with mean 70.
R check
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.
Independent practice
Scenario
A metric is flagged below 40 or above 60 for N(50,5).
R check
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.
Independent practice
Scenario
A continuous threshold says at most 12.
R check
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.
Independent practice
Scenario
Z is standard normal.
R check
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.
Independent practice
Scenario
X~N(100,15).
R check
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.
R connection
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.
Reflection
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.
Explanation 1
z score: Signed distance from the mean in standard-deviation units.
Explanation 2
Left tail: Cumulative area below a threshold.
Explanation 3
Right tail: Complementary area above a threshold.
Assistant
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.
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%.