Quantile: The x value with a specified cumulative probability. Tail conversion: Right-tail targets must be converted to left-tail area. Central interval: Split excluded probability equally when symmetry is intended.

x=mu+z_p sigma, where Phi(z_p)=p

1. گام 1Convert the wording to left-tail p.
2. گام 2Find z_p or call qnorm(p,mu,sigma).
3. گام 3Verify by substituting the threshold into the CDF.

متغیر

Define X from the scenario and state its measurement or count units before choosing a model.

انتخاب مدل

Use inverse normal only when the support, shape, and process assumptions match the scenario.

پارامترها

Symbols: p is a left-tail probability with 0<p<1; z_p is its standard-normal quantile; mu is the mean; sigma>0 is the standard deviation; x is the requested threshold. Units: x, mu, and sigma use the measurement unit; p and z_p are unit-free.

بیان احتمال

Find a threshold met 90% of the time.

محاسبه

x=mu+z_p sigma, where Phi(z_p)=p. Use qnorm(0.90,18,3).

تفسیر

Use qnorm(0.90,18,3).

بررسی معقول‌بودن

Confirm that probabilities lie from 0 to 1, thresholds use the requested units, and the direction of the event matches the business question.

Use qnorm(0.90,18,3).

متغیر

Define X from the scenario and state its measurement or count units before choosing a model.

انتخاب مدل

Use inverse normal only when the support, shape, and process assumptions match the scenario.

پارامترها

Symbols: p is a left-tail probability with 0<p<1; z_p is its standard-normal quantile; mu is the mean; sigma>0 is the standard deviation; x is the requested threshold. Units: x, mu, and sigma use the measurement unit; p and z_p are unit-free.

بیان احتمال

Find the top 2.5% boundary.

محاسبه

x=mu+z_p sigma, where Phi(z_p)=p. Use left-tail area 0.975.

تفسیر

Use left-tail area 0.975.

بررسی معقول‌بودن

Confirm that probabilities lie from 0 to 1, thresholds use the requested units, and the direction of the event matches the business question.

Use left-tail area 0.975.

متغیر

Define X from the scenario and state its measurement or count units before choosing a model.

انتخاب مدل

Use inverse normal only when the support, shape, and process assumptions match the scenario.

پارامترها

Symbols: p is a left-tail probability with 0<p<1; z_p is its standard-normal quantile; mu is the mean; sigma>0 is the standard deviation; x is the requested threshold. Units: x, mu, and sigma use the measurement unit; p and z_p are unit-free.

بیان احتمال

Find limits covering the middle 80%.

محاسبه

x=mu+z_p sigma, where Phi(z_p)=p. Use cumulative areas 0.10 and 0.90.

تفسیر

Use cumulative areas 0.10 and 0.90.

بررسی معقول‌بودن

Confirm that probabilities lie from 0 to 1, thresholds use the requested units, and the direction of the event matches the business question.

Use cumulative areas 0.10 and 0.90.

متغیر

Define X from the scenario and state its measurement or count units before choosing a model.

انتخاب مدل

Use inverse normal only when the support, shape, and process assumptions match the scenario.

پارامترها

Symbols: p is a left-tail probability with 0<p<1; z_p is its standard-normal quantile; mu is the mean; sigma>0 is the standard deviation; x is the requested threshold. Units: x, mu, and sigma use the measurement unit; p and z_p are unit-free.

بیان احتمال

Find the 5th percentile.

محاسبه

x=mu+z_p sigma, where Phi(z_p)=p. Use qnorm(0.05,40,5).

تفسیر

Use qnorm(0.05,40,5).

بررسی معقول‌بودن

Confirm that probabilities lie from 0 to 1, thresholds use the requested units, and the direction of the event matches the business question.

Use qnorm(0.05,40,5).

متغیر

Define X from the scenario and state its measurement or count units before choosing a model.

انتخاب مدل

Use inverse normal only when the support, shape, and process assumptions match the scenario.

پارامترها

Symbols: p is a left-tail probability with 0<p<1; z_p is its standard-normal quantile; mu is the mean; sigma>0 is the standard deviation; x is the requested threshold. Units: x, mu, and sigma use the measurement unit; p and z_p are unit-free.

بیان احتمال

Choose capacity exceeded only 1% of days.

محاسبه

x=mu+z_p sigma, where Phi(z_p)=p. Use the 99th percentile and discuss model risk.

تفسیر

Use the 99th percentile and discuss model risk.

بررسی معقول‌بودن

Confirm that probabilities lie from 0 to 1, thresholds use the requested units, and the direction of the event matches the business question.

Use the 99th percentile and discuss model risk.

qnorm() returns a quantile from a left-tail probability; verify the result by passing it to pnorm(). 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.

Name the assumption, the evidence you would seek, and how the recommendation could change.

Reverse the normal CDF to turn coverage targets into operational thresholds. The reliable workflow is: define X and units, justify the model, state the event, calculate, verify, interpret, and disclose limitations.

Use the short exit check after instruction and practice. Reach 70% to complete this lesson; explanations appear only after submission.