Turning Normal Calculations into a Complete Response
In Common Errors with Normal Calculations, we practiced checking that the event, model, and calculator inputs match. An exam response needs to make that reasoning visible, too. A correct calculator result is only part of the answer: you also need to identify the random variable, explain why a normal model is reasonable when the question calls for justification, show the calculation, and interpret the result in context.
This tutorial works through free-response questions that combine normal areas and cutoffs. As in Interpreting Normal Probabilities in Context, an area describes a probability for a specified random variable and event. As in Finding a Value from a Percentile with invNorm, a cutoff is a value associated with a specified cumulative area. The new skill is combining these ideas into one organized, complete response.
Worked Example: Area and Upper-Tail Cutoff for Delivery Times
Worked Example: Area and Upper-Tail Cutoff for Delivery Times
A fictional delivery company models the time \(X\), in minutes, for a randomly selected route as normal with mean 32 minutes and standard deviation 4.5 minutes. Records from earlier routes show a roughly symmetric, single-peaked distribution, and a normal probability plot is approximately linear. Find the probability that a route takes between 27 and 38 minutes. Then find the cutoff time for the slowest 10% of routes. Show and interpret both results.
State. Let \(X\) be the delivery time, in minutes, for a randomly selected route. The model is \(X\sim N(32,4.5)\), where 32 is the mean and 4.5 minutes is the standard deviation. The first requested event is \(27\leq X\leq38\). The second request is for a time with 10% of routes above it.
Plan. The stated distribution evidence supports using a normal model: the observed shape is roughly symmetric and single-peaked, and the normal probability plot is approximately linear. For the area, use normalcdf with the two time bounds and the model’s mean and standard deviation, all in minutes. For the slowest 10%, the desired cutoff has 90% of the model’s area to its left, so use 0.90 as the left-tail area in invNorm.
Do. For the between-values probability, enter the lower bound, upper bound, mean, and standard deviation in that order:
As a check on the setup, the standardized bounds are \((27-32)/4.5\approx-1.1111\) and \((38-32)/4.5\approx1.3333\). The standard normal area is about \(0.9088-0.1333=0.7755\), consistent with the calculator result.
For the slowest 10%, the cutoff has 90% of the area to its left. Using invNorm with the model parameters gives:
The cutoff is also consistent with the standard-normal percentile: the 90th-percentile z-score is about 1.2816, so \(32+1.2816(4.5)\approx37.77\) minutes.
Conclude. Under the normal model, the probability that a randomly selected route takes between 27 and 38 minutes is about 0.7755, or 77.55%. The slowest 10% of routes take longer than about 37.77 minutes; equivalently, about 90% of routes take 37.77 minutes or less. The model evidence justifies using normal calculations as an approximation for these route times.
What Makes the Response Verifiable?
The example’s answer is more than two calculator outputs. It connects each result to a clearly stated question. For the area, the bounds match the event \(27\leq X\leq38\). For the cutoff, the stated upper-tail proportion of 10% is converted to the left-tail area 0.90 before using invNorm. These details let a reader check whether the calculations answer the requested questions.
In a free-response setting, a short justification should use evidence supplied in the question. If the question describes a roughly symmetric, single-peaked distribution and an approximately straight normal probability plot, cite those features as support for a normal model. Do not claim that the model is proven exactly normal. If the problem explicitly states that the variable follows a normal model, identify that given model rather than inventing additional evidence.
Keep the model parameters and units visible. Here, the mean, standard deviation, bounds, and cutoff are all in minutes. A calculator entry using a standard deviation of 4.5 is appropriate; entering a variance in its place would not be. The previous tutorial, Common Errors with Normal Calculations, develops that calculation check in more detail.
Worked Example: A Probability Between Two Fill Amounts
Worked Example: A Probability Between Two Fill Amounts
A fictional beverage plant models the amount \(X\) in a randomly selected bottle as normal with mean 500 milliliters and standard deviation 6 milliliters. Find the probability that a bottle contains between 492 and 509 milliliters. Give a contextual interpretation.
State. \(X\) is the fill amount, in milliliters, for a randomly selected bottle, with \(X\sim N(500,6)\). The event is \(492\leq X\leq509\).
Plan. Use the given normal model and calculate the area between the two fill amounts. The bounds, mean, and standard deviation are all measured in milliliters, so normalcdf can use them directly.
Do. Enter the lower bound, upper bound, mean, and standard deviation:
To check, standardize the endpoints: \((492-500)/6\approx-1.3333\) and \((509-500)/6=1.5\). The area between them is approximately \(0.9332-0.0912=0.8420\), rounded to four decimal places.
Conclude. According to the model, the probability that a randomly selected bottle contains between 492 and 509 milliliters is about 0.8420, or 84.20%. This probability describes the proportion of bottles the model predicts will fall within that fill range.
Worked Example: Finding a Cutoff from an Upper-Tail Percentage
Worked Example: Finding a Cutoff from an Upper-Tail Percentage
A fictional community website models the time \(X\), in seconds, that a randomly selected page takes to load as normal with mean 18 seconds and standard deviation 4.2 seconds. Find the cutoff for the slowest 15% of page loads. Interpret the cutoff.
State. \(X\) is the page-load time, in seconds, for a randomly selected page load. We want a value \(c\) such that 15% of the model’s values exceed \(c\).
Plan. The slowest 15% is the right-tail area. Therefore, the area to the left of the cutoff is \(1-0.15=0.85\). Use that left-tail area in invNorm, along with the mean and standard deviation in seconds.
Do. The calculator setup and equivalent z-score calculation are:
Conclude. Under the normal model, about 15% of page loads take longer than 22.35 seconds, and about 85% take 22.35 seconds or less. The cutoff is a modeled value, not a guarantee that exactly 15% of any particular small group of page loads will exceed it.
Common Mistakes and AP Exam Tips
A complete response makes the direction of an area or cutoff clear. Several errors can produce a numerical answer that looks reasonable but addresses a different question.
- Using the upper-tail percentage as the invNorm area. invNorm takes the area to the left. For the slowest 10%, enter 0.90, not 0.10. State the conversion so the setup is clear.
- Reporting an area without its event. A decimal by itself does not identify which values it describes. Write the probability notation and interpret the result using the random variable and context.
- Calling a cutoff a probability. A cutoff is a value in the variable’s units, such as 37.77 minutes. The probability or percentile describes the proportion of model values on one side of that value.
- Using calculator inputs in the wrong order. For normalcdf, enter lower bound, upper bound, mean, and standard deviation. For invNorm, enter the left-tail area, mean, and standard deviation.
- Giving an unsupported model justification. Refer to the information provided, such as a roughly symmetric, single-peaked shape or an approximately linear normal probability plot. Do not say a normal model is justified merely because a calculator can calculate an area.
- Leaving off units or rounding too early. Cutoffs should have units from the original measurement, and probabilities have no units. Keep calculator precision through the calculation and round the reported result sensibly.
For full credit, a reader should be able to identify the random variable, verify that the event or tail matches the wording, follow the calculator setup, and understand the conclusion in context. If a question asks for justification, include the relevant evidence in the Plan or conclusion; a correct numerical result alone does not supply that justification.
Key Takeaway
Normal-model free-response problems ask you to connect the context, model, calculation, and interpretation. A visible setup makes it easier to check that an area or cutoff answers the question, and a concise justification explains why the model is suitable when evidence is provided.
Check Your Understanding
For each question, show the setup and write a contextual interpretation.
- A fictional plant models the length \(X\) of a randomly selected stem as \(N(24,3)\), in centimeters. Find \(P(21\leq X\leq28)\) using normalcdf.
- A normal model has mean 70 and standard deviation 8. What left-tail area should be entered into invNorm to find the cutoff for the highest 5%? Explain.
- A fictional battery life is modeled as \(N(9.5,1.2)\), in hours. Find the 25th-percentile battery life using invNorm and interpret it.
- A response gives an area of 0.12 for a question asking for the slowest 12% of observations. What additional value is needed to find the cutoff, and why is 0.12 not the correct left-tail area?
- A question says the observed distribution is approximately symmetric and single-peaked, with an approximately linear normal probability plot. How could those details support a normal-model calculation?