From a Standard Normal Table to normalcdf
In Finding Area with a Standard Normal Table, you used z-scores and Table A to find areas under a standard normal curve. A calculator can find normal areas directly. For a between-values question, the normalcdf command uses the lower value, upper value, mean, and standard deviation as its inputs.
Suppose \(X\) follows a normal model with mean \(\mu\) and standard deviation \(\sigma\). If the question asks for the proportion of values between \(a\) and \(b\), where \(a<b\), the event is \(a<X<b\). The calculator finds the area under the model’s curve between those bounds. You do not need to calculate z-scores first, although doing so is a useful way to check your answer.
The input order matters: lower bound, upper bound, mean, standard deviation. Bounds are in the original units of \(X\), and the last input is the standard deviation—not the variance. For instance, if a variable is measured in minutes, the bounds, mean, and standard deviation should all be entered in minutes.
On a TI-84, you can find normalcdf in the DISTR menu. The command is written as \(\operatorname{normalcdf}(\text{lower},\text{upper},\text{mean},\text{SD})\). If your calculator opens a screen with named fields instead of a command line, fill in those four fields in the same order.
A Routine for Between-Values Problems
Before calculating, sketch or picture the requested region. Identify which endpoint is lower and which is upper, and make sure both are values of the variable in its original units. Then enter the four numbers in the command. A result near 0 means little of the curve is between the bounds; a result near 1 means most of it is.
Identify what \(X\) measures, its units, and the normal model’s mean and standard deviation.
Translate “between” into bounds \(a\) and \(b\), with the smaller value first.
Use \(\operatorname{normalcdf}(a,b,\mu,\sigma)\); keep every value in the original units.
Confirm the result is between 0 and 1 and plausible for the shaded region. State what proportion of modeled values lies in the interval.
A normal variable is continuous, so the probability of being exactly equal to one endpoint is 0. Thus the area between endpoints is the same whether the event uses strict inequalities or includes the endpoints. As in the earlier tutorial on sketching and shading normal curves, the picture helps you confirm that the bounds and requested region match.
Worked Example: A Middle Range of Assessment Scores
Worked Example: A Middle Range of Assessment Scores
A fictional school uses a normal model with mean 70 points and standard deviation 8 points for scores on a practice assessment. Find the proportion of scores between 62 and 82 points.
State. Let \(X\) be a practice assessment score, in points. The model is \(X\sim N(70,8)\), and the event is \(62<X<82\).
Plan. This is a between-values question. Enter the lower score first, followed by the upper score, the mean, and the standard deviation.
Do. Enter the values in the calculator:
As a check, standardize each bound using \(z=(x-\mu)/\sigma\). The lower bound has \(z=(62-70)/8=-1\), and the upper bound has \(z=(82-70)/8=1.5\). Table A gives left areas of about 0.1587 at \(-1\) and 0.9332 at 1.5. Their difference is \(0.9332-0.1587=0.7745\), which agrees with the calculator result to four decimal places.
Conclude. About 0.7745, or 77.45%, of scores in this model are between 62 and 82 points.
Worked Example: Fill Volumes in a Production Model
Worked Example: Fill Volumes in a Production Model
For a fictional production line, the fill volume of a container is modeled as normal with mean 500 milliliters and standard deviation 4 milliliters. Find the proportion of containers with fill volumes between 496 and 503 milliliters.
State. Let \(X\) be a container’s fill volume, in milliliters. The model is \(X\sim N(500,4)\), and the event is \(496<X<503\).
Plan. Both bounds are already in milliliters, the same units as the mean and standard deviation. Enter 496 as the lower bound and 503 as the upper bound.
Do. The calculator input and result are:
To check, the standardized lower bound is \(z=(496-500)/4=-1\), and the standardized upper bound is \(z=(503-500)/4=0.75\). The corresponding left areas are about 0.1587 and 0.7734. Subtracting gives \(0.7734-0.1587=0.6147\), matching the calculator output after rounding.
The interval is not centered at the mean: it extends 4 milliliters below the mean and 3 milliliters above it. The calculator does not require a symmetric interval; it finds the area between any correctly ordered pair of bounds.
Conclude. About 61.47% of container fill volumes in this model are between 496 and 503 milliliters.
Worked Example: An Interval That Is Not Centered at the Mean
Worked Example: An Interval That Is Not Centered at the Mean
A fictional greenhouse models the length of a particular type of stem as normal with mean 24 centimeters and standard deviation 4 centimeters. Find the proportion of modeled stem lengths between 22 and 29 centimeters.
State. Let \(X\) be stem length, in centimeters. The model is \(X\sim N(24,4)\), and the event is \(22<X<29\).
Plan. Enter the original stem-length values as the lower and upper bounds, followed by the model’s mean and standard deviation. Then check the result by finding the corresponding standard normal area.
Do. The calculator command gives:
For a separate check, standardize the endpoints. At 22 centimeters, \(z=(22-24)/4=-0.5\). At 29 centimeters, \(z=(29-24)/4=1.25\). Table A gives left areas of about 0.3085 and 0.8944. The difference is \(0.8944-0.3085=0.5859\), which is about 0.5858 when calculated with unrounded areas. The small difference here comes from rounding the table entries before subtracting.
This check also makes the interval’s location visible: its lower bound is half a standard deviation below the mean, while its upper bound is 1.25 standard deviations above. Since the interval includes the mean and extends farther to the right than to the left, an area greater than one-half is plausible.
Conclude. About 58.58% of stem lengths in this model are between 22 and 29 centimeters.
Common Mistakes and AP Exam Tips
The calculator can return a number even when the inputs do not match the question. Use the event, units, and model parameters to verify the setup before reporting the result.
- Reversing the bounds. Put the smaller value first. If the question asks for values from 22 to 29, enter 22 before 29. Reversing the bounds can produce an invalid or misleading result.
- Entering z-scores with the original mean and SD. If you enter original-value bounds such as 62 and 82, also enter the original mean and standard deviation. Do not mix those values with standardized bounds. If you choose to work with z-scores instead, use a standard normal model with mean 0 and standard deviation 1.
- Entering the variance instead of the standard deviation. If the standard deviation is 4, enter 4—not \(4^2=16\). The command’s final input is SD.
- Mixing units. A bound measured in milliliters cannot be combined with a mean measured in liters unless you first convert the values so that all inputs use the same units.
- Reporting a decimal without its meaning. A result such as 0.6147 should be interpreted in context as about 61.47% of modeled fill volumes between the stated bounds—not as 0.6147 milliliters.
- Rounding too early during a check. Table A entries are rounded. If their subtraction differs slightly from the calculator output, keep more calculator precision during the calculation and round the final area consistently.
For a full-credit response, make the event and the calculator setup visible, report the area to a sensible number of decimal places, and interpret it using the variable’s context and units. A calculator value by itself does not explain which region was found or what the probability represents.
Key Takeaway
For a between-values problem, normalcdf finds the area directly from the original bounds and the normal model’s parameters. The order is lower bound, upper bound, mean, and standard deviation. Standardizing the bounds and subtracting Table A areas provides a useful check, but is not required before using the calculator.
Check Your Understanding
For each question, identify the event and write the normalcdf input order before calculating. Interpret the resulting area in context.
- A fictional race’s finishing times follow \(N(42,5)\), in minutes. Write the command for the proportion of times between 37 and 48 minutes.
- A fictional device’s battery life follows \(N(10,1.2)\), in hours. Which four inputs would you use to find the proportion between 9 and 11 hours?
- For \(X\sim N(30,6)\), a student enters \(\operatorname{normalcdf}(24,36,30,36)\). Identify the input error and write the corrected command.
- A fictional plant-height model is \(N(18,2)\), in centimeters. Explain why the bounds 16 and 20 should be entered before 18 and 2.
- For \(X\sim N(50,10)\), describe how you could use z-scores and Table A to check the result from \(\operatorname{normalcdf}(40,60,50,10)\).