Source: FRAPPY Student Samples, Principles of Statistics
Tags: normal distribution, z-score, normalcdf, binomial probability, binompdf, AP Stats, normal probability calculation, combining normal and binomial
Difficulty: Intermediate Prerequisites: Basic probability rules, understanding of mean and standard deviation, familiarity with the standard normal (z) table or calculator functions.
Big picture: This material sits at the heart of introductory statistics. You are learning to calculate how likely it is that a single observation falls in a given range when the data follow a normal distribution, and then to take that probability and feed it into a binomial model when you have multiple independent trials. If you have been away for a few weeks, make sure you are comfortable with what a normal distribution looks like (bell-shaped, symmetric) and what mean and standard deviation control (centre and spread). This is the machinery you will use for inference later in the course.
TL;DR: Use z-scores to convert any normal distribution question into a standard normal probability. Once you have the probability for a single trial, you can plug it into the binomial formula to find the probability of a specific number of successes across multiple independent trials.
Normal distribution
A continuous, symmetric, bell-shaped distribution fully described by its mean (μ) and standard deviation (σ). In simple terms, most values cluster near the centre, and extreme values become increasingly rare the further you move from the mean.
Z-score
The number of standard deviations a value sits above or below the mean: z = (x – μ) / σ. Think of it as a universal ruler that lets you compare values from any normal distribution on the same scale.
normalcdf (calculator function)
A TI-83/84 function that returns the area (probability) under the normal curve between a lower bound and an upper bound, given a mean and standard deviation. In simple terms, it answers "what proportion of the data falls between these two values?"
Binomial distribution
A discrete distribution that counts the number of successes in a fixed number of independent trials, each with the same probability of success. Think of it as the "how many out of n?" distribution.
binompdf (calculator function)
A TI-83/84 function that returns the probability of getting exactly k successes in n trials. In simple terms, it answers "what is the chance of exactly this many?"
The scenario: Buckley Farms potato chip bags are labelled "16 ounces." Bag weights follow a normal distribution with μ = 16.15 oz and σ = 0.12 oz. The question asks for the probability that one randomly selected bag weighs less than 16 oz.
Step 1 – State the distribution. X ~ N(16.15, 0.12).
Step 2 – Calculate the z-score. z = (16 – 16.15) / 0.12 = –0.15 / 0.12 = –1.25
Step 3 – Find the probability. Using a z-table: look up z = –1.25 to get P(Z < –1.25) ≈ 0.1057. Using a calculator: normalcdf(–9999, 16, 16.15, 0.12) ≈ 0.1057.
Result: About 10.57% of bags weigh less than 16 oz.
Two valid approaches appear in the student samples:
Sample 1 used normalcdf directly with the raw values (lower: –9999, upper: 16, mean: 16.15, SD: 0.12).
Sample 2 computed the z-score first, drew a diagram marking 16 and 16.15 on the axis, then looked up the probability.
Both are correct. The z-score approach shows more of your reasoning and is generally safer on free-response exams because graders can see each step.
The follow-up question: if you randomly select 10 bags, what is the probability that exactly 2 weigh less than 16 oz?
Recognise the setup. Each bag is an independent trial. "Success" = bag weighs less than 16 oz. The probability of success on a single trial is p = 0.1057 (from part a). The number of trials is n = 10.
Apply the binomial formula. P(X = 2) = C(10, 2) × (0.1057)² × (1 – 0.1057)⁸
Using a calculator: binompdf(n: 10, p: 0.1057, x: 2) ≈ 0.2057.
Using the formula by hand (as Sample 2 did, though they omitted the combination coefficient): C(10, 2) = 45 (0.1057)² × (0.8943)⁸ ≈ 0.01117 × 0.4089 ≈ 0.004571 45 × 0.004571 ≈ 0.2057
Key point: Sample 2 wrote (0.1057)²(1 – 0.1057)⁸ = 0.004571 but did not multiply by C(10, 2). That is a common and costly mistake on the AP exam. The binomial formula requires the combination term because there are multiple arrangements of which 2 out of 10 bags are the light ones.
Z-score formula:
z = (x – μ) / σ
Binomial probability formula:
P(X = k) = C(n, k) × p^k × (1 – p)^(n – k)
where C(n, k) = n! / [k!(n – k)!]
Calculator syntax (TI-83/84):
normalcdf(lower, upper, μ, σ)
binompdf(n, p, k)
Quality control in manufacturing relies on exactly this kind of calculation. A factory sets a target fill weight and a tolerance, then uses normal probabilities to predict how many units will fall outside spec. The binomial extension lets them forecast how many defective items appear in a batch of a given size.
Students often forget the combination term C(n, k) when computing binomial probabilities by hand. The calculator handles it automatically with binompdf, but if you write the formula out, you must include it or you will get the wrong answer.
Students sometimes confuse normalcdf (area under the curve, i.e. a probability) with normalpdf (the height of the curve at a point, which is not a probability). On the exam, you almost always want normalcdf.
Some students set the lower bound of normalcdf to 0 instead of a very large negative number (like –9999 or –1E99). Because the normal distribution extends infinitely in both directions, 0 is not the correct lower bound unless your problem context truly starts at 0.
Students occasionally use the z-score as the final answer instead of converting it into a probability. The z-score is an intermediate step, not the answer to "what is the probability?"
⚠️ Free-response graders look for three things on normal probability questions: (1) stated distribution, (2) correct z-score or calculator input, (3) correct probability with interpretation. Missing any one of these costs points.
⚠️ When a problem says "exactly k out of n," that is a binomial cue. Confirm the conditions (fixed n, independent trials, same p, two outcomes) before applying the formula.
⚠️ Sample 2's error (omitting C(10,2)) is one of the most common on the AP exam. Always double-check that your binomial calculation includes the "choose" term.
True or false: A z-score of –1.25 means the value is 1.25 standard deviations above the mean.
A: False. A negative z-score means below the mean.
Fill in the blank: normalcdf(–9999, 16, 16.15, 0.12) returns the probability that a bag weighs ________ 16 ounces.
A: less than
True or false: binompdf gives you the probability of "at most k" successes.
A: False. binompdf gives exactly k successes. binomcdf gives "at most k."
Q: A machine fills bottles with a mean of 500 ml and a standard deviation of 4 ml. What is the z-score for a bottle containing 492 ml?
A: z = (492 – 500) / 4 = –8 / 4 = –2.00
Q: Using the z-score above, what is the probability that a randomly selected bottle contains less than 492 ml?
A: P(Z < –2.00) ≈ 0.0228, or about 2.28%.
Q: If you select 8 bottles independently and the probability that any single bottle is underfilled is 0.0228, what is the probability that exactly 1 bottle is underfilled?
A: P(X = 1) = C(8, 1) × (0.0228)¹ × (0.9772)⁷ ≈ 8 × 0.0228 × 0.8520 ≈ 0.1554.
Q: A student calculates (.1057)²(1 – .1057)⁸ and reports that as the binomial probability for exactly 2 successes in 10 trials. What did they forget?
A: They forgot to multiply by C(10, 2) = 45. The correct answer is 45 times larger.
This material connects directly to sampling distributions (Chapter 7 in most AP Stats texts), where you will use the normal model not for individual observations but for sample means. The z-score formula adapts to z = (x̄ – μ) / (σ / √n), and the logic is identical.
The binomial connection reappears in hypothesis testing for proportions, where the number of successes in n trials is modelled as binomial and then approximated by a normal distribution when n is large enough.
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