TI-84 Plus CE lesson 11: DISTR: normal, binomial, Poisson, tests and intervals

Probability distributions are on every calculator paper. On the TI-84 Plus CE they are all in one menu, DISTR. The marks are lost on the details: the order of μ and σ, a bound of 1E99, and turning "at least" into a binomcdf.

Lesson 11 of 12
About 55 minutes
TI-84 Plus CE (OS 5.x)
Free, no sign-in

Objective

Find normal, binomial and Poisson probabilities and inverse normal values from the DISTR menu, and run a χ² goodness of fit test and a t-interval.

By the end, students can:

use normalcdf(lower, upper, μ, σ) with ±1E99 for one-sided probabilities
use invNorm with the Tail setting to find a value from a probability
use binompdf and binomcdf, writing "at least" as 1 − binomcdf
use poissonpdf and poissoncdf
run χ²GOF-Test and TInterval and read p-values written in E notation

Which courses this lesson suits

IB Maths AA (SL and HL): Papers 2 and 3
IB Maths AI (SL and HL): every paper
IB Maths AI HL
A Level Maths (UK and IAL): calculator allowed in every paper

Steps 1–5 are for every course. Steps 6–7 (the χ² goodness of fit test and the t-interval) are IB AI HL; A Level students meet tests through the binomial and normal distributions instead.

The key presses, step by step

Every key is shown as it is printed on the calculator, like 2nd or enter. A second function (printed above a key) is 2nd then that key; a menu item is chosen by its number, or by moving to it and pressing enter.

2nd vars opens DISTR. 1: normalpdf( is the height of the curve: not a probability. 2: normalcdf( is the one you want for P(a < X < b). Choose 2.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
X ~ N(172, 6.5²). For P(165 < X < 180): lower 165, upper 180, μ 172, σ 6.5. Move to Paste, enter, enter: 0.7500380904.
For a one-sided probability use 1E99 as the missing bound: E is 2nd , (EE). P(X > 200) = normalcdf(200, 1E99, 172, 6.5) = 8.248332531E−6.
8.248332531E−6 means about 0.00000825: read the E−6 at the end before you write the answer. The TI order is μ then σ, the opposite of the Casio.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
Inverse normal: 2nd vars 3: invNorm(. area 0.95, μ 172, σ 6.5, Tail: LEFT, Paste: 182.6915486. This is the k with P(X < k) = 0.95.
For P(X > k) = 0.1, choose Tail: RIGHT with area 0.1, or use LEFT with area 0.9. Both give the same k.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
Binomial, X ~ B(15, 0.22). Scroll down DISTR to A: binompdf( (exactly) and B: binomcdf( (at most). binompdf(15, 0.22, 3) = P(X = 3) = 0.2457053831. binomcdf(15, 0.22, 4) = P(X ≤ 4) = 0.7805296924.
At least: P(X ≥ 5) = 1 − P(X ≤ 4) = 1 − binomcdf(15, 0.22, 4) = 0.2194703076.
binomcdf always means "less than or equal to". Rewrite the question in that form first: X > 4, X ≥ 5 and "at least 5" are all 1 − binomcdf(…, 4).
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
Poisson, X ~ Po(2.8): C: poissonpdf(2.8, 2) = P(X = 2) = 0.2383754455; D: poissoncdf(2.8, 3) = P(X ≤ 3) = 0.6919374326; P(X > 4) = 1 − poissoncdf(2.8, 4) = 0.152323943.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
AI HL: a χ² goodness of fit test. A spinner with 4 equal sectors is spun 100 times: 28, 35, 22, 15. Put the observed counts in L1 and the expected counts (25 each) in L2. stat, TESTS tab, D: χ²GOF-Test…. Observed L1, Expected L2, df 3, Calculate.
df is the number of categories minus 1 (minus any parameters you estimated from the data).
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
AI HL: a t-interval. Eight masses, 4.2, 3.8, 4.5, 4.9, 4.1, 3.6, 4.4, 4.7, go in L1. stat, TESTS tab, 8: TInterval…. Inpt: Data, List L1, Freq 1, C-Level 0.95, Calculate.
C-Level is a decimal, 0.95. With summary statistics instead of data, choose Inpt: Stats and type x̄, Sx and n.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).

Your turn: check your understanding

Try these straight after the key presses, on your own calculator, before looking at the answers.

X ~ N(50, 8²). Find P(X < 60).
X ~ B(10, 0.4). Find P(X ≥ 6).
A test gives p = 4.5E−3. Is this less than 0.05? Write p as an ordinary decimal.

Show the answers

normalcdf(−1E99, 60, 50, 8) = 0.894 (3 s.f.)
1 − binomcdf(10, 0.4, 5) = 0.166 (3 s.f.)
p = 0.0045, which is less than 0.05. E−3 moves the decimal point 3 places left.

Exam efficiency

Write the probability statement first, e.g. X ~ N(172, 6.5²), P(165 < X < 180) = …, not "normalcdf(165, 180, 172, 6.5)": calculator syntax is not working.
Store a probability you will reuse (sto→ alpha letter) instead of retyping a rounded value.
For discrete distributions, turn every inequality into ≤ before using a cdf: P(X < 5) = P(X ≤ 4).
Read the end of every answer for E−: 8.25E−6 is not 8.25.
In a test, write the hypotheses, the p-value and the conclusion in context.

Worked exam-style example

Question. Adult heights X cm follow N(172, 6.5²). (a) Find the probability that a randomly chosen adult is taller than 185 cm. (b) 15 adults are chosen at random. Find the probability that at least one is taller than 185 cm.

(a) normalcdf(185, 1E99, 172, 6.5) enter: 0.0227501319. Store it at once: sto→ alpha math (A) enter.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
(b) Y ~ B(15, p) with p = A. At least one: 1 − P(Y = 0) = 1 − binompdf(15, A, 0) enter: 0.291916773.
Using A keeps every digit of p. With p rounded to 0.0228, the last figure can change.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).

What to write down

(a) P(X > 185) = 0.0228

(b) Y ~ B(15, 0.0228); P(Y ≥ 1) = 1 − P(Y = 0) = 0.292

Practice

Do these on your own calculator. Write the answers to 3 significant figures unless the task says otherwise.

X ~ N(80, 12²). Find (a) P(X < 65), (b) P(70 < X < 95).
X ~ N(80, 12²). Find k such that P(X > k) = 0.1.
X ~ B(20, 0.35). Find (a) P(X = 7), (b) P(X ≥ 10).
X ~ Po(4.5). Find (a) P(X = 3), (b) P(X > 6).
AI HL. Find a 95% confidence interval for the mean from the sample 12.1, 11.6, 12.8, 12.3, 11.9, 12.5.
AI HL. Observed counts 18, 30, 32, 20 are tested against equal expected counts of 25. Find χ² and the p-value, and say whether the counts fit at the 5% level.

Show the practice answers

(a) 0.106 (b) 0.692
k = 95.4 (invNorm with area 0.1 and Tail: RIGHT, or area 0.9 and LEFT)
(a) 0.184 (b) 1 − binomcdf(20, 0.35, 9) = 0.122
(a) 0.169 (b) 1 − poissoncdf(4.5, 6) = 0.169
(11.7, 12.7) with x̄ = 12.2
χ² = 5.92, df = 3, p = 0.116: more than 0.05, so there is not enough evidence that the counts do not fit.

Common mistakes

σ and μ the wrong way round: the TI order is μ then σ.
Using normalpdf for a probability. It gives the height of the curve.
Typing −1E99 with the subtraction key: use (−).
P(X ≥ 5) as 1 − binomcdf(…, 5): that is P(X ≥ 6).
Reading 8.25E−6 as 8.25, or missing the E altogether.
Variance instead of standard deviation: N(172, 6.5²) has σ = 6.5, not 42.25.

1-minute quick check

Answer these without looking back. Most need no calculation: they check that the key presses have stuck.

Which keys open DISTR?
In normalcdf, which comes first, μ or σ?
What do you type as the upper bound for P(X > 30)?
Write P(X ≥ 4) for a binomial X using binomcdf.
In which menu is χ²GOF-Test?

Show the quick-check answers

2nd vars
μ
1E99 (1 2nd , 99)
1 − binomcdf(n, p, 3)
stat, TESTS tab

Teaching this lesson (about 55 minutes)

0–5 min	Starter: P(X ≥ 5), P(X > 5), P(X < 5): write each as P(X ≤ something).
5–18 min	Steps 1–3: normalcdf with 1E99, reading E notation, invNorm with Tail.
18–28 min	Steps 4–5: binomial and Poisson. First two "your turn" tasks.
28–36 min	Worked example: store p, then the binomial. Discuss why storing matters.
36–48 min	Practice 1–4. AI HL: steps 6–7 and practice 5–6 instead of 4.
48–52 min	Third "your turn" task: reading p-values.
52–55 min	1-minute quick check.

Students need their own TI-84 Plus CE. With a projector, the calculator-skills guides below have a Show on the board view and an animated walkthrough of the key presses.