HP 21S and TI-84 Plus CE: Normal Distributions and Hypothesis Tests

HP 21S and TI-84 Plus CE: Normal Distributions and Hypothesis Tests

Let’s compare how calculations involving the normal distribution are done between a classic calculator and a current one. The classic calculator is the rare HP 21S calculator from 1988 and the current calculator is the TI-84 Plus CE.

In 2017 (how time flies!) I wrote a review for the HP 21S (along with it’s cousin HP 20S):

https://edspi31415.blogspot.com/2017/04/retro-review-hewlett-packard-hp-20s-and.html.

Normal Distribution Calculations

z = z-score, point; p = probability, area

Assumptions: The mean is assumed to be 0 while the standard deviation is 1. (μ = 0, σ = 1).

Lower Tail (-∞, z)

	HP 21S	TI-84 Plus CE*	Example
Given z, find p	Keystrokes: 1 [ - ] z [ ←| ] (Q(z)) [ = ]	normalcdf(-1E99, z)	Input: z = 0.77 Result: p ≈ 0.7794
Given p, find z	Keystrokes: [ ( ] 1 [ - ] p [ ) ] [ |→ ] (zp)	invNorm(p)	Input: p = 0.77 Result: z ≈ 0.7388

* also includes TI-83 Plus, TI-84 Plus, TI-83 Premium (Python), TI-82 Advanced, and I do believe it is the same syntax for TI-89/Nspire family

Upper Tail (z, +∞)

	HP 21S	TI-84 Plus CE*	Example
Given z, find p	Keystrokes: z [ ←| ] (Q(z))	normalcdf(z, 1E99)	Input: z = 0.49 Result: p ≈ 0.3121
Given p, find z	Keystrokes: p [ |→ ] (zp)	InvNorm(1 - p)	Input: p = 0.49 Result: z ≈ 0.0251

Two Tail (-z, z)

	HP 21S**	TI-84 Plus CE*	Example
Given z, find p	Keystrokes: z [ STO ] [ 0 ] [ +/- ] [ ←| ] (Q(z)) [ - ] [ RCL ] 0 [ ←| ] (Q(z)) [ = ]	normalcdf(-z, z)	Input: z = 1 Result: p ≈ 0.6827
Given p, find z	Keystrokes: p [ ÷ ] 2 [ = ] [ STO ] 0 Lower Limit: [ ( ] 0.5 [ - ] [ RCL ] 0 [ ) ] [ |→ ] (zp) Upper Limit: [ ( ] 0.5 [ + ] [ RCL ] 0 [ ) ] [ |→ ] (zp)	InvNorm(p, 0, 1, CENTER)	Input: p = 0.25 Result: Upper: z ≈ 0.3186 Lower: z ≈ -0.3186

** HP 21S: A memory register is needed for this particular algorithm. I use register 0 for this example.

Sample Test of the Mean: Is the Proposed Mean the True Mean? (μ0 = μ)

Perform a significant test of whether a proposed mean (μ0) is the true mean (μ) given data from a sample:

x-bar: arithmetic mean of a sample

σ: population deviation of the sample

n: sample size

In this test, the null or default hypothesis is μ =μ0, while the alternate hypothesis is μ ≠ μ0.

Your confidence level and critical level, α, are complimentary. For example, if you want a 95% confidence level, your critical level, or α, is: α = 1 – 0.95 = 0.05 (5%). This is a two-tail test.

HP 21S	TI-84 Plus CE
[ ←| ] [ ← ] (LOAD) (A) (1-Stat) n [ XEQ ] D μ [ R/S ] σ [ R/S ] Test Method # 1: α [ ÷ ] 2 [ = ] [ |→ ] (zp) (= critical value) μ0 [ XEQ ] B (= test value) Null: μ0 = x-bar, Alternative: μ0 ≠ x-bar If test value < critical value, do not reject null hypothesis If test value > critical value, reject null hypothesis and accept the alternate hypothesis Test Method # 2: μ0 [ XEQ ] B [ ←| ] [ 1 ] (Q(z)) (= p) Null: μ0 = x-bar, Alternative: μ0 ≠ x-bar If p > α / 2, do not reject null hypothesis If p < α / 2, reject null hypothesis and accept the alternate hypothesis Note: LBL D: store summary statistics for the test of one sample mean or probability. For this test, the standard error that is shown will not be used. LBL B: calculate the z-score: z = (x-bar – μ0) / (σ / √n)	[ stat ], TESTS, 1: Z-Test… Inpt: Stats (Input shown as Inpt) μ0: enter μ0 σ: enter σ x-bar: enter x-bar n: enter n Select μ ≠ μ0 (alternate hypothesis) Select Calculate Test Method # 1: critical value: normalcdf(α/2, 1E99) test value: [ vars ], 5: Statistics…, TEST, 2: z Null: μ0 = μ, Alternative: μ0 ≠ μ If test value < critical value, do not reject null hypothesis If test value > critical value, reject null hypothesis and accept the alternate hypothesis Test Method # 2: Compare p to α Null: μ0 = x-bar, Alternative: μ0 ≠ x-bar If p > α, do not reject null hypothesis If p < α, reject null hypothesis and accept the alternate hypothesis One line command: Z-Test(μ0, σ, x-bar, n, alt) alt: -1 for μ0 < μ, 0 for μ0 ≠ μ, 1 for μ0 > μ
Example: n = 515, μ = 7.65, σ = 3.56 α = 5% = 0.05, μ0 = 7.7, α / 2 = 0.025 [ ←| ] [ ← ] (LOAD) (A) 515 [ XEQ ] D 7.65 [ R/S ] 3.56 [ R/S ] 0.05 [ ÷ ] 2 [ = ] [ |→ ] (zp) (critical value = 1.9600) 7.7 [ XEQ ] B (z ≈ -0.3187 < 1.9600) 7.7 [ XEQ ] B [ ←| ] [ 1 ] (Q(z)) (p ≈ 0.6250 > 0.025) Do not reject null hypothesis	Example: n = 515, μ = 7.65, σ = 3.56 α = 5% = 0.05, μ0 = 7.7 [ stat ], TESTS, 1: Z-Test… Inpt: Stats μ0: 7.7 σ: 3.56 x-bar: 7.65 n: 515 Select μ ≠ μ0 Select Calculate Results: μ≠7.7 z=-0.3187304977 p=0.7499310153 x-bar=7.65 n=515 p ≈ 0.7499 > 0.05 z ≈ -0.3187 < 1.9600 Do not reject null hypothesis

This is one of several z-tests that can be performed.

Confidence Interval

Determine a confidence level of where the true mean is located given data from a sample:

x-bar: arithmetic mean of a sample

σ: population deviation of the sample

n: sample size

Confidence Mean:

x-bar ± (z * σ) / √n

z: absolute value of the z-score where the two tail probability equals c.

Using TI-84 Plus CE commands: invNorm(c, 0, 1, CENTER)

HP 21S	TI-84 Plus CE
[ ←| ] [ ← ] (LOAD) (A) n [ XEQ ] D μ [ R/S ] σ [ R/S ] Confidential Interval: [ ( ] 1 [ - ] c [ ) ] [ ÷ ] 2 [ = ] [ |→ ] [ 1 ] (zp) [ XEQ ] C → precision [ R/S ] → lower limit [ R/S ] → upper limit Note: LBL C: Confidence interval precision = (z * σ) / √n lower limit = x-bar – precision upper limit = x-bar + precision	[ stat ], TESTS, 7: Zinterval… Inpt: Stats (Input is shown as Inpt) σ: enter σ x-bar: enter x-bar n: enter n C-Level: enter confidence level as a decimal Calculate One line command: ZInterval σ, x-bar, n, c
n = 389, μ = 88.84, σ = 10.25 c = 90% = 0.9 [ ←| ] [ ← ] (LOAD) (A) 389 [ XEQ ] D 88.84 [ R/S ] 10.25 [ R/S ] [ ( ] 1 [ - ] 0.9 [ ) ] [ ÷ ] 2 [ = ] [ |→ ] [ 1 ] (zp) [ XEQ ] C → 0.8548 [ R/S ] → 87.9852 [ R/S ] → 89.6948	n = 389, μ = 88.84, σ = 10.25 c = 90% = 0.9 Inpt: Stats σ: 10.25 x-bar: 88.84 n: 389 C-Level: 0.9 Result: (87.985, 89.695) x-bar = 88.84 n = 389

Sources

CalcBlog “Hypothesis Testing using Z-Test on the TI-83 Plus, TI-84 Plus, TI-89, and Voyage 200” February 6, 2011. https://www.calcblog.com/hypothesis-testing-z-test-on-ti84-ti89-graphing-calculator/ Accessed January 30, 2025.

Hewlett Packard HP-21 Stat/Math Calculator: Owner’s Manual Edition 3. June 1990.

TI-Basic Developer “The Z-Test( Command” http://tibasicdev.wikidot.com/z-test Accessed January 30, 2025.

Enjoy! Until next time,

Eddie

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Fun with the TI-81: Part II

Fun with the TI-81:  Part II

Please check out yesterday's blog entry for Part I.  Now let's continue.

TI-81 Decimal to Fraction:  DECTOFRAC
(148 bytes)

This program converts a number X to fraction as an approximation.  Each successful approximation is displayed until the absolute value of the error falls below 10^-12.  This program is based on a program written for the Radio Shack TRS-80 (see source). 

Variables:
X = Number
H = numerator
I = denominator
Y = error
Used: A, B, C, D, E, F, G, J

Program:
Disp "X="
Input X
IPart X → A
1 → B
X - A → C
0 → D
1 → E
1 → F
0 → G
Lbl 2
A * F + D → H
A * G + E → I
H / I - X → Y
ClrHome
Disp H
Disp "/"
Disp I 
Disp "ERR="
Disp Y
Pause
If abs Y<1e-12 font="">
Stop
C → J
B - A * C → C
J → B
F → D
H → F
G → E 
I → G
Goto 2

Example:
X = 4.7995
4 / 1,  ERR = -.7995
5 / 1,  ERR = .2005
13 / 4, ERR = -.0495
24 / 5,  ERR = 5E-4
1915 / 399, ERR = -1.253133E-6
9599 / 2000, ERR = 0

4.7995 = 9599/2000

Source:

Craig, John Clark  119 Practical Programs for the TRS-80 Pocket Computer  Tabs Books Inc.:  Blue Ridge, PA.  1982 ISBN 0-8306-0061-1 (Paperback)

TI-81 Simple Logistic Regression:  LOGISFIT 
(71 bytes)

The program LOGISFIT fits the statistical data to the equation:

y = 1 / (a + b*e^x)

This program uses the linear regression fit with the following translations:

x' = e^(-x), y' = 1/y

This fit will is good for all data except when y = 0.

Instructions: 
1.  Enter the data through the Stat Edit menu. 
2.  Run LOGISFIT.  The data will be altered. 

Program:
1 → I
Lbl 1
e^( -{x}(I) ) → {x}(I)
{y}(I)⁻¹ → {y}(I)
IS>( I, Dim{x} )
Goto 1
LinReg
Disp "Y=1/(a+be^(X))"
Disp "a"
Disp a
Disp "b"
Disp b

Example:
x1 = 0.5
y1 = 0.384
x2 = 1
y2 = 0.422
x3 = 1.5
y3 = 0.45
x4 = 2
y4 = .468
x5 = 2.5
y5 = .48

Results:
a = 2.001859259
b = .9942654005

Equation:
y = 1 / (2.001859259 + .9942654005*e^x)

Source:

Shore, Edward.  "HP Prime and TI-84 Plus CE: Simple Logistic Regression"  Eddie's Math and Calculator Blog. 2017.  http://edspi31415.blogspot.com/2017/04/hp-prime-and-ti-84-plus-ce-simple.html
Retrieved August 17, 2019

TI -81 Confidence Intervals: INTERVAL
(184 bytes)

The program INTERVAL calculates a confidence interval given the sample's mean (M), variance (V), and number of data points (N).  A Z scored is selected when the user selects one of three confidence levels: 

99%  (0.5% on each side of the curve, Z = 2.575829586)
95%  (2.5% on each side of the curve, Z = 1.959963986)
90%.  (5% on each side of the curve, Z = 1.644853627)

The interval lies between ( M - Z * V/√N,  M + Z * V/√N )

Notes:
1.  Z is used as an control variable and the Z score.
2.  The percent symbol is built of three characters, the degree symbol (°), the forward slash by pressing the [ ÷ ] key (/), the decimal point (.).

Program:
0 → Z
Disp "MEAN="
Input M
Disp "VAR="
Input V
Disp "N="
Input N
Lbl 0
ClrHome
Disp "1. 99°/."
Disp "2. 95°/."
Disp "3. 90°/."
Input P
If P=1
2.575829586 → Z
If P=2
1.959963986 → Z
If P=3
1.644853627 → Z
If Z=0
Goto 0
M + Z * V / √N → U
M - Z * V / √N → V
Disp "INTERVAL"
Disp U
Disp V

Example:
Input:  n = 100, M = 156.39, V = 10.94, 99% confidence interval
Results: 
162.2079576
156.5720424

Source:
Kelly, Kathy A., Robert E. Whitsitt II, M. Deal LaMont, Dr. Ralph A. Olivia, et all.  Scientific Calculator Sourcebook   Texas Instruments Inc.  1981.  (no ISBN number is given)

TI-81 Fresnel Polarization:  MICROPOL
(120 bytes)

Given a microwave transferring from one medium to another with the initial angle with respect to the plane surface that separates the mediums, the following are calculated:

1.  Angle of refraction, θt
2.  Fresnel Horizontal Polarization, R_H
3.  Fresnel Vertical Polarization, R_V

The Law of Refraction:
n1 sin θi = n2 sin θt

Fresnel Horizontal Polarization:
R_H = sin(θ_i - θ_t) / sin(θ_i + θ_t)

Fresnel Vertical Polarization:
R_V = tan(θ_i - θ_t) / tan(θ_i + θ_t)

Variables:
N = n_1  (index of refraction of medium 1)
M = n_2  (index of refraction of medium 2)
θ = θ_i  (angle of incidence)
Z = θ_t (angle of refraction)
H = R_H (Fresnel horizontal polarization)
V = R_V (Fresnel vertical polarization)

Note:  Angles are in degrees

Program:
Deg
Disp "N1="
Input N
Disp "θ="
Input θ
Disp "N2="
Input M
sin⁻¹ (Nsin θ / M) → Z
sin(θ-Z) / sin(θ+Z) → H 
tan (θ-Z) / tan (θ+Z) → V
Disp "REFRACT θ=" 
Disp Z
Disp "H-POLAR="
Disp H
Disp "V-POLAR="
Disp V

Example:
Inputs:  N1 = 1.001, θ = 40°, N2 = 1.333
Results:
REFRACT θ = 28.86146514°
H-POLAR = .2071186671
V-POLAR = .0761259908

Source:
Barue, Geraud  Microwave Engineering:  Land & Space Communications  John Wiley & Sons: Hoboken, NJ 2008.  ISBN 978-0-470-08996-5

TI-81 Hyperbolic Circles: Circumference and Area:  HYPCIRCL
(61 bytes)

The program HYPCIRCL calculates the circumference and area of a circle in hyperbolic space.  Note that this not the same as (normal, regular, everyday) circles in Euclidean space. 

Circumference of a hyperbolic circle: C = 2 π sinh(R)

Area of a hyperbolic circle:  A = 4 π sinh(R/2)^2

Program:
Disp "HYP CIRCLE"
Disp "R="
Input R
2π sinh R → C
4 π (sinh(R/2))² → A
Disp "C="
Disp C
Disp "A="
Disp A

Example:
Input: R = 3
Results: 
C = 62.94416455
A = 56.97380062

Source:
Series, Caroline  "Hyperbolic Geometry MA 448"  2010. Edited Jan. 4, 2013

Eddie

All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.