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.

Casio fx-3900Pv: Linear System, Poisson Kernel, Normal Distribution, 3 x 3 Matrix Determinant

Casio fx-3900Pv: Linear System, Poisson Kernel, Normal Distribution, 3 x 3 Matrix Determinant

Today, we feature four programs for the classic Casio fx-3900Pv calculator. The fx-3900Pv has 300 steps.

My review from 2023: https://edspi31415.blogspot.com/2023/05/retro-review-casio-fx-3900pv.html

One of the great features of the early “inexpensive” algebraic keystroke programmable calculators (1980s) is that the programs can be seen and edited.

The programs listed will require inputs to be stored in to the memory registers prior to running the program by using the [ Kin ] key.

Casio fx-3900Pv: 2 x 2 Systems

This program solves 2 x 2 linear systems. The Constant memory registers (K1 – K6) are mapped as follows:

K4 * x + K5 * y = K6

K1 * x + K2 * y = K3

The equations are solved as follows:

x = (K2 * K6 – K5 * K3) / M

y = (-K1 * K6 + K4 * K3) / M

where M = (K4 * K2 – K1 * K5)

This program takes 36 steps.

Code:

Kout 4 (determinant)

×

Kout 2

-

Kout 1

×

Kout 5

=

Min

HLT (Pause program, shows determinant. We want a nonzero-determinant.)

(

Kout 2

×

Kout 6

-

Kout 5

×

Kout 3

)

÷

MR

=

HLT (Solve for x)

(

Kout 1

+/-

×

Kout 6

+

Kout 4

×

Kout 3

)

÷

MR

= (Solve for y)

Examples

Example 1

x – y = 5

x + 3 * y = 9

Store the parameters: 1 Kin 4, -1 Kin 5, 5 Kin 6; 1 Kin 1, 3 Kin 2, 9 Kin 3

Results: det = 4, x = 6, y = 1

Example 2

3 * x + 16 * y = 49

5 * x – 2 * y = 22

Store the parameters: 3 Kin 4, 16 Kin 5, 49 Kin 6; 5 Kin 1, -2 Kin 2, 22 Kin 3

Results: det = -86, x = 5.23255814, y = 2.081395349

Casio fx-3900Pv: Poisson Kernel

The program calculates the kernel using the formula:

Pr(θ) = (1 – r^2) / (1 – 2 * r * cos θ + r^2),

0 ≤ r < 1, -π < θ ≤ π

This program takes 22 steps.

Store r in register 1, θ in register 2.

Code:

Rad (Mode 5)

(

1

-

Kout 1

x^2

)

÷

(

1

-

2

×

Kout 1

×

Kout 2

cos

+

Kout 1

x^2

)

=

Examples

Example 1:

r = 0.5, θ = 2.1

0.5 Kin 1, 2.1 Kin 2

Result: 0.4273879049

Example 2:

r = 0.268, θ = 0.842

0.268 Kin 1, 0.842 Kin 2

Result: 1.298397221

Source:

“Poisson Kernel” Wikipedia. May 28, 2024. https://en.wikipedia.org/wiki/Poisson_kernel

Retrieved November 26, 2024

Casio fx-3900Pv: Normal Distribution (Integration)

The program, which is ran in Integration Mode (Mode 1):

∫( e^(-t^2 / 2) / √(2 * π) dt, a, b)

To calculate, really approximate the integral:

MODE 1 P#

Store a level n (1-9), SHIFT RUN

Lower limit RUN

Upper limit RUN

The level n corresponds to the integration intervals 2^n. The higher n is, the longer the calculation takes but the accurate the integral is.

The integrated variable is stored in memory M. Start the function using Min.

Code:

Min

(

MR

x^2

+/-

÷

2

)

e^x

÷

(

2

×

π

)

√

=

Examples

The following examples will use n = 4.

Example 1: lower limit = 0, upper limit = 3. Result: 0.49865

(actual ≈ 0.4986501019)

Example 2: lower limit = -1, upper limit = 1. Result: 0.6827

(actual ≈ 0.6826894921)

Example 3: lower limit = -2, upper limit = 1.5. Result: 0.910443

(actual ≈ 0.910 4426667)

Casio fx-3900Pv: 3 x 3 Determinant

This program calculates the determinant of a 3 x 3 matrix. How are we to do this with only seven memory registers?

The program sets the matrix as:

[ [ 1^st input, 2^nd input, 3^rd input ] [ K4, K5, K6 ], [ K1, K2, K3 ] ]

The user will be stop execution three times. At each time, enter the element corresponding to the top row.

For example:

[ [ 1, 2, 3 ] [ 4, 5, 6 ] [ 7, 8, 9 ] ]

Store 4 in K4, 5 in K5, 6 in K6; 7 in K1, 8 in K2, 9 in K3. While running, enter 1, 2, then 3 during program execution.

The determinant is stored to memory M. This program takes 40 steps.

Code:

(

Kout 5

×

Kout 3

-

Kout 2

×

Kout 6

)

×

ENT # (enter a legitimate number to continue, prompt for row 1, column 1)

=

Min

(

Kout 4

×

Kout 3

-

Kout 6

×

Kout 1

)

×

ENT # (prompt for row 1, column 2)

=

M-

(

Kout 4

×

Kout 2

-

Kout 5

×

Kout 1

)

×

ENT # (prompt for row 1, column 3)

=

M+

MR

Examples

Example 1:

[ [ 1, 5, 6 ] [ 2, 7, 8 ] [ 11, 4, 3 ] ]

Store:

(2^nd row) 2 Kin 4, 7 Kin 5, 8 Kin 6

(3^rd row) 11 Kin 1, 4 Kin 2, 3 Kin 3

Run the program (P1-P4): 1 RUN, 5 RUN, 6 RUN

Result: -15

Example 2:

[ [ -3, 0, 9 ] [ 0 , -3, 9 ] [ 9, -3, 0 ] ]

Store:

(2^nd row) 0 Kin 4, -3 Kin 5, 9 Kin 6

(3^rd row) 9 Kin 1, -3 Kin 2, 0 Kin 3

Run the program (P1-P4): -3 RUN, 0 RUN, 9 RUN

Result: 162

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.

Spotlight: Texas Instruments TI-30 ECO RS: The Solar TI-30Xa

Spotlight: Texas Instruments TI-30 ECO RS: The Solar TI-30Xa

Quick Facts

Model: TI-30 ECO RS

Company: Texas Instruments

Timeline: 2015 – present (Europe

Type: Scientific calculator

Power: Solar

Display: One line, 10 digits, 2 digit exponents

Introduction

The TI-30 ECO RS is a European all-solar powered version of the battery-powered, specifically the LR44 battery, powered TI-30Xa.

Why did I get the TI-30 ECO RS? Truth be told, I am partial to solar-powered scientific calculators. I still dream of the day a solar powered (even if it is hybrid with battery) graphing calculator, that’s yet to happen. Blue is my favorite color.

Per Texas Instruments, the TI-30 ECO RS was awarded the Blue Angel award, which is a certification for the product being environmental friendly. Along with the calculator being completely solar powered, the TI-30 ECO RS is made from recycled plastic.

Features

The TI-30 ECO RS is a standard and simple scientific calculator that follows the classic TI-30 line:

* One line display, up to 10 digits. Display modes include floating pint, fixed decimal, scientific notation, and engineering notation.

* Arithmetic, powers, roots, exponential and logarithm functions

* Trigonometric functions and inverses

* Hyperbolic functions and inverses

* Fraction calculations, including fraction/decimal conversions. The maximum denominator is 999.

* One-variable statistics including mean, standard deviation, sums

* Combinations, permutations, and factorial of positive integers (up to 69)

* Polar/Rectangular and Decimal/Decimal-Minute-Second conversions

* Three memory registers (M1, M2, M3)

The TI-30 ECO RS follows the algebraic operating system and the standard order of operations is followed with a maximum of four pending operations. There is no implied multiplication.

The calculator uses postfix notation, where some one-number operations are entered after the number. For example, to calculate the sine of 30 degrees, press 30 [ SIN ] (in degrees mode). For the anti-log of 2.6, press 2.6 [ 2^nd ] [ LOG ] <10^x>.

TI-30 ECO RS vs. TI-30Xa

Even though the feature set of the TI-30 ECO RS and the TI-30Xa are the same, there are some minor differences.

* The TI-30 ECO RS is solar powered and will never require a separate battery. This is not the first time Texas Instruments implemented the solar only design, as previous models include the TI-108, TI-31, TI-30 SLR+, the classic one-line TI-36X Solar, the classic TI-36 Solar, and the BA-Solar from 1987 are some examples. In the 1980s and 1990s, Texas Instruments used technology named Anylite Technology (TM).

* Pressing the [ON/AC] button clears everything including the memory and statistics registers and resets the calculator to Degrees mode. This is consistent with classic one-line TI solar powered calculators. If you intend to store constants long-term, you may want to consider the battery-powered TI-30Xa instead.

* Calculations on the solar powered TI-30 ECO RS are slightly slower than the TI-30Xa. I am guessing that it is due to the power source.

* Since 2015, both the TI-30 ECO RS and the TI-30Xa are free the logarithm bug (Datamath).

* The TI-30 ECO RS is sold in Europe. Outside of Europe, we have to buy a TI-30 ECO RS online. I was fortunately enough to buy one from an e-bay seller in Virginia.

Source

Woerner, Joerg. “Texas Instruments TI-30 ECO RS (2015)” Datamath Calculator Museum.

Updated May 17, 2016. Accessed June 6, 2024. http://www.datamath.org/Sci/Modern/TI-30ecRS_2015.htm

“Scientific calculator TI-30 ECO RS” Texas Instruments Calculators. 1995-2024. Accessed June 7, 2024. The page is in German. https://education.ti.com/de/produkte/taschenrechner/wissenschaftliche-rechner/ti-30-eco-rs

Eddie

All original content copyright, © 2011-2024. 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.