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

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