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.

HP 21S: Gompertz Curve

HP 21S:  Gompertz Curve

Introduction

This program fits sequential data to the Gompertz Curve:

y = c * a ^ (b ^ x)

where:

b = ( (S3 - S1) / (S2 - S1) ) ^ 1/m

a = e^( (S2 - S1) / b * (b - 1) / (b^m - 1)^2 )

c = e^( 1 / m * (S1 * S3 - S2^2) / (S1 + S3 - 2 * S2) )

where:

m = number of data points / 3

S1 = Σ ln x_i of the first tier 

S2 = Σ ln x_i of the second tier

S3 = Σ ln x_i of the third tier

HP 21S Program Code: Gompertz Curve

Labels:

A:  initialize the program

B:  enter data

C:  calculate b, a, c

Registers:

R1 = natural log of the sum of the first-third of data points

R2 = natural log of the sum of the second-third of data points

R3 = natural log of the sum of the third-third of data points

R4 = n = number of data points ÷ 3

R5 = a

R6 = b

R7 = c

Program Code:

step #: key code; key

01:  61, 41, A;  LBL A

02:  51, 75; CLRG

03:  61, 26;  RTN

04:  61, 41, b; LBL B

05:  13;  LN

06:  21, 75, 1; STO+ 1

07:  26;  R/S

08:  13;  LN

09:  21, 75, 2; STO+ 2

10:  26; R/S

11:  13; LN

12:  21, 75, 3; STO+ 3

13:  1;  1

14:  21, 75, 4; STO+ 4

15:  22, 4;  RCL 4

16:  61, 26;  RTN

17:  61, 41, C;  LBL C

18:  33;  (

19:  22, 3;  RCL 3

20:  65;  ×

21:  22, 2;  RCL 2

22:  34;  )

23:  45;  ÷

24:  33;  (

25:  22, 2;  RCL 2

26:  65;  -

27:  22, 1; RCL 1

28:  34;  )

29:  74;  =

30:  14;  y^x

31:  22, 4; RCL 4

32:  15;  1/x

33:  74;  =

34:  21, 6;  STO 6

35:  26;  R/S

36:  33;  (

37:  22, 2; RCL 2

38:  65;  -

39:  22, 1; RCL 1

40:  34;  )

41:  55;  ×

42:  33;  (

43:  22, 6;  RCL 6

44:  65;  -

45:  1;  1

46:  34;  )

47:  45;  ÷

48:  22, 6;  RCL 6

49:  45;  ÷

50:  33; (

51:  22, 6;  RCL 6

52:  14;  y^x

53:  22, 4;  RCL 4

54:  65;  -

55:  1;  1

56:  34;  )

57:  51, 11;  x^2

58:  74;  =

59:  12;  e^x

60:  21, 5;  STO 5

61:  26;  R/S

62:  33;  (

63:  22, 1; RCL 1

64:  55;  ×

65:  22, 3; RCL 3

66:  65;  -

67:  22, 2;  RCL 2

68:  51, 11;  x^2

69:  34;  )

70:  45;  ÷

71:  33;  (

72:  22, 4; RCL 4

73:  55;  ×

74:  33;  (

75:  22, 1;  RCL 1

76:  75;  +

77:  22, 3;  RCL 3

78:  65;  -

79:  2;  2

80:  55;  ×

81:  22, 2;  RCL 2

82:  34;  )

83:  34;  )

84:  74;  =

85:  12;  e^x

86:  21, 7;  STO 7

87:  61, 26;  RTN

Example

Fit the following data into a Gompertz curve:

(1, 16)

(2, 18)

(3, 19)

(4, 25)

(5, 28)

(6, 29)

(7, 32)

(8, 36)

(9, 37)

First, order the data points. Second, divide the data into three equal parts.  For this example, we have 9 data points.  From our example:

16, 18, 19, 25, 28, 29, 32, 36, 37

is divided into:

Group I:  16, 18, 19

Group II: 25, 28, 29

Group III: 32, 36, 37

Initialize the program:  XEQ A

Enter the data:

16 XEQ B, 25 R/S, 32 R/S

18 XEQ B, 28 R/S, 36 R/S

19 XEQ B, 29 R/S, 37 R/S

Calculate the parameters:  XEQ C

b:  0.82711002676, R/S

a:  2.33690515E-1, R/S

c:  48.2137954914

The Gompertz Curve is 

y = 48.2137954914  * 2.33690515E-1 ^ (0.82711002676 ^ x)

Predict the 10th point:

Key strokes:  RCL 6, y^x ,10, =, RCL 5, y^x, LAST, ×, RCL 7 

Result:  38.776391586

Source

HP-37E & HP-38E/38C:  Marketing and Forecasting Applications  Hewlett Packard.  May 1979.

Until next time,

Eddie

All original content copyright, © 2011-2023.  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. 

Intensity/Illumination, Days Since Jan. 1, Derivatives: HP 20S and 21S

Intensity/Illumination, Days Since Jan. 1, Derivatives: HP 20S and 21S

Table of contents

1.  Intensity and Illumination
2.  Days Since January 1
3.  Numerical Derivative

Disclaimer: I believe the key codes for the programs in this blog entry are the all the same even though the HP 20S and HP 21S have slightly different codes.  Format:   Step  Key: Key Code.  I took turns programming the HP 20S and HP 21S.

1.  Intensity and Illumination

The follow equation relates the luminous intensity (measured in candelas, cd) and illuminance (measured in lux) of a light source.  The equation assumes the light source radiates a spherical matter.

E = I / R^2

E = illuminance
I = luminous intensity
R = radius of the sphere's light (meters)

LBL A:  Solve for E
LBL B:  Solve for I
LBL C:  Solve for R

Registers:
R0 = E
R1 = I
R2 = R

Store the following values in the register and execute the appropriate label.

Program:

01  LBL A:  61,41,A
02 RCL 1:  22, 1
03 ÷:  45
04 RCL 2:  22, 2
05 x^2:  51, 11
06 =: 74
07 STO 0:  21, 0
08 R/S:  26
09 LBL B:  61,41,B
10  RCL 0: 22,0
11  *:  55
12  RCL 2: 22,2
13  x^2:  51,11
14  =:  74
15  STO 1: 21, 1
16  R/S:  26
17  LBL C: 61,41,C
18  RCL 1: 22,1
19 ÷: 45
20 RCL 0: 22,0
21  =:  74
22 √:  11
23 STO 2: 21, 2
24 R/S:  26

Example 1:
R1 = I = 400, R2 = R = 2.
Solve for E,  XEQ A returns 100

Example 2:
R0 = E = 180, R2 = R = 3
Solve for I, XEQ B returns 1620

Example 3:
R1 = I =420, R0 = E = 195
Solve for R, XEQ C returns 1.467598771

2.  Days Since January 1

Calculate the number of days since January 1.  For more information, please see:  http://edspi31415.blogspot.com/2019/03/ti-84-plus-and-hp-41c-number-of-days.html

Input:
R1:  day
R2: month
R3: 0 if we are working in a non-leap year, 1 if we are working in a leap year

Output:
R4:  number of days since January 1

Program:

01 LBL A: 61,41,A
02 RCL 1:  22,1
03 STO 4: 21, 4
04 3:  3
05 5:  5
06 STO - 4:  21,65,4
07 RCL 2: 22,2
08  INPUT:  31
09  2:  2
10  x ≤ y?:  61,42
11  GTO 2:  51,41,2
12  RCL 2: 22,2
13  *:  55
14  3:  3
15  0:  0
16  . : 73
17 6:  6
18  +:  75
19  1:  1
20  .  : 73
21  6:  6
22  =:  74
23  IP:  51, 45
24  STO + 4:  21,75,4
25  RCL 3:  22,3
26  STO + 4:  21,75,4
27  RCL 4: 22,4
28  RTN:  61, 26
29  LBL 2:  61,41,2
30  RCL 2:  22,2
31   *:  55
32  3:  3
33  0:  0
34  .  : 73
35  6:  6
36  +:  75
37  3:  3
38  6:  6
39  8:  8
40:  .  : 73
41  8:  8
42 =: 74
43  IP:  51,45
44  STO + 4:  21,75,4
45  3:  3
46  6:  6
47  5:  5
48  STO - 4:  21,65,4
49  RCL 4:  22,4
50  RTN:  61,26

Example 1:
1/1/2019 - 5/7/2019  (non-leap year)
R1:  7,  R2:  5,  R3:  0
Result:  R4 = 126

Example 2:
1/1/2020 - 11/14/2020  (leap year)
R1:  14,  R2:  11, R3:  1
Result:  R4 = 318

3.  Numerical Derivative

f'(x0) ≈ ( f(x0 + h) - f(x0 - h) ) / ( 2*h )

x = point
h = small change of x, example h = 0.0001

LBL A:  Main Progam
LBL F:  f(X), where R0 acts as X

Input variables:
R1 = h
R2 = point x0

Used variables:
R0 = x   (use R0 for f(x), LBL F)

Calculated Variables:
R3 = f'(x)

Radians mode will be set.

Program:

01  LBL A:  61,41A
02  RAD:  61,24
03  RCL 2:  22,2
04  +:  75
05  RCL 1:  22, 1
06  =:  74
07  STO 0:  21,0
08 XEQ F:  41,F
09  STO 3:  21, 3
10  RCL 2: 22,2
11 -:  65
12 RCL 1: 22,1
13 =:  74
14 STO 0: 21,0
15 XEQ F:  41,F
16 STO - 3:  21,65,3
17 2:  2
18 STO ÷ 3:  21,45,3
19  RCL 1:  22,1
20 STO ÷ 3: 21,45,3
21  RCL 3:  22,3
22 R/S:  26
23 LBL F:  61,41,F
...
xx  RTN:  61,26  (end f(X) with RTN)

Example:  e^x * sin x

LBL F
RCL 0
e^x
*
RCL 0
SIN 
= 
RTN

R1 = 0.0001
R2 = x0 = 0.03
Result:  1.060899867

R1 = 0.0001
R2 = x0 = 1.47
Result:  4.7648049


Note:  I am going on vacation this week and I have jury duty in June. So far, I have blog entries scheduled to be posted throughout June 22.  I plan to have a weekly post every Saturday in June.  - E

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.

HP 20S and HP 21S: Precession

HP 20S and HP 21S: Precession 

Introduction

The program PRECESS estimate the new position (right ascension (RA), declination (δ)) of a celestial object given their position in Epoch 2000 with the object’s proper notion. 

Estimation formulas:

Change in RA and δ before accounting for proper notion

ΔRA = m + n * sin RA * tan δ  (in seconds)

        = 3.07496 + 1.33621 * sin RA * tan δ  (for epoch 2000)

Δδ = (15 * n) * cos RA   (in arcseconds)

     = 20.0431 cos RA  (epoch 2000)

With m = 3.07496 seconds and n = 1.33621 seconds (for epoch 2000)

For other epochs:

1900:  m = 3.0731, n = 1.33678

2100:  m = 3.07682, n = 1.33564

Then:

RA_new = (RA_old + Y * (ΔRA + RA_proper) / 3600 ) / 15    (hours)

δ_new = (δ_old + Y * (Δδ + δ_proper) / 3600    (degrees)

Y = years from 2000  (or the appropriate epoch)

For the HP Prime and TI-84 Plus CE versions, click here:  http://edspi31415.blogspot.com/2018/09/hp-prime-and-ti-84-plus-ce-precession.html

HP 20S and HP 21S Program: Precession 

The key codes for both calculators are the same.

STEP  KEY     KEY CODES

01    LBL D   61, 41, d

02    DEG     61, 23

03    RCL 0   22, 0

04    →HR     51, 54

05    ×       55

06    1       1

07    5       5

08    =       74

09    STO 0   21, 0

10    RCL 1   22, 1

11    →HR     51, 54

12    STO 1   21, 1

13    3       3

14    .       73

15    0       0

16    7       7

17    4       4

18    9       9

19    6       6

20    +       75

21    1       1

22    .       73

23    3       3

24    3       3

25    6       6

26    2       2

27    1       1

28    ×       55

29    RCL 0   22, 0

30    SIN     23

31    ×       55

32    RCL 1   22, 1

33    TAN     25

34    =       74

35    STO 5   21, 5

36    2       2

37    0       0

38    .       73

39    0       0

40    4       4

41    3       3

42    1       1

43    ×       55

44    RCL 0   22, 0

45    COS     24

46    =       74

47    STO 6   21, 6

48    RCL 0   22, 0

49    +       75

50    RCL 4   22, 4

51    ×       55

52    (       33

53    RCL 5   22, 5

54    +       75

55    RCL 3   22, 3

56    )       34

57    ÷       45

58    3       3

59    6       6

60    0       0

61    0       0

62    =       74

63    ÷       45

64    1       1

65    5       5

66    =       74

67    →HMS    61, 54

68    STO 7   21, 7

69    R/S     26

70    RCL 1   22, 1

71    +       75

72    RCL 4   22, 4

73    ×       55

74    (       33

75    RCL 6   22, 6

76    +       75

77    RCL 3   22, 3

78    )       34

79    ÷       45

80    3       3

81    6       6

82    0       0

83    0       0

84    =       74

85    →HMS    61, 54

86    STO 8   21, 8

87    RTN     61, 26

Instructions

Store the following values in the registers:

R0:  Initial RA in HH.MMSSSS format

R1:  Initial δ in DD.MMSSSS format

R2:  RA proper notion

R3:  δ proper notion

R4:  number of years from 2000.  For 2022, store 22.  For 1978, store -22.  

Result:  

R7:  New RA in HH.MMSSSS format, press [ R/S ] to get

R8:  New δ in DD.MMSSSS format

Examples

Estimate the RA and δ of Regulus (Alpha Leonis) and Sadalmelik (Alpha Aquarii) for 2020 (Y = 20).  (data from Wikipedia)

Regulus (Leo the Lion  ♌)

Epoch 2000: RA = 10h 8m 22.311s, δ = +11° 58’ 0.195”

Proper Notion:  RA_prop = -0.016582 arcsec/yr, δ_prop = 0.00556 arcsec/yr

 (arcsec = “)

Results:

RA_2020 ≈ 10h 8m 26.56557s  (shown as 10°08’26.56557”)

‘δ_2020 ≈ +11° 52’ 6.06118”

Sadalmelik (Aquarius the Water Bearer ♒)

Epoch 2000:  RA = 22h 5m 47.03593s, δ = -0° 19’ 11.4568”

Proper Notion:  RA_prop = 1.216667 * 10^-3 arcsec/yr, δ_prop = -0.00939 arcsec/yr

Result:

RA_2020 ≈ 22h 5m 51.14225s

δ ≈ -0° 13’ 19.5407”

Convert mas/yr to arcsec/yr:

For RA:  (x/15) /1000

For δ:  x/1000

Sources:

Jones, Aubrey.  Mathematical Astronomy with a Pocket Calculator  John Wiley & Sons: New York.  Printed in Great Britain. 1978.  ISBN 0 470 26552 3

Meeus, Jean.  Astronomical Algorithms  William-Bell, Inc.  Richmond, VA 1991.  ISBN 0-943396-35-2

Eddie

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