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Chi-Square Evaluation for the HP-67

This program is Copyright © 1976 by Hewlett-Packard and is used here by permission. This program was originally published in the HP-67 Math Pac 1.

This program is supplied without representation or warranty of any kind. Hewlett-Packard Company and The Museum of HP Calculators therefore assume no responsibility and shall have no liability, consequential or otherwise, of any kind arising from the use of this program material or any part thereof.

Card Labels
Chi-Square Evaluation
Shift Oi (Σ+) Oh (Σ-) →X22 →E  
Label Start P? Oi↑Ei(Σ+) Ok↑Ek(Σ-) →X12
Key A B C D E

Overview

This program calculates the value of the X2 statistic for the goodness of fit test by the equation

             X12 = ∑(i=1..n)(Oi - Ei)2/Ei

where
         Oi = observed frequency
         Ei = expected frequency

If the expected values are equal
          (E = Ei = (ΣOi)/n for all i)

then   X22 = (nΣOi2)/(ΣOi) - ΣOi

Note: In order to apply the goodness of fit test to a set of given data, combining some classes may be necessary to make sure that each expected frequency is not too small (say, not less than 5).

Reference: Mathematical Statistics. J. E. Freund. Prentice Hall. 1962

Instructions

Step Instructions Input Data/Units Keys Output Data/Units
1 Load side 1 and side 2.      
2 Initialize   A 0.00
3 To set print mode. *   B 1.00
4 For equal expected values, go to 10      
5 Do 6~7 for i = 1,2, ...., n      
6 Input: Oi Oi ENTER  
       Ei Ei C i
7 If you made a mistake in inputting Ok and Ek, then correct by → Ok ENTER↑  
    Ek D i-1
8 Calculate: X12   E X12
9 For a new case, go to 2      
10 Do 11~12 for i = 1,2, . . ., n      
11 Input: Oi Oi f A i
12 If you made a mistake in inputting Oh, then correct by → Oh f B i-1
13 Calculate: X22   f C X22
       E   f D E
         
  *Note: to clear print mode press →   CLF 0  

Example

Find the value of x2 statistic for the goodness of fit for the following data set:

   Oi   8    50    47    56    5     14
   Ei  9.6  46.75 51.85 54.4  8.25  9.15

X12 = 4.84

Keystrokes Outputs

A                               0.00
B                               1.00
8 ENTER↑ 9.6 C                  8.00 ***
                                9.60 ***
                                1.00 ***

50 ENTER↑ 46.75 C              50.00 ***
                               46.75 ***
                                2.00 ***

47 ENTER↑ 51.85 C              47.00 ***
                               51.85 ***
                                3.00 ***

56 ENTER↑ 54.4 C               56.00 ***
                               54.40 ***
                                4.00 ***

5 ENTER↑ 8.25 C                 5.00 ***
                                8.25 ***
                                5.00 ***

100 ENTER↑ 100 C               100.00 *** (error)
                               100.00 ***
                                 6.00 ***

100 ENTER↑ 100 D               100.00 *** (correction)
                               100.00 ***
                                 5.00 ***

14 ENTER↑ 9.15 C               14.00 ***
                                9.15 ***
                                6.00 ***

E                               4.84 *** (X12)

The following table shows the observed frequencies in tossing a die 120 times. X2 can be used to test if the die is fair. Note: Assume that the expected frequencies are equal.

Number        1  2  3  4  5  6
frequency Oi 25 17 15 23 24 16

X22 = 5.00
E = 20.00

Keystrokes                     Outputs
A                               0.00
B                               1.00
25 f A                          25.00 ***
                                 1.00 ***

17 f A                          17.00 ***
                                 2.00 ***

19 f A                          19.00 *** (error)
                                 3.00 ***

19 f B                          19.00 *** (correction)
                                 2.00 ***

15 f A                          15.00 ***
                                 3.00 ***

23 f A                          23.00 ***
                                 4.00 ***

24 f A                          24.00 ***
                                 5.00 ***

16 f A                          16.00 ***
                                 6.00 ***

f C                              5.00 *** (x22)
f D                             20.00 *** (E)

The Program

LINE  KEYS
001  *LBL A     Initilize
002   CF 0
003   CF 1
004   0
005   STO 1
006   STO 2
007   STO 3
008   RTN
009  *LBL C
010   F0?
011   GSB 9
012   STO 3     Σ +
013   -
014   X2
015   RCL 3
016   ÷
017   STO + 2
018   RCL 1
019   1
020   +
021   STO 1
022   GSB 8
023   RTN
024  *LBL E
025   F1?
026   GTO 1     X12
027   RCL 2
028   GSB 7
029   GSB 8
030   GSB 7
031   RTN
032  *LBL 1
033   1
034   RCL 1
035   RCL 3     X22
036   ×
037   RCL 2
038   ÷
039   RCL 2
040   -
041   GSB 7
042   GSB 8
043   RTN
044  *LBL d
045   RCL 2
046   RCL 1
047   ÷
048   GSB 8
049   GSB 7
050   RTN
051  *LBL D     For correction Oj, Ej.
052   F0?
053   GSB 9
054   STO 3
055   -
056   X2
057   RCL 3
058   ÷
059   STO - 2
060   RCL 1
061   1
062   -
063   STO 1
064   GSB 8
065   RTN
066  *LBL a     For equal expected frequency.
067   GSB 8
068   STO + 2
069   X2
070   STO + 3
071   RCL 1
072   1
073   +
074   STO 1
075   GSB 8
076   GSB 7
077   SF 1
078   RTN
079  *LBL b
080   GSB 8     For correction Oj.
081   STO - 2
082   X2
083   STO - 3
084   RCL 1
085   1
086   -
087   STO 1
088   GSB 8
089   GSB 7
090   RTN
091  *LBL B     Set flag 0 for print.
092   SF 0
093   1
094   RTN
095  *LBL 9
096   PRT SPC   Print Oj, Ej.
097   X⇔Y
098   PRTX
099   X⇔Y
100   PRTX
101   RTN
102  *LBL c
103   SF 1      Set flag 1 for equally expected frequency.
104   GTO E
105   RTN
106  *LBL 8     Subroutine for print x.
107   F0?
108   PRTX
109   RTN
110  *LBL 7
111   F0?       Subroutine for space.
112   PRT SPC
113   RTN

Register Use

R1  n
R2  Used
R3  Used

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