APPENDIX A COMPUTATIONALLY GENERATED RANDOM DIGITS 748 APPENDIX C CHI-SQUARE RIGHT-HAND TAIL PROBABILITIES 754

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1 IV Appendices APPENDIX A COMPUTATIONALLY GENERATED RANDOM DIGITS 748 APPENDIX B RANDOM NUMBER TABLES 750 APPENDIX C CHI-SQUARE RIGHT-HAND TAIL PROBABILITIES 754 APPENDIX D LINEAR INTERPOLATION 755 APPENDIX E STANDARD NORMAL DENSITY LEFT-HAND TAIL PROBABILITIES 757 APPENDIX F T-DISTRIBUTION RIGHT-HAND TAIL PROBABILITIES 759 APPENDIX G CUMULATIVE BINOMIAL PROBABILITIES 760 APPENDIX H F-DISTRIBUTION RIGHT-HAND TAIL PROBABILITIES 768 APPENDIX I BONFERRI SIMULTANEOUS CONFIDENCE INTERVALS 771 APPENDIX J CUMULATIVE POISSON PROBABILITIES

2 A COMPUTATIONALLY GENERATED RANDOM DIGITS Computers (and some calculators) can produce random digits rapidly and in very large quantities. Often it is not practical to require such devices to store long lists of random data. Therefore it is usually better to generate such random digits only as they are needed. Formulas have been invented to compute random digits. At first, you might think this is impossible, because if you compute a number by a formula then you will know what the next digit will be, and thus such digits cannot be random. This is true, but although these integers are generated deterministically, they nonetheless appear to be true random digits for all practical purposes. Digits produced by such formulas are therefore sometimes called pseudo-random. They are pseudo-random because they are not random. For most purposes, such pseudo-random digits work extremely well in that they appear in every observable way to be random and are very convenient to use. Some such formulas are very complicated; others are surprisingly simple. You can usually find out from a user manual or a consultant the formula that a particular computer program uses for generating its random numbers. There is one particularly easy method for producing pseudo-random numbers that you can use on a calculator or computer with little or no programming. It has some flaws and hence is not often used, but it shows us how a deterministic approach can produce digits that appear to be random. It is called mid-square method, and it was suggested by the mathematician John von Neumann. Take an arbitrary number of five (or more) digits (you could use the last five digits of your telephone number or social security number, for example.) Suppose we start with This is not a part of your list of random digits it is only the starting value for the procedure. We square this number: (63537) 2 = We now take the middle five digits, 69503, shown in boldface in the above equation. These are our first five random digits. Next we square these: (69503)2 = Again, we take the middle five digits and square: (30667)2 = We repeat this process as long as desired. We then write down the string of random digits. For our example, we get This string of digits may be regarded as a set of random digits. The advantage is that it has been mechanically produced. In this case, we used an ordinary calculator to do the job. A

3 Computationally Generated Random Digits 749 caution is necessary about this method: after a while it can sometimes start generating all 0s. Nobody uses this method for actual production of random number tables. We have presented it because it illustrates the idea that a simple computation done repeatedly can produce digits that appear to be random. In addition, you can certainly use it if you wish.

4 B RANDOM NUMBER TABLES Table B.1 Random Number Table for Tossing a Fair Coin:{0, 1}

5 Random Number Tables 751 Table B.2 Random Number Table for Rolling a Fair Six-Sided Die:{1, 2, 3, 4, 5, 6}

6 752 RANDOM NUMBER TABLES Table B.3 Random Number Table for Rolling a Fair 10-Sided Die: {1, 2, 3, 4, 5, 6, 7, 8, 9, 0}* *This 10-digit case is what is usually referred to when one refers to a random number table.

7 Random Number Tables 753 Table B.4 Random Number Table for a Standard Normal Variable

8 C CHI-SQUARE RIGHT-HAND TAIL PROBABILITIES Probability p Table C.1 Right-Hand Tail Chi-Square Density Critical Values x Right-Hand Tail Probability p df Table entry is the value of x (critical value) corresponding to P(x 2 df x) = p for the chosen probability p (column) and number of df (row).

9 D LINEAR INTERPOLATION Interpolation can be used to help find a value that is not provided in a table. The procedure is called linear interpolation, since it assumes that the relations between the values involved are (or are nearly) linear that is, they are straight-line relationships. This assumption gives very good estimates of desired values for the exercises in this book. Example 1 Find P(χ ). The chi-square table in Appendix C does not have an entry of 20.0 for a chi-square with 10 degrees of freedom. So interpolation must be used. The chi-square values of and are entries in the table, with corresponding probabilities (areas) of 0.05 and 0.025, respectively (that is, P(χ ) = 0.05 and P(χ ) = 0.025). So P(χ ) is somewhere between 0.05 and Note that = 2.17 and = Now consider the following proportionality argument. The following table shows the proportions we are dealing with. Chi-square Probability (area) d 20.0 P(χ ) The unknown probability P(χ ) is proportionally decreased from 0.05 by an amount d according to the proportionality equation Thus, Thus, P(χ 2 10 d = d = (0.025) )

10 756 LINEAR INTERPOLATION Example 2 Find the value of a in the following: The chi-square table in Appendix C does not have a column for a probability of However, there are columns for 0.05 and 0.025, so we can use interpolation to find the required chi-square value a with 12 degrees of freedom. Chi-square Probability (area) d 0.02 a Thus the unknown a is proportionally increased from by an amount d according to the proportionality equation So we have d 2.31 = = 0.8. d = 0.8(2.31) The unknown value a is thus a d = Therefore, P(χ ) 0.03.

11 E STANDARD NORMAL DENSITY LEFT-HAND TAIL PROBABILITIES Table E.1 Standard Normal Probabilities to Left of Given z Probability p z 0 z Table entry is the left-hand tail probability corresponding to P(Z z) = p for the chosen value of z given by the chosen row (tenths) and column (hundreths). For any z 3.5, the area is approximately 0; indeed, for any z 4, the area is 0 to four decimal places.

12 758 STANDARD NORMAL DENSITY LEFT-HAND TAIL PROBABILITIES Table E.1 Standard Normal Probabilities to Left of Given z (Continued) z Probability p 0 z Table entry is the left-hand tail probability corresponding to P(Z z) = p for the chosen value of z given by the chosen row (tenths) and column (hundreths). For any z 3.5, the area is approximately 1; indeed, for any z 4, the area is 1 to four decimal places.

13 Probability p F t* T-DISTRIBUTION RIGHT-HAND TAIL PROBABILITIES Table F Table entry for p and C is the critical value t with probability p lying to its right and probability C lying between t and t Upper tail probability p df % 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9% This row lists the corresponding standard normal critical values. Confidence level C

14 G CUMULATIVE BINOMIAL PROBABILITIES The table gives the cumulative probability of obtaining x or fewer successes in n independent Bernoulli trials, namely P(X x), where p = probability of success in a single trial. Table G Cumulative Binomial Probabilities n x p

15 Cumulative Binomial Probabilities 761 Table G Cumulative Binomial Probabilities (Continued) n x p

16 762 CUMULATIVE BINOMIAL PROBABILITIES Table G Cumulative Binomial Probabilities (Continued) n x p

17 Cumulative Binomial Probabilities 763 Table G Cumulative Binomial Probabilities (Continued) n x p

18 764 CUMULATIVE BINOMIAL PROBABILITIES Table G Cumulative Binomial Probabilities (Continued) n x p

19 Cumulative Binomial Probabilities 765 Table G Cumulative Binomial Probabilities (Continued) n x p

20 766 CUMULATIVE BINOMIAL PROBABILITIES Table G Cumulative Binomial Probabilities (Continued) n x p

21 Cumulative Binomial Probabilities 767 Table G Cumulative Binomial Probabilities (Continued) n x p