THE UNIVERSITY OF NEW SOUTH WALES

SCHOOL OF ECONOMICS

ECON1203/ECON2292 (ARTS) QUANTITATIVE METHODS B

FINAL EXAMINATION

Session1, 2006

Time allowed: Three hours.

Total marks: 65 marks.

There are FIVE questions in this examination.

Answer ALL five questions.

The marks assigned to each question are NOT of equal value.

The value of each sub-question is indicated in brackets.

On the front of your answer book, write the number of each question you have attempted.

Statistical tables and useful formulae are attached to this examination paper.

Electronic calculators may be used.

The examination paper may be retained by the candidate.

Answers must be written in ink. Pencils may be used only for drawing, sketching orgraphical work.

Show the working steps in your answers.

Question 1 [13 marks in total]

(a) Suppose that the annual percentage change in advertisement spending for arandomly-selected company listed in a large stock exchange for the year 2001approximately follows a normal distribution. The mean and standard deviation ofa random sample of changes in ad spending for 16 companies turn out to be 5%and 12% respectively.

(i) Assume that the population standard deviation is known to be 12%. Aconfidence interval for the mean change in ad spending is constructed. Itsupper confidence limit is 9.935%. What is the level of confidence?[2 marks]

(ii) Assume that the population standard deviation is unknown. Find the 95%confidence interval for the mean change in ad spending. [3 marks]

(b) A financial adviser uses the variance of returns from the set of stocks sherecommends as a measure of investment risk. The set consists of many stocks andthe distribution of the stock annual returns is approximately normal. She has arandom sample with 15 stock returns from the recommended set. The observedsample variance is 0.0256.

(i) Construct the 90% confidence interval for the population variance of thereturns. [2 marks]

(ii) Interpret the numerical result in (i) in terms of the definition of the 90%confidence interval. [2 marks]

(c) To decide whether or not a soccer game series should be aired, a commercial TVchannel needs to estimate the proportion of households that are interested in thesoccer series. Suppose that the TV channel's management wishes to have a 95%confidence interval for the proportion, of which the width is 0.06 (i.e. B = 0.03).

(i) Suppose that you are able to interview households about whether or notthey will watch the soccer series. Outline how would you construct therequired confidence interval? [2 marks]

(ii) Determine the minimum sample size that satisfies the above requirement.[2 marks]

2

Question 2 [10 marks in total]

(a) A watch running more than one minute faster or slower than the standard time ina year is regarded as defective. QWatches, a digital watch manufacturer, claimsthat the probability of its watch being defective is 0.05. Having received somecomplaints, a consumer protection agency plans to conduct a survey about thewatches made by QWatches.

(i) If QWatches' claim is correct, what is the probability that more than onewatch out of 10 surveyed consumers are defective? [1 mark]

(ii) If QWatches' claim is correct, use an appropriate approximation to find theprobability that more than 3 watches out of 80 surveyed consumers aredefective. [2 marks]

(iii) If QWatches' claim is correct, use an appropriate approximation to find theprobability that more than 8 watches out of 160 surveyed consumers aredefective. [2 marks]

(b) The City of Wildwood issues fine tickets to individuals who park in no-parkingzones. The fine depends on the day of week and the time of day. Historical dataindicate that 40%, 30%, 20% and 10% of the fine tickets are respectively for $20,$40, $60 and $80. Let Xbe the fine on a randomly-selected fine ticket. Find theexpected value and the standard deviation ofX. [2 marks]

(c) Anna, Brad and Candy work for the billing division of an electricity supplier,respectively handling 30%, 50% and 20% of the monthly bills. Historical datashow that 1%,2% and 3% of the bills handled respectively by Anna, Brad andCandy contain errors. Suppose that the division receives a complaint about abilling error. What is the probability that the error is made by Brad? [3 marks]

3

Question 3 [12 marks in total]

(a) Define what is meant by an unbiased estimator. Give two examples of estimatorsthat are unbiased. [2 marks]

(b) State the Central Limit Theorem. [2 marks]

(c) The number of hits received by a web-page during the hour 9am-l Oam varies from oneday to another. Below is a sample of the hits (already ranked) for the hour over 16 days.

25

36

36

47

47

58

59

517

(i) Find the median and the mode of this data set. [2 marks]

(ii) Find the first and third quartiles of this data set. [2 marks]

(d) In a corporation, a very small number of employees have extremely high salaries,whereas the majority of employees receive much lower salaries. Ifyou were thebargaining agent for the union, what statistic would you use to illustrate that theemployees have a low level of salaries? Why? [2 marks]

(e) A sample of marks from 12 students for a test in an economics subject yields amean of 25 and a standard deviation of 4. Suppose that the sample is enlarged to14 data points by including two additional marks with a common value of25.Find the mean and standard deviation of the enlarged sample. [2 marks]

4

Question 4 [15 marks in total]

(a) In hypothesis testing, data are used to examine a null hypothesis about a populationparameter against an alternative hypothesis. In deciding whether or not the nullhypothesis should be rejected,

(i) what is a Type I error and what is a Type II error? [I mark](ii) why is the level of significance usually chosen as a small positive

number? [I mark]

(b) People have different views on a reform for the national health care system. A surveyconsisting of 200 adults has been carried out. The survey data are summarised in the tablebelow. David is interested in the hypothesis that a person's view on the reform isindependent of the person's gender. By using the technique of hypothesis testing (X2 test),he found that the hypothesis could not be rejected. He used the 5% level of significancein his test.

Views on the reformGender For Against NeutralFemale 58 30 15Male 41 40 16

Repeat David's test to check ifhis conclusion is correct. Write down thehypotheses, the test statistic, the decision rule and your conclusion. [5 marks]

(c) Fund managers aim to achieve high returns for their funds. A newspaper article claimsthe mean return achieved by the fund managers in 2003 is 5%. A random sampleconsisting of the returns in 2003 from 16 fund managers indicates that the mean is -I %and the standard deviation is 8%. Test the null hypothesis that the mean return in 2003 is5% against the alternative hypothesis that the mean return in 2003 is less than 5%,assuming that the return is approximately normally distributed. Use the 5% level ofsignificance. [3 marks]

(d) A government spokesman claims that 55% of voters support a recent tax cut.Sam, an accounting graduate, decides to test the spokesman's claim by using thefollowing procedure. First, 10 independent voters will be interviewed about thetax cut. Second, the number of voters supporting the tax cut will be recorded.Finally, the claim should be rejected if the number of supporters is less than 4.

(i) Write down the null and alternative hypotheses Sam are interested in.[1 mark]

(ii) Find the level of significance ofSam's test. [2 marks](iii) Find the power of Sam's test when the proportion of supporters is 45%.

[2 marks]

5

Question 5 [15 marks in total]

(a) For the simple linear regression model r; = Po + PjX; + 8; to be valid, the error

term c; must satisfy certain conditions. What are these conditions? [2 marks]

(b) The following summary output is from a regression equation estimated usingMicrosoft Excel. The dependent variable (Y) is the interest rate on a short-termgovernment bond and the independent variable (X) is the inflation rate. Both theinterest rate and the inflation rate are expressed as percent per year. The model is

r; = Po + PjX; + 8; .

Quarterly data from March 2000 to March 2005 comprising twenty-oneobservations (i.e.,n = 21) were used to estimate the regression equation.

Regression StatisticsR SquareStandard Error of the RegressionObservations

0.27500.5444

21

InterceptX Variable

Coefficients Standard Error4.4602 0.32190.2408 0.0897

t Stat13.85752.6845

P-value0.00000.0147

(i) Find the 95% confidence interval for PI. [2 marks](ii) Test the null hypothesis that PI = 0 against the alternative that P, > 0 at the

5% level of significance. Write down your decision rule and conclusion.[2 marks]

(iii) Suppose that the inflation rate (X) is 5%. Make a point "prediction" for theinterest rate (Y) by using the estimated linear regression model. [2 marks]

(iv) The "R Square" or "coefficient of determination" is 0.275. Explain themeaning of"R Square". [2 mark]

(v) The p-value "0.0147" in the above table is a probability. Exactly, what isthis probability? [2 marks]

(c) To find the relationship between a company's expenditure on advertisements (X)and sales (Y), an analyst collected 6 Quarterly observations on X and Y of thecompany. The means and sums of squares for the two variables from the sampleare computed:

x=5.17, Y=143.33, SSx =7.47, SSy:=963.89, SS<y =66.11.

Test the null hypothesis that X and Y are uncorrelated against the alternative thatthey are positively correlated, using the 5% level of significance. Write downyour decision rule and conclusion. [3 marks]

6

USEFUL FORMULAE

2 1 ~ 2Population Variance: () = - L.. (X; -I-')N i=1

Sample Mean:- 1 n

x=- IX;n ;=1

Sample Variance:

Population Mean:

Additive Law of Probability: peA u B) =peA) + P(B) - peA 11 B)

Multiplicative Law of Probability: peA 11 B) = p(AIB)P(B) = p(BIA)P(A)

Bayes'theorem:

Binomial Distribution:

P(E; )p(AIE;)P(E lA) - =----'-=--

; - Ip(Ek)P(AIEk)k

Poisson Distribution:

E(X) = np;

-!-! xe f..1.

P(X = x) =--'--­x!

Var(X) = npq

E(X) = f..1.; Var(X) = f..1.

t = _(X_-=I-':-)s/.j;; ,Standardising transformations:

Confidence intervals for J.1:

Confidence interval for p:

Confidence interval for (f2:

z=(X-I-'),()

x2=(n-l)s2(}2

- () - ()

X - z~ .j;; < I-' < X + z~.j;; , cr known

- s - sX - tal _ I < I-' < X + tal _ I' cr unknown

12,n 1 "1/ n 12,n 1 "1/ n

P-Z'7i~:q <P<P+Z'7i~:q, where q=(l-p)

(n -1)s2 2 (n -1)s22 < () < -'--:-2----''-

X a I2,n-1 XI-aI2,n-1

Goodness of Fit Test:

Independence Test:

2 IK(0 _e)2

X- I J- ,

1=1 e j

7

df= K-l

df = (K-l )(H-l)

Sample Correlation Coefficient:

r = SSXY = IXiYi - nXY

~SSx ~SSy ~IX; _nX 2 ~IY? _ny 2

Coefficient of Determination: R2 = 1 _ SSE = 1 _ SSESST SSy

Hypothesis tests on Correlation Coefficient:r.Jn - 2

t =--===-~

Least Squares Estimators of f31 and f30 in the equation: 1'; = fJ 0 + fJ 1X j + e j

fil = SSxr = I(X; -xXy; -y) = IX;Y; -nXY = nIX;Y; -(IxJIY;)SSx I(X; -XY IX

j

2 _nX2 nIX;2 -(IxJ

fio =Y - fil X

Forecast Intervals:

for Yp

n 1\2

Le;s = ;=1 or s =

e (n-2) e

is the estimated standard error of the regression.

Ly;2 - fioLY; - filLX;Y;(n-2)

and

8

BINOMIAL PROBABILITIES: P(X =x Ip,n)

pn " 0.05 0.10 0.15 0.10 0.25 0..30 OJ5 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.110 0.85 0.90 0.951 0 0.9500 0.9000 0.8500 0.8000 0.7500 0.7000 0.6500 0.6000 0.5'00 0.5000 0.4500 0.4000 0.3500 0.3000 0.2500 0.2000 0.1500 0.1000 0.0500

1 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.4000 0.4500 0.5000 0.5500 0.6000 0.6500 0.7000 0.7500 0.8000 0.8500 0.9000 0.9500

2 0 0.9025 0.8100 0.7225 0.6400 0.5625 0.4900 0.4225 0.3600 OJ025 01500 0.2025 0.1600 0.1225 0.0900 0.0625 0.0400 0.0125 0.0100 0.00251 0.0950 0.1800 0.2550 OJ200 0.3750 0.4200 0.4550 0.4800 0.4950 0.5000 0.4950 0.4800 0.4550 0.4200 0.3750 0.3200 0.2550 0.1800 0.09502 0.0025 0.0100 0.0225 0.0400 0.0625 0.0900 0.1225 0.1600 0.2025 01500 0.3015 0.3600 0.4225 0.4900 0.5625 0.6400 0.7225 0.8100 0.9025

3 0 0.8574 0.7290 0.6141 0.5110 0.4219 0.3430 0.2746 0.2160 0.1664 0.1250 0.0911 0.0640 0.0429 0.0270 0.0156 0.0080 0.Q034 0.0010 0.00011 0.1354 0.2430 0.3251 0.3840 0.4219 0.4410 0.4436 0.4320 0.4084 03750 0.3341 0.2880 0.2389 0.1890 0.1406 0.0960 0.0574 0.0270 0.00712 0.0071 0.0270 0.0574 0.0960 0,1406 0.1890 0.2389 6.2880 0.3341 03750 0.4084 0.4320 0.4436 0.4410 0.4219 0.3840 03251 01430 0.1354J 0.0001 0.0010 0.0034 0.0080 0.0156 0.0270 0,0429 0.0640 0.0911 0.1250 0.1664 0.2160 0.2746 0.3430 0.4219 0.5120 0.6141 0.7290 0.8574

4 0 0.8145 0.6561 0.5220 0.4096 0.3164 0.2401 0.1785 0.1296 0.0915 0.0625 0.0410 0.0256 0.0150 0.0081 0.0039 0.0016 0.0005 0.0001 0.00001 0.1715 0.2916 0.3685 0.4096 0.4219 0.4116 0.3845 0.3456 0.2995 0.2500 0.2005 0.1536 0.1115 0.0756 0.0469 0.0156 0.0115 0.0036 0.00052 0.0135 0.0486 0.0975 0.1536 0.2109 0.2646 0.3105 0.3456 03675 03750 03675 0.3456 0.3105 0.2646 0.2109 0.1536 0.0975 0.0486 0.0135J 0.0005 0.0036 0.0115 0.0256 0.0469 0.0756 0.1115 0.1536 0.2005 0.2500 0.2995 0.3456 0.3845 0.4116 0.4119 0.4096 03685 0.2916 0.17154 0.0000 0.0001 0.0005 0.0016 0.0039 0.0081 0.0150 0.0256 0.0410 0.0625 0.0915 0.1296 0.1785 0.2401 0.3164 0.4096 0.5220 0.6~1 0.8145

5 0 0.7738 0.5905 0,4437 03117 0.2373 0.1681 0.1160 0.0778 0.0503 0.0312 0.0185 0.0101 0.0053 0.0014 0.0010 0.0003 0.0001 0.0000 0.00001 0.2036 03280 03915 0.4096 0.3955 0.3602 0.3124 0.2582 02059 0.1563 0.1128 0.0768 0.0488 0.0184 0.0146 0.0064 O.Q012 0.0005 0.00002 0.0114 0.0729 0.1382 0.2048 0.2637 0.3087 0.3364 0.3456 0.3369 03125 0.2757 0.2304 0.1811 0.1323 0.0879 0.0512 0.Q244 0.0081 0.0011J 0.0011 0.0081 0.0244 0.0512 0,0879 0,1323 0.1811 0.2304 02757 03125 0.3369 0.3456 0.3364 0.3087 0.2637 0.2048 0.1382 0.0129 0.02144 0.0000 0.0005 0.0012 0.0064 0.0146 0.0284 0.0488 0.0768 0.1128 0.1563 0.2059 0.2592 0.3124 0.3602 0.3955 0.4096 0.3915 0.3280 0.20365 0.0000 0.0000 0,0001 0.0003 0.0010 0,0014 0.0053 0.0102 0.0185 0.0312 0.0503 0.0778 0.1160 0.1681 0.2373 0.3177 0.4437 0.5905 0.7738

6 0 0.7351 0.5314 0.3771 0.2621 0.1780 0.1176 0.0754 0.0467 0.0117 0.0156 0.0083 0.0041 0.0018 0.0007 0.0002 0.0001 0.0000 0.0000 0.0000I 0.2321 0.3543 0.3993 0.3932 0.3560 0.3025 0.2437 0.1866 0.1359 0.0938 0.0609 0.0369 0.0205 0.0102 0.0044 0.0015 0.Q004 0.0001 0.00002 O.oJ05 0.0984 0.1762 0.2458 0,2966 0.3241 0.3280 0.3110 02780 02344 0.1861 0.1382 0.0951 0.0595 0.0330 0.0154 0.0055 0.0012 0.0001J 0.0011 0.0146 0.0415 0.0819 0.1318 0.1852 0.2355 0.2765 0.3032 0.3125 0.3032 0.2765 0.2355 0.1852 0.1318 0.0819 0.0415 0.0146 0.00214 0.0001 0.0012 0.0055 0.0154 0.0330 0.0595 0.095 I 0.1382 0.1861 0.2344 02780 0.3110 0.3280 0.3241 0.2966 0.2458 0.1762 0.0984 0.03055 0.0000 0.0001 0.0004 0.0015 0.0044 0.0102 0.0105 0.0369 0.0609 0.0938 0.1359 0.1866 0.2437 0.3015 0.3~0 0.3932 0.3993 0.3543 0.23216 0.0000 0.0000 0.0000 0.0001 0.0002 0.0007 0.0018 0.0041 0.0083 0.0156 0.0277 0.0467 0.0754 0.1176 0.1780 0.2621 0.3771 0.5314 0.7351

7 0 0.6983 0.4783 0.3206 02097 0.1335 0.0824 0.0490 0.0280 0.0152 0.0078 0.0037 0.0016 0.0006 0.0002 0.0001 0.0000 0.0000 0.0000 0.00001 0.2573 0.3720 0.3960 0.3670 0,3115 0.2411 0.1848 0.1306 0.0872 0.0547 0.0320 0.0172 0.0084 0.0036 0.0013 0.0004 0.0001 0.0000 0.00002 0.0406 0.1240 02097 02753 0.3115 0.3177 0.2985 0.2613 0.2140 0.1641 0.1172 0.0774 0.0466 0.0250 0.0115 0.0043 0.0012 0.0002 0.0000J 0.0036 0.0230 0.0617 0.1147 0,1730 0.2269 0.2679 0.2903 0.2918 02734 02388 0.1935 0.1442 0.0972 0.0577 0.0187 0.0109 0.0016 0.00024 0.0002 0.0026 0.0109 0.0287 0.0577 0.0972 0.1442 0.1935 02388 02734 0.2918 0.2903 0.2679 0.2269 0.1730 0.1147 0.0617 0.0230 0.00365 0.0000 0.0002 0.0012 0.0043 0.0115 0.0250 0.0466 0.0774 0.1172 0.1641 02140 0.2613 0.2985 0.3177 0.3115 0.1753 02097 0.1240 0.04066 0.0000 0.0000 0.0001 0.0004 0.0013 0.0036 0.0084 0.0172 0.0320 0,0547 0.0872 0.1306 0.1848 0.2471 0.3115 0.3670 0.3960 0.3720 025737 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0006 0.0016 0.0037 0.0078 0.0152 0.0280 0.0490 0.0824 0.1335 0.2097 0.3206 0.4783 0.6983

8 0 0.6634 0.4305 0.2725 0.1678 0.1001 0.0576 0.0319 0.0168 0.0084 0.0039 0.0017 0.0007 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.00001 0.2793 OJ826 0.3847 0.3355 0.2670 0.1977 0.1373 0.0896 0.0548 0.0313 0.0164 0.0079 0.0033 0.0012 0.0004 0.0001 0.0000 0.0000 0.00002 0.0515 0.1488 0.2376 0.2936 0.3115 0.2965 0.2587 0.2090 0.1569 0.1094 0.0703 0.0413 0.0211 0.0100 0.0038 0.0011 O.llOO1 0.0000 0.0000J 0.0054 0.0331 0.0839 0.1468 0.2076 0.2541 0.2786 0.2787 02568 01188 0.1719 0.1239 0.0808 0.0467 0.0131 0.0092 0.0026 0.0004 0.00004 0.0004 0.0046 0.0185 0.0459 0.0865 0.1361 0.1875 0.2322 02627 0.2734 0.2627 0.2322 0.1875 0.1361 0.0865 0.0459 0.0185 0.0046 0.00045 0.0000 0.0004 0.0026 0.0092 0.0131 0.0467 0.0806 0.1239 0.1719 02188 0.2~8 0.2787 0.2786 0.2541 0.2076 0.1468 0.0839 0.0331 0.00546 0.0000 0.0000 0.0002 0.0011 0.0038 0.0100 0.0217 0.0413 0.0703 0.1094 0.1~9 0.2090 0.2587 0.2965 0.3115 0.2936 0.2376 0.1488 0.05157 0.0000 0.0000 0.0000 0.0001 0.0004 0.0012 0.0033 0.0079 0.0164 0.0313 0.0548 0.0896 0.1373 0.1977 0.2670 0.3355 0.3847 0.3826 027938 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0007 0.0017 0.0039 0.0084 0.0168 0.0319 0.0576 0.1001 0.1678 0.2725 0.4305 0.6634

9 0 0.6302 0.3874 02316 0.1342 0.0751 0.0404 0.0107 0.0101 0.0046 0.0010 0.0008 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000I 0.2985 0.3874 0.3679 0.3010 0.2253 0.1556 0.1004 0.0605 0.0339 0.0116 0.0083 0.0035 0.0013 0.0004 0.0001 0.0000 0.0000 0.0000 0.00002 0.0629 0.1722 01597 0.3010 0.3003 0.2668 0.2162 0.1612 0.1110 0.0703 0.0407 0.0112 0.0098 0.0039 0.0012 0.0003 0.0000 0.0000 0.00003 0.0077 0.0446 0.1069 0.1762 0.2336 0.2668 0.2716 0.2508 02119 0.1641 0.1160 0.0743 0.0424 0.0110 0.0087 0.0028 0.0006 0.0001 0.00004 0.0006 0.0074 0.0283 0.0661 0.1168 0.1715 0.2194 0.2508 0.2600 02461 02128 0.1672 0.1181 0.0735 0.0389 0.0165 0.0050 0.0008 0.00005 0.0000 0.0008 0.0050 0.0165 0.0389 0.0735 0.1181 0.1672 02128 01461 02600 0.2506 0.2194 0.1715 0.1168 0.0661 0.Q283 0.0074 0.00066 0.0000 0.0001 0.0006 0.0018 0.0087 0.0210 0.0424 0.0743 0.1160 0.1641 0.2119 0.2506 0.2716 0.2668 0.2336 0.1762 0.1069 0.0446 0.00777 0.0000 0.0000 0.0000 0.0003 0.0012 0.0039 0.0098 0.0212 0.0407 0.0703 0.1110 0.1612 0.2162 0.2668 0.3003 0.3010 0.2597 0.1722 0.06298 0.0000 0.0000 0.0000 0.0000 0.0001 0.0004 0.0013 0.0035 0.0083 0.0176 0.0339 0.0605 0.1004 0.1556 0.2253 0.3010 0.3679 0.3874 0.29869 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0003 0.0008 0.0020 0.0046 0.0101 0.0207 0.0404 0.0751 0.1342 02316 0.3874 0.6301

10 0 0.5987 0.3487 0.1969 0.1074 0.0563 0.0282 0.0135 0.0060 0.0025 0.0010 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.00001 0.3151 0.3874 0.3474 02684 0.1877 0.1211 0.0725 0.0403 0.0207 0.0098 0.0042 0.0016 0.0005 0.0001 0.0000 0.0000 0.0000 0.0000 0.00002 0.0746 0.1937 02759 0.3010 0.2816 0.2335 0.m7 0.1209 0.0763 0.0439 0.0229 0.0106 0.0043 0.0014 0.0004 0.0001 0.0000 0.0000 0.00003 0.0105 0.0574 0.1298 02013 0.2503 0.2668 0.2522 0.2150 0.1665 0.1172 0.0746 0.0425 0.0112 0.0090 0.0031 0.0008 0.0001 0.0000 0.00004 0.0010 0.QI11 0.0401 0.0881 0.1460 0.2001 0.2377 0.2508 0.2384 01051 0.1596 0.1115 0.0689 0.0368 0.0162 0.0055 0.0012 0.0001 0.00005 0.0001 0.0015 0.0085 0.0264 0.0584 0.1019 0.1536 0.2007 02340 0.2461 02340 0.2007 0.1536 0.1019 0.0584 0.0164 0.0085 0.0015 0.00016 0.0000 0.0001 0.0012 0.0055 0.0162 0.0368 0.0689 0.1115 0.1596 02051 02384 0.2506 0.2377 0.2001 0.1460 0.0881 0.Q401 0.0112 0.00107 0.0000 0.0000 0.0001 0.0008 0.0031 0.0090 0.0112 0.0425 0.0746 0.1172 0.1665 0.2150 0.2522 0.2668 0.2503 0.2013 0.1298 0.0574 0.01058 0.0000 0.0000 0.0000 0.0001 0.0004 0.0014 0.0043 0.0106 0.0129 0.0439 0.0763 0.1209 0.1757 0.2335 0.2816 0.3020 02759 0.1937 0.07469 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0005 0.0016 0.0042 0.0098 0.0207 0.0403 0.0725 0.1211 0.1877 0.2684 0.3474 0.3874 0.315110 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0003 0,0010 0.0025 0.0060 0.0135 0.0182 0.0563 0.1074 0.1969 0.3487 0.5987

CUMULATIVE BINOMIAL PROBABILITIES: P(X $; x Ip,n)

pn x 0.05 OJO 0.15 0.20 0.25 O.JO 0.J5 0.40 D.45 0.50 0.55 0.60 0.65 0.70 0.75 0.ll0 0.ll5 0.90 0.95

1 0 0.9500 0.9000 0.8500 0.8000 0.7500 0.7000 0.6500 0.6000 0.5500 0.5000 0.4500 0.4000 0.3500 0.3000 0.2500 0.2000 0.1500 0.1000 0.0500

1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

2 0 0.9025 0.8100 0.7225 0.6400 0.5625 0.4900 0.4225 0.3600 0.3025 01500 0102.5 0.1600 0.1225 0.0900 0.0625 0.0400 0.0225 0.0100 0.0025] 0.9975 0.9900 09775 0.9600 0.9375 0.9100 0.8775 0.8400 0.7975 0.7500 0.6975 0.6400 0.5775 0.5100 0.4375 0.3600 01775 0.1900 0.0975

2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

J 0 0.8574 0.7290 0.6141 0.5120 0.4219 0.3430 0.2746 0.2160 0.1664 0.1250 0.0911 0.0640 0.0429 0.0270 0.0156 0.0080 0.0034 0.0010 0.0001

1 0.9928 09720 09393 0.8960 0.8438 0.7840 0.7183 0.6480 0.5748 0.5000 0.4252 0.3520 0.2817 0.2160 0.1562 0.1040 0.0607 0.0280 0.0072

2 0.9999 09990 09966 09920 0.9844 0.9730 0.9571 0.9360 09089 0.8750 0.8336 0.7840 0.7254 0.6570 0.5781 0.4880 0.3859 0.2710 0.1426

J 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

4 0 0.8145 0.6561 0.5220 0.4096 0.3164 0.2401 0.1785 0.1296 0.0915 0.0625 0.0410 0.0256 0.0150 0.0081 0.0039 0.0016 0.0005 0.0001 0.0000

1 0.9860 09477 0.8905 0.8192 0.7383 0.6517 0.5630 0.4752 0.3910 0.3125 01415 0.1792 0.1265 0.0837 0.0508 0.0272 0.0120 0.0037 0.0005

2 0.9995 0.9963 09880 0.9728 0.9492 0.9163 0.8735 0.8208 0.7585 0.6875 0.6090 0.5248 0.4370 0.3483 0.2617 0.1808 0.1095 0.0523 0.0140

J 1.0000 0.9999 09995 09984 0.9961 0.9919 0.9850 09744 0.9590 0.9375 09085 0.8704 0.8215 0.7599 0.6836 0.5904 0.4780 0.3439 0.1855

4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

5 0 0.7738 0.5905 0.4437 0.3277 0.2373 0.1681 0.1160 0.0778 0.0503 0.0312 0.0185 0.0102 0.0053 0.0024 0.0010 0.0003 0.0001 0.0000 0.0000] 0.9774 09185 0.8352 0.7373 0.6328 0.5282 0.4284 0.3370 01562 0.1875 0.1312 0.0870 0.0540 0.0308 0.0156 0.0067 0.0022 0.0005 0.00002 0.9988 0.9914 09734 09421 0.8965 0.8369 0.7648 0.6826 0.5931 0.5000 0.4069 0.3174 0.2352 0.1631 0.1035 0.0579 0.0266 0.0086 0.0012

3 1.0000 09995 0.9978 0.9933 0.9844 0.9692 0.9460 0.9130 0.8688 0.8125 0.7438 0.6630 0.5716 0.4718 0.3672 0.2627 0.1648 0.0815 0.0226

4 1.0000 1.0000 09999 0.9997 0.9990 0.9976 0.9947 0.9898 0.9815 0.9688 09497 0.9222 0.8840 0.8319 0.7627 0.6723 0.5563 0.4095 0.2262

5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

6 0 0.7351 0.5314 0.3771 01621 0.1780 0.1176 0.0754 0.0467 0.0277 0.0156 0.0083 0.0041 0.0018 0.0007 0.0002 0.0001 0.0000 0.0000 0.0000] 0.9672 0.8857 0.7765 0.6554 0.5339 0.4202 0.3191 0.2333 0.1636 0.1094 0.0692 0.0410 0.0223 0.0109 0.0046 0.0016 0.0004 0.0001 0.0000

2 0.9978 09842 09527 0.9011 0.8306 0.7443 0.647 I 0.5443 0.4415 0.3438 01553 0.1792 0.1174 0.0705 0.0376 0.0170 0.0059 0.0013 0.0001J 0.9999 0.9987 0.9941 0.9830 0.9624 0.9295 0.8826 0.8208 0.7447 0.6562 0.5585 0.4557 0.3529 0.2557 0.1694 0.0989 0.0473 0.0158 0.0022

4 1.0000 09999 0.9996 0.9984 0.9954 0.9891 0.9777 0.9590 0.9308 0.8906 0.8364 0.7667 0.6809 0.5798 0.4661 0.3446 01235 0.1143 0.03285 1.0000 1.0000 1.0000 0.9999 0.9998 0.9993 0.9982 0.9959 0.9917 0.9844 0.9723 0.9533 0.9246 0.8824 0.8220 0.7379 0.6229 0.4686 0.26496 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

7 0 0.6983 0.4783 0.3206 01097 0.1335 0.0824 0.0490 0.0280 0.0152 0.0078 0.0037 0.0016 0.0006 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000] 0.9556 0.8503 0.7166 0.5767 0.4449 0.3294 0.2338 0.1586 0.1024 0.0625 0.0357 0.0188 0.0090 0.0038 0.0013 0.0004 0.0001 0.0000 0.00002 0.9962 0.9743 0.9262' 0.8520 0.7564 0.6471 0.5323 0.4199 0.3164 0.2266 0.1529 0.0963 0.556 0.0288 0.0129 0.0047 0.0012 0.0002 0.0000J 0,9998 0.9973 0.9879 0.9667 0.9294 0.8740 0.8002 0.7102 0.6083 0.5000 0.3917 0.2898 0.1998 0.1260 0.0706 0.0333 0.0121 0.0027 0.00024 1.0000 0.9998 0.9988 0.9953 0.9871 0.9712 0.9444 0.9037 0,8471 0.7734 0.6836 0.5801 0.4677 0.3529 0.2436 0.1480 0.0738 0.0257 0.00385 1.0000 1.0000 0.9999 0.9996 0,9987 0.9962 0.9910 0.9812 0.9643 0.9375 0,8976 0.8414 0.7662 0.6706 0.5551 0.4233 0.2834 0.1497 0,04446 1.0000 1.0000 1.0000 1,0000 0,9999 0.9998 0.9994 0.9984 0.9963 09922 0.9848 0.9720 0.9510 0.9176 0.11665 0.7903 0.6794 0.5217 0.30177 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1,0000 1.0000

8 0 0.6634 0.4305 0.2725 0.1678 0.1001 0,0576 0.0319 0.0168 0.0084 0.0039 0.0017 0.0007 0.0002 0.0001 0.0000 0.0000 0,0000 0.0000 0.0000] 0.9428 0.8131 0.6572 0.5033 0.3671 0.2553 0.1691 0.1064 0.06J2 0.0352 0.0181 0,0085 0.0036 0.0013 0.0004 0.0001 0.0000 0.0000 0.00002 0.9942 0.9619 0.8948 0.7969 0.6785 0,5518 0.4278 0.3154 01201 0.1445 0.0885 0.0498 0.0253 0.0113 0.0042 0.0012 0.0002 0.0000 0.0000J 0.9996 0.9950 0.9786 0.9437 0.8862 0.8059 0.7064 0.594 I 0.4770 0.3633 01604 0.1737 0.1061 0,0580 0.0273 0.0104 0.0029 0.0004 0.00004 1.0000 09996 0.9971 0.9896 0.9727 0,9420 0.8939 0.8263 0.7396 0.6367 0.5230 0.4059 0.2936 0.1941 0.1138 0.0563 0.0214 0.0050 0.00045 1.0000 1.0000 0.9998 0.9988 0.9958 0.9887 0.9747 0.9502 0.9115 0.8555 0.7799 0.6846 0.5722 0.4482 0.3215 0.2031 0.1052 0.0381 0.00586 ],0000 1.0000 1.0000 09999 0.9996 0.9987 0.9964 0.9915 0.9819 0.9648 0.9368 0.8936 0.8309 0.7447 0,6329 0.4967 0.3428 0.1869 0.05727 1.0000 1.0000 1.0000 1.0000 1.0000 0,9999 0.9998 0.9993 0.9983 0.9961 0.9916 0.9832 0,9681 0.9424 0.8999 0.8322 0.7275 0.5695 0.33668 1.0000 1.0000 1.0000 1.0000 1.0000 1,0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

9 0 0.6J02 0.3874 0.2J16 0.1342 0,0751 0,0404 0.0207 0.0101 0.0046 0.0020 0.0008 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000] 0.9288 0.7748 0.5995 0.4362 0.3003 0,1960 0.1211 0.0705 0.0385 0.0195 0.0091 0.0038 0.0014 0.0004 0.0001 0.0000 0.0000 0.0000 0.00002 0.9916 0.9470 0.8591 0,7382 0.6007 0.4628 0.J373 0.2318 0.1495 0.0898 0.0498 0.0250 0.0112 0.0043 0.0013 0,0003 0.0000 0.0000 0.0000J 0.9994 0.9917 0.9661 0.9144 0.8343 0.7297 0.6089 0.4826 0.3614 0.2539 0,1658 0.0994 0.0536 0.0253 0.0100 0.0031 O.tJ006 0.0001 0.00004 1.0000 0.9991 0.9944 0.9804 0.9511 0.9012 0.8283 0.7334 0.6214 0.5000 0.3786 0.2666 0.1717 0.0988 0.0489 0.0196 0.0056 0.0009 0.00005 1.0000 0.9999 0.9994 0.9969 0.9900 0.9747 0.9464 0,9006 0.8342 0.7461 0.6386 0.5174 0.3911 0.2703 0.1657 0.0856 0.0339 0.0083 0.00066 1.0000 1.0000 1.0000 0.9997 0.9987 0.9957 0.9888 0.9750 09502 0.9102 0.8505 0.7682 0,6627 0.5372 0.3993 0.2618 0.1409 0,0530 0.00847 1.0000 1.0000 1,0000 1.0000 0.9999 0.9996 0.9986 0.9962 0.9909 0.9805 0.9615 0.9295 0.8789 0.8040 0.6997 0.5638 0.4005 0.2252 0.07128 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 09992 0.9980 0.9954 0.9899 0.9793 0,9596 0.9249 0.8658 0.7684 0.6126 0.36989 1.0000 1.0000 1,0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1,0000 1.0000 1.0000 1.0000

]0 0 0.5987 0.3487 0.1969 0.1074 0.0563 0.0287 0.0135 0.0060 0.0025 0.0010 0.0003 0.0001 0.0000 0,0000 0.0000 0.0000 0.0000 0.0000 0.0000] 0.9139 0.7361 0.5443 0.3758 0.2440 0.1493 0.0860 0,0464 0.0233 0.0107 0.0045 0.0017 0.0005 0.0001 0.0000 0,0000 0.0000 0.0000 0.00002 0.9885 0.9298 0.8202 0.6778 0.5256 0.3828 0.2616 0.1673 0.0996 0.0547 0.0274 0.0123 0.0048 0.0016 0.0004 0.0001 0.0000 0.0000 0.0000J 0.9990 0.9872 0.9500 0.8791 0.7759 0.6496 0.5138 0.3823 0.2660 0.1719 0.1020 0.0548 0.0260 0.0106 0.0035 0,0009 0.0001 0.0000 0.00004 0.9999 0.9984 0.9901 0.9672 0.9219 0.8497 0.7515 0.6331 0.5044 0.3770 0.2616 0.1662 0.0949 0.0473 0,0197 0.0064 0.0014 0.0001 0.00005 1.0000 0.9999 0.9986 0.9936 0.9803 0.9527 0.9051 0.8338 0.7384 0.6230 0.4956 0.3669 0.2485 0.1503 0,0781 0.0328 0.0099 0,0016 0.00016 1.0000 1,0000 0.9999 0.9991 0.9965 0.9894 0.9740 0.9452 0.8980 0.8281 0.7340 0,6177 0.4862 0.3504 0.2241 0.1209 0.0500 0,0128 0.00107 1.0000 1.0000 1.0000 09999 0.9996 0.9984 0.9952 0.9877 0.9726 0.9453 0.9004 0.8327 0.7384 0.6172 0.4744 0.3222 0.1798 0.0702 0.01158 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9983 0.9955 0.9893 0.9767 0.9536 0.9140 0.8507 0.7560 0.6242 0.4557 0.2639 0.08619 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9990 0.9975 0.9940 0.9865 0.9718 0.9437 0.ll926 0.8031 0.6513 0.401310 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

to

CUMULATIVE POISSON PROBABILITIES

A -l' x

The values in the table show P(X ~ k) =L~ ,rounded to 3 decimal places.x=O xl

~

k .10 I .20 I .30 I .40 I 0.50 I 1.0 I 1.5 I 2.0 I 2.5 I 3.0 I 3.5 [ 4.0 1 4.5 I 5.0 I 5.5 I 6.0

0 .905 .819 .741 .670 .607 .368 .223 .135 .082 .050 .030 .018 .Oll .007 .004 .002

I .995 .982 .963 .938 .910 .736 .558 .406 .287 .199 .136 .092 .061 .040 .027 .017

2 1.000 .999 .996 .992 .986 .920 .809 .677 .544 .423 .321 .238 .174 .125 .088 .062

3 1.000 1.000 1.000 .999 .998 .981 .934 .857 .758 .647 .537 .433 .342 .265 .202 .151

4 1.000 1.000 1.000 1.000 1.000 .996 .981 .947 .891 .815 .725 .629 .532 .440 .358 .285

5 .999 .996 .983 .958 .916 .858 .785 .703 .616 .529 .446

6 1.000 .999 .995 .986 .966 .935 .889 .831 .762 .686 .606

7 1.000 .999 .996 .988 .973 .949 .913 .867 .809 .744

8 1.000 .999 .996 .990 .979 .960 .932 .894 .847

9 1.000 .999 .997 .992 .983 .968 .946 .916

10 1.000 .999 .997 .993 .986 .975 .957

11 1.000 .999 .998 .995 .989 .980

12 1.000 .999 .998 .996 .991

13 1.000 .999 .998 .996

14 1.000 .999 .999

15 1.000 .999

16 1.000

17181920

I..lk 6.5 I 7.0 I 7.5 I 8.0 I 8.5 I 9.0 I 9.5 I 10 I 11 I 12 T 13 T 14 T 15

0 .002 .001 .001 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000

1 .Oll .007 .005 .003 .002 .001 .001 .000 .000 .000 .000 .000 .0002 .043 .030 .020 .014 .009 .006 .004 .003 .001 .001 .000 .000 .0003 .1l2 .082 .059 .042 .030 .021 .015 .010 .005 .002 .001 .000 .0004 .224 .173 .132 .100 .074 .055 .040 .029 .015 .008 .004 .002 .001

5 .369 .301 .241 .191 .150 .116 .089 .067 .038 .020 .011 .006 .0036 .527 .450 .378 .313 .256 .207 .165 .130 .079 .046 .026 .014 .0087 .673 .599 .525 .453 .386 .324 .269 .220 .143 .090 .054 .032 .0188 .792 .729 .662 .593 .523 .456 .392 .333 .232 .155 .100 .062 .0379 .877 .830 .776 .717 .653 .587 .522 .458 .341 .242 .166 .109 .070

10 .933 .901 .862 .816 .763 .706 .645 .583 .460 .347 .252 .176 .11811 .966 .947 .921 .888 .849 .803 .752 .697 .579 .462 .353 .260 .18512 .984 .973 .957 .936 .909 .876 .836 .792 .689 .576 .463 .358 .26813 .993 .987 .978 .966 .949 .926 .898 .864 .781 .682 .573 .464 .36314 .997 .994 .990 .983 .973 .959 .940 .917 .854 .772 .675 .570 .46615 .999 .998 .995 .992 .986 .978 .967 .951 .907 .844 .764 .669 .56816 1.000 .999 .998 .996 .993 .989 .982 .973 .944 .899 .835 .756 .66417 1.000 .999 .998 .997 .995 .991 .986 .968 .937 .890 .827 .74918 1.000 .999 .999 .998 .996 .993 .982 .963 .930 .883 .81919 1.000 .999 .999 .998 .997 .991 .979 .957 .923 .87520 1.000 1.000 .999 .998 .995 .988 .975 .952 .91721 1.000 .999 .998 .994 .986 .971 .94722 1.000 .999 .997 .992 .983 .96723 1.000 .999 .996 .991 .98124 .999 .998 .995 .98925 1.000 .999 .997 .99426 1.000 .999 .99727 .999 .99828 1.000 .99929 1.000

, I

AREAS UNDER THE STANDARD NORMAL DISTRIBUTION

The table below gives areas under the standardnonnal distribution between 0 and z.

o z z

z 0 I 1 I 2 I 3 r 4 I 5 I 6 I 7 I 8 I 90.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359

0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0754

0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141

0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517

0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879

0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

0.6 .2258 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2518 .2549

0.7 .2580 .2612 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

0.8 .2881 .2910 .2939 .2967 .2996 .3023 .3051 .3078 .3106 .3133

0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621

1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .38301.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .40151.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177

1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319

1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .44411.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .45451.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .46331.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .47061.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .47672.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .48172.1 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .48572.2 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .48902.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .49162.4 .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .49362.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .49522.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .49642.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .49742.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .49812.9 .4981 .4982 .4982 .4983 .4984 .4984 .4985 .4985 .4986 .49863.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .49903.1 .4990 .4991 .4991 .4991 .4992 .4992 .4992 .4992 .4993 .49933.2 .4993 .4993 .4994 .4994 .4994 .4994 .4994 .4995 .4995 .49953.3 .4995 .4995 .4995 .4996 .4996 .4996 .4996 .4996 .4996 .49973.4 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .49983.5 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .49983.6 .4998 .4998 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .49993.7 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .49993.8 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .4999 .49993.9 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000

12.

CRITICAL VALUES OF THE t DISTRIBUTION

The table below gives critical values of t forgiven probability levels

o ta.,u tu

Example: The value of t with 8 degrees of freedom that cuts otT 0.5% of the area in the righthand tail is 1.860: to.05,8 =1.860.

Degrees of Critical Values ta.Freedom, U

to.1, to.05 I to.015 I to.01 I to.005

1 3.078 6.314 12.706 31.821 63.6572 1.886 2.920 4.303 6.965 9.9253 1.638 2.353 3.192 4.541 5.8414 1.533 2.132 2.776 3.747 4.6045 1.476 2.015 2.571 3.365 4.0326 1.440 1.943 2.447 3.143 3.7077 1.415 1.895 2.365 2.998 3.4998 1.397 1.860 2.306 2.896 3.3559 1.383 1.833 2.262 2.821 3.25010 1.372 1.812 2.228 2.764 3.16911 1.363 1.796 2.201 2.718 3.10612 1.356 1.782 2.179 2.681 3.05513 1.350 1.771 2.160 2.650 3.01214 1.345 1.761 2.145 2.624 2.97715 1.341 1.753 2.131 2.602 2.94716 1.337 1.746 2.120 2.583 2.92117 1.333 1.740 2.110 2.567 2.89818 1.330 1.734 2.101 2.552 2.87819 1.328 1.729 2.093 2.539 2.86120 1.325 1.725 2.086 2.528 2.84521 1.323 1.721 2.080 2.518 2.83122 1.321 1.717 2.074 2.508 2.81923 1.319 1.714 2.069 2.500 2.80724 1.318 1.711 2.064 2.492 2.79725 1.316 1.708 2.060 2.485 2.78726 1.315 1.706 2.056 2.479 2.77927 1.314 1.703 2.052 2.473 2.77128 1.313 1.701 2.048 2.467 2.76329 1.311 1.699 2.045 2.462 2.75630 1.310 1.697 2.042 2.457 2.75040 1.303 1.684 2.012 2.423 2.70460 1.296 1.671 2.000 2.390 2.660120 1.290 1.661 1.984 2.358 2.626

<Xl 1.282 1.645 1.960 2.326 2.576From: Mernngton, Maxine, "Table of Percentage Points of the t-distibution", Biometrika, Vol.32, 1941, p. 300.

{ '3

CRITICAL VALUES OF THE CID-SQUARE DISTRIBUTION

2la.,u

The table below gives critical values of X1 for

given probability levels

oExample: The value of 1. 2 with 8 degrees of freedom that cuts ofT 0.5% of the area in the right

hand tail is 15.5073: X~OS.8 =15.5073.

Degrees Critical Values X~of 2 I 1 I 1 I 1

I1

I1

I1

I1 I 1

I1

Freedom 1..995 1..99 1..975 1..95 1..90 1..10 1..05 1..025 1..01 X.OOS

1 0,0000393 0.0001571 0.0009821 0.0039321 0.0157908 2.70554 3.84146 5.02389 6.63490 7.87944

2 0.0100251 0.0201007 0.0506356 0.102587 0.210720 4.60517 5.99147 7.37776 9.21034 10.5966

3 0.0717212 0.114832 0.215795 0.351846 0.584375 6.25139 7.81473 9.34840 11.3449 12.8381

4 0.206990 0.297110 0.484419 0.710721 1.063623 7.77944 9.48773 11.1433 13.2767 14.8602

5 0.411740 0.554300 0.831211 1.145476 1.61031 9.23635 11.0705 12.8325 15.0863 16.7496

6 0.675727 0.872085 1.237347 1.63539 2.20413 10.6446 12.5916 14.4494 16.8119 18.5476

7 0.989265 1.239043 1.68987 2.16735 2.83311 12.0170 14.0671 16.0128 18.4753 20.2777

8 1.344419 1.646482 217973 273264 3.48954 13.3616 15.5073 17.5346 20.0902 21.95509 1.734926 2.087912 2.70039 3.32511 4.16816 14.6837 .16.9190 19.0228 21.6660 23.589310 2.15585 2.55821 3.24697 3.94030 4.86518 15.9871 18.3070 20.4831 23.2093 25.1882

11 2.60321 3.05347 3.81575 4.57481 5.57779 17.2750 19.6751 21.9200 24.7250 26.7569

12 3.07382 3.57056 4.40379 5.22603 6.30380 18.5494 21.0261 233367 26.2170 28.299513 3.56503 4.10691 5.00874 5.89186 7.04150 19.8119 22.3621 24.7356 27.6883 29.819414 4.07468 4.66043 5.62872 6.57063 7.78953 21.0642 23.6848 26.1190 29.1413 31319315 4.60094 5.22935 6.26214 7.26094 8.54675 22.3072 24.9958 27.4884 30.5779 32.801316 5.14224 5.81221 6.90766 7.96164 9.31223 23.5418 26.2962 28.8454 31.9999 34.2672

17 5.69724 6.40776 7.56418 8.67176 10.0852 24.7690 27.5871 30.1910 33.4087 35.718518 6.26481 7.01491 8.23075 9.39046 10.8649 25.9894 28.8693 31.5264 34.8053 37.156419 6.84398 7.63273 8.90655 10.1170 11.6509 27.2036 30.1435 32.8523 36.1908 38.582220 7.43386 8.26040 9.59083 10.8508 12.4426 28.4120 31.4104 34.1696 37.5662 39.996821 8.03366 8.89720 10.28293 11.5913 13.2396 29.6151 32.6705 35.4789 38.9321 41.401022 8.64272 9.5249 10.9823 123380 14.0415 30.8133 33.9244 36.7807 40.2894 42.795623 9.26042 10.19567 11.6885 13.0905 14.8479 32.0069 35.1725 38.0757 41.6384 44.181324 9.88623 10.8564 12.4011 13.8484 15.6587 33.1963 36.4151 39.3641 42.9798 45.558525 10.5197 11.5240 13.1197 14.6114 16.4734 34.3816 37.6525 40.6465 44.3141 46.927826 11.1603 12.1981 13.8439 15.3791 17.2919 35.5631 38.8852 41.9232 45.6417 48.289927 11.8076 12.8786 14.5733 16.1513 18.1138 36.7412 40.1133 43.1944 46.9630 49.644928 12.4613 13.5648 15.3079 16.9279 18.9392 37.9159 41.3372 44.4607 48.2782 50.993329 13.1211 14.2565 16.0471 17.7083 19.7677 39.0875 42.5569 45.7222 49.5879 52.335630 13.7867 14.9535 16.7908 18.4926 20.5992 40.2560 43.7729 46.9792 50.8922 53.672040 20.7065 22.1643 24.4331 26.5093 29.0505 51.8050 55.7585 59.3417 63.6907 66.765950 27.9907 29.7067 32.3574 34.7642 37.6886 63.1671 67.5048 71.4202 76.1539 79.490060 35.5346 37.4848 40.4817 43.1879 46.4589 74.3970 79.0819 83.2976 88.3794 91.951770 43.2752 45.4418 48.7576 51.7393 55.3290 85.5271 90.5312 95.0231 100.425 104.21580 51.1720 53.5400 57.1532 60.3915 64.2778 96.5782 101.879 106.629 112329 116.32190 59.1963 61.7541 65.6466 69.1260 73.2912 107.565 113.145 118.136 124.116 128.299100 67.3276 70.0648 74.2219 77.9295 82.3581 118.498. 124.342 129.561 135.807 140.169

From: Thompson, C. M., "Tables of the Percentage Points of the Xl -Distribution",Biometrika, Vol. 32, 1941, pp. 188-89.

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