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Faculty of Engineering, Mathematics and Science

School of Computer Science & Statistics

Junior Freshman, MathematicsJunior/Senior Freshman, TSM

Trinity Term 2016

ST1252: Introduction to Statistics II

14 May 2016 Goldsmith Hall 09:30 – 11:30

Prof. Arthur White

Instructions to Candidates:

Answer ALL questions. All questions carry equal marks.

You may not start this examination until you are instructed to do so by the

Invigilator.

Materials permitted for this examination:

Special Statistical Tables are attached.

Non-programmable calculators are permitted for this examination – please indicate the

make and model of your calculator on each answer book used.

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1. Patients undergoing cardiac bypass surgery were randomised into one of two

groups: in Group I, patients received a 50% nitrous oxide and 50% oxygen

mixture continuously for 24 hours, while patients in Group II received a 50%

nitrous oxide and 50% oxygen mixture only during the operation. The table

below shows red cell folate levels for the two groups 24 hours later.

Group I (n = 8) Group II (n = 8)243 206251 210275 226291 249347 255354 273380 285392 295

Mean 316.6 249.9SD 58.7 33.6

(a) Carry out a formal test of the hypothesis that the population means are the

same, at a significance level of α = 0.05.

(b) Calculate and interpret a 95% confidence interval for the difference between

the population means.

(c) Discuss the relationship between the results in parts (a) and (b).

(d) An F -test was carried out to decide whether or not the long term standard

deviations of Groups I and II were the same. Calculate the test statistic for

this test. The resultant p-value from the test was 0.16. Interpret this result.

(e) What assumptions underlie the analyses you have carried out?

(f) In the context of statistical hypothesis/significance testing, explain the

terms: test statistic; sampling distribution; significance level; Type I and

Type II error. Use the numerical example to illustrate your discussion.

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2. Each week during the football season, the BBC sport website features an article

where resident expert Mark Lawrenson is asked to predict the outcome of

upcoming fixtures, i.e., which teams will win, lose or draw. These predictions are

then compared to those of a different weekly guest, who does not have a

footballing background. The following table compares Lawrenson’s predictions

against those of his guests, aggregated over five weeks.

Expert Guest TotalCorrect Prediction 33 28 61

Incorrect Prediction 38 43 81Total 71 71 142

Proportion Correct 0.46 0.39 0.43

(a) Use a moscaic plot to graphically depict the table. Comment on this plot.

(b) A χ2-test was performed on the table. State the hypothesis being tested in

this case. The test statistic was found to be χ2 = 0.4598. Explain how this

statistic was calculated. (You do not have to perform the full calculation

explicitly.) Illustrate your explanation by showing how the expected value

corresponding to the observed value of 33 was calculated. What is the

critical value for a significance level of α = 0.05? Interpret the result of the

test.

(c) For the same table, carry out a Z -test, and calculate and interpret a 95%

confidence interval based on the difference between the sample proportions.

Comment on the relationship between Z and χ2-tests. Which test do you

consider to be more appropriate for this dataset? Explain why.

(d) Calculate and interpret the odds-ratio of the table.

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3. A lecturer is interested in determining whether attendance at lectures and

tutorials is related to student performance. To investigate this, students taking

her statistics module were put into four groups of decreasing order, such that

students in Group 1 had the highest overall attendance, and Group 4 the

poorest. She then took a sample of 50 students from each group, and compared

their final overall course mark. The mean and standard deviation of these marks

for each group is shown in the table below, followed by output obtained by

running an ANOVA in R:

Group 1 Group 2 Group 3 Group 4Mean 55.46 56.75 53.30 28.25Std. dev. 14.34 11.96 10.55 13.62N 50 50 50 50

anova( lm( final.mark ~ group ) )

Analysis of Variance Table

Df Sum Sq Mean Sq F value Pr(>F)

group 3 27462 9154 56.7

Areas under the standard Normal curve

The table gives the area left of z. For example, if z = 1.23, the area, shaded in the illustration below, is .8907.

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359

0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753

0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141

0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517

0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879

0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224

0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549

0.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852

0.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133

0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389

1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621

1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830

1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015

1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177

1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319

1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441

1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545

1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633

1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706

1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767

2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817

2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857

2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890

2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916

2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936

2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952

2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964

2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974

2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981

2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986

3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990

3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993

3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995

3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997

3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998

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Selected critical values for the t-distribution

α is the proportion of values in a t distribution with ν degrees of freedom which exceed in magnitude the tabled value. For example, 25% of the values in a t distribution with 1 degree of freedom are outside ±2.41.

α .25 .10 .05 .02 .01 .002 .001

ν = 1 2.41 6.31 12.71 31.82 63.66 318.32 636.61 2 1.60 2.92 4.30 6.96 9.92 22.33 31.60 3 1.42 2.35 3.18 4.54 5.84 10.22 12.92 4 1.34 2.13 2.78 3.75 4.60 7.17 8.61 5 1.30 2.02 2.57 3.36 4.03 5.89 6.87 6 1.27 1.94 2.45 3.14 3.71 5.21 5.96 7 1.25 1.89 2.36 3.00 3.50 4.79 5.41 8 1.24 1.86 2.31 2.90 3.36 4.50 5.04 9 1.23 1.83 2.26 2.82 3.25 4.30 4.78 10 1.22 1.81 2.23 2.76 3.17 4.14 4.59 12 1.21 1.78 2.18 2.68 3.05 3.93 4.32 15 1.20 1.75 2.13 2.60 2.95 3.73 4.07 20 1.18 1.72 2.09 2.53 2.85 3.55 3.85 24 1.18 1.71 2.06 2.49 2.80 3.47 3.75 30 1.17 1.70 2.04 2.46 2.75 3.39 3.65 40 1.17 1.68 2.02 2.42 2.70 3.31 3.55 60 1.16 1.67 2.00 2.39 2.66 3.23 3.46 120 1.16 1.66 1.98 2.36 2.62 3.16 3.37 ∞ 1.15 1.64 1.96 2.33 2.58 3.09 3.29

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Selected critical values for the chi-squared distribution

α is the proportion of values in a chi-squared distribution with ν degrees of freedom which exceed the tabled value. For example, 20% of the values in a chi-squared distribution with 1 degree of freedom exceed 1.64.

α .2 .1 .05 .025 .01 .005 ν = 1 1.64 2.71 3.84 5.02 6.64 7.88 2 3.22 4.61 5.99 7.38 9.21 10.60 3 4.64 6.25 7.82 9.35 11.35 12.84 4 5.99 7.78 9.49 11.14 13.28 14.86 5 7.29 9.24 11.07 12.83 15.09 16.75 6 8.56 10.65 12.59 14.45 16.81 18.55 7 9.80 12.02 14.07 16.01 18.48 20.28 8 11.03 13.36 15.51 17.54 20.09 21.96 9 12.24 14.68 16.92 19.02 21.67 23.59 10 13.44 15.99 18.31 20.48 23.21 25.19 12 15.81 18.55 21.03 23.34 26.22 28.30 15 19.31 22.31 25.00 27.49 30.58 32.80 20 25.04 28.41 31.41 34.17 37.57 40.00 24 29.55 33.20 36.42 39.36 42.98 45.56 30 36.25 40.26 43.77 46.98 50.89 53.67 60 68.97 74.40 79.08 83.30 88.38 91.96 120 132.81 140.23 146.57 152.21 158.95 163.65

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Selected critical values for the F distribution with ν1 numerator and ν2 denominator degrees of freedom

For example, 10% of the values in an F distribution with 1 numerator and 2 denominator degrees of freedom exceed 8.5.

10% critical values for the F distribution

ν1 1 2 3 4 5 6 7 8 10 12 24 ∞

ν2

1 39.9 49.5 53.6 55.8 57.2 58.2 58.9 59.4 60.2 60.7 62.0 63.3 2 8.5 9.0 9.2 9.2 9.3 9.3 9.3 9.4 9.4 9.4 9.4 9.5 3 5.5 5.5 5.4 5.3 5.3 5.3 5.3 5.3 5.2 5.2 5.2 5.1 4 4.5 4.3 4.2 4.1 4.1 4.0 4.0 4.0 3.9 3.9 3.8 3.8 5 4.1 3.8 3.6 3.5 3.5 3.4 3.4 3.3 3.3 3.3 3.2 3.1 6 3.8 3.5 3.3 3.2 3.1 3.1 3.0 3.0 2.9 2.9 2.8 2.7 7 3.6 3.3 3.1 3.0 2.9 2.8 2.8 2.8 2.7 2.7 2.6 2.5 8 3.5 3.1 2.9 2.8 2.7 2.7 2.6 2.6 2.5 2.5 2.4 2.3 9 3.4 3.0 2.8 2.7 2.6 2.6 2.5 2.5 2.4 2.4 2.3 2.2

10 3.3 2.9 2.7 2.6 2.5 2.5 2.4 2.4 2.3 2.3 2.2 2.1 12 3.2 2.8 2.6 2.5 2.4 2.3 2.3 2.2 2.2 2.1 2.0 1.9 15 3.1 2.7 2.5 2.4 2.3 2.2 2.2 2.1 2.1 2.0 1.9 1.8 20 3.0 2.6 2.4 2.2 2.2 2.1 2.0 2.0 1.9 1.9 1.8 1.6 40 2.8 2.4 2.2 2.1 2.0 1.9 1.9 1.8 1.8 1.7 1.6 1.4

120 2.7 2.3 2.1 2.0 1.9 1.8 1.8 1.7 1.7 1.6 1.4 1.2 ∞ 2.7 2.3 2.1 1.9 1.8 1.8 1.7 1.7 1.6 1.5 1.4 1.0

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5% critical values for the F distribution

ν1 1 2 3 4 5 6 7 8 10 12 24 ∞

ν2

1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 241.9 243.9 249.1 254.3 2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.5 19.5 3 10.1 9.6 9.3 9.1 9.0 8.9 8.9 8.8 8.8 8.7 8.6 8.5 4 7.7 6.9 6.6 6.4 6.3 6.2 6.1 6.0 6.0 5.9 5.8 5.6 5 6.6 5.8 5.4 5.2 5.1 5.0 4.9 4.8 4.7 4.7 4.5 4.4 6 6.0 5.1 4.8 4.5 4.4 4.3 4.2 4.1 4.1 4.0 3.8 3.7 7 5.6 4.7 4.3 4.1 4.0 3.9 3.8 3.7 3.6 3.6 3.4 3.2 8 5.3 4.5 4.1 3.8 3.7 3.6 3.5 3.4 3.3 3.3 3.1 2.9 9 5.1 4.3 3.9 3.6 3.5 3.4 3.3 3.2 3.1 3.1 2.9 2.7

10 5.0 4.1 3.7 3.5 3.3 3.2 3.1 3.1 3.0 2.9 2.7 2.5 12 4.7 3.9 3.5 3.3 3.1 3.0 2.9 2.8 2.8 2.7 2.5 2.3 15 4.5 3.7 3.3 3.1 2.9 2.8 2.7 2.6 2.5 2.5 2.3 2.1 20 4.4 3.5 3.1 2.9 2.7 2.6 2.5 2.4 2.3 2.3 2.1 1.8 30 4.2 3.3 2.9 2.7 2.5 2.4 2.3 2.3 2.2 2.1 1.9 1.6 40 4.1 3.2 2.8 2.6 2.4 2.3 2.2 2.2 2.1 2.0 1.8 1.5

120 3.9 3.1 2.7 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.6 1.3 ∞ 3.8 3.0 2.6 2.4 2.2 2.1 2.0 1.9 1.8 1.8 1.5 1.0

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2.5% critical values for the F distribution

ν1

1 2 3 4 5 6 7 8 10 12 24 ∞

ν2 1 647.8 799.5 864.2 899.6 921.8 937.1 948.2 956.6 968.6 976.7 997.3 1018.3 2 38.5 39.0 39.2 39.2 39.3 39.3 39.4 39.4 39.4 39.4 39.5 39.5 3 17.4 16.0 15.4 15.1 14.9 14.7 14.6 14.5 14.4 14.3 14.1 13.9 4 12.2 10.6 10.0 9.6 9.4 9.2 9.1 9.0 8.8 8.8 8.5 8.3 5 10.0 8.4 7.8 7.4 7.1 7.0 6.9 6.8 6.6 6.5 6.3 6.0 6 8.8 7.3 6.6 6.2 6.0 5.8 5.7 5.6 5.5 5.4 5.1 4.8 7 8.1 6.5 5.9 5.5 5.3 5.1 5.0 4.9 4.8 4.7 4.4 4.1 8 7.6 6.1 5.4 5.1 4.8 4.7 4.5 4.4 4.3 4.2 3.9 3.7 9 7.2 5.7 5.1 4.7 4.5 4.3 4.2 4.1 4.0 3.9 3.6 3.3 10 6.9 5.5 4.8 4.5 4.2 4.1 3.9 3.9 3.7 3.6 3.4 3.1 12 6.6 5.1 4.5 4.1 3.9 3.7 3.6 3.5 3.4 3.3 3.0 2.7 15 6.2 4.8 4.2 3.8 3.6 3.4 3.3 3.2 3.1 3.0 2.7 2.4 20 5.9 4.5 3.9 3.5 3.3 3.1 3.0 2.9 2.8 2.7 2.4 2.1 30 5.6 4.2 3.6 3.2 3.0 2.9 2.7 2.7 2.5 2.4 2.1 1.8 40 5.4 4.1 3.5 3.1 2.9 2.7 2.6 2.5 2.4 2.3 2.0 1.6 120 5.2 3.8 3.2 2.9 2.7 2.5 2.4 2.3 2.2 2.1 1.8 1.3 ∞ 5.0 3.7 3.1 2.8 2.6 2.4 2.3 2.2 2.0 1.9 1.6 1.0

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1% critical values for the F distribution

ν

1 1 2 3 4 5 6 7 8 10 12 24 ∞

ν2 1 4052.2 4999.3 5403.5 5624.3 5764.0 5859.0 5928.3 5981.0 6055.9 6106.7 6234.3 6365.6 2 98.5 99.0 99.2 99.3 99.3 99.3 99.4 99.4 99.4 99.4 99.5 99.5 3 34.1 30.8 29.5 28.7 28.2 27.9 27.7 27.5 27.2 27.1 26.6 26.1 4 21.2 18.0 16.7 16.0 15.5 15.2 15.0 14.8 14.5 14.4 13.9 13.5 5 16.3 13.3 12.1 11.4 11.0 10.7 10.5 10.3 10.1 9.9 9.5 9.0 6 13.7 10.9 9.8 9.1 8.7 8.5 8.3 8.1 7.9 7.7 7.3 6.9 7 12.2 9.5 8.5 7.8 7.5 7.2 7.0 6.8 6.6 6.5 6.1 5.6 8 11.3 8.6 7.6 7.0 6.6 6.4 6.2 6.0 5.8 5.7 5.3 4.9 9 10.6 8.0 7.0 6.4 6.1 5.8 5.6 5.5 5.3 5.1 4.7 4.3 10 10.0 7.6 6.6 6.0 5.6 5.4 5.2 5.1 4.8 4.7 4.3 3.9 11 9.6 7.2 6.2 5.7 5.3 5.1 4.9 4.7 4.5 4.4 4.0 3.6 12 9.3 6.9 6.0 5.4 5.1 4.8 4.6 4.5 4.3 4.2 3.8 3.4 14 8.9 6.5 5.6 5.0 4.7 4.5 4.3 4.1 3.9 3.8 3.4 3.0 16 8.5 6.2 5.3 4.8 4.4 4.2 4.0 3.9 3.7 3.6 3.2 2.8 18 8.3 6.0 5.1 4.6 4.2 4.0 3.8 3.7 3.5 3.4 3.0 2.6 20 8.1 5.8 4.9 4.4 4.1 3.9 3.7 3.6 3.4 3.2 2.9 2.4 25 7.8 5.6 4.7 4.2 3.9 3.6 3.5 3.3 3.1 3.0 2.6 2.2 30 7.6 5.4 4.5 4.0 3.7 3.5 3.3 3.2 3.0 2.8 2.5 2.0 40 7.3 5.2 4.3 3.8 3.5 3.3 3.1 3.0 2.8 2.7 2.3 1.8 120 6.9 4.8 3.9 3.5 3.2 3.0 2.8 2.7 2.5 2.3 2.0 1.4 ∞ 6.6 4.6 3.8 3.3 3.0 2.8 2.6 2.5 2.3 2.2 1.8 1.0

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Critical  values  for  the  Studentized  range  distribution  

Family confidence coefficient: 1–α= 0.95

Number of means 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 DF

1 18.0 27.0 32.8 37.1 40.4 43.1 45.4 47.4 49.1 50.6 52.0 53.2 54.3 55.4 56.3 57.2 58.0 58.8 59.6 2 6.08 8.33 9.80 10.9 11.7 12.4 13.0 13.5 14.0 14.4 14.7 15.1 15.4 15.7 15.9 16.1 16.4 16.6 16.8 3 4.50 5.91 6.82 7.50 8.04 8.48 8.85 9.18 9.46 9.72 9.95 10.2 10.3 10.5 10.7 10.8 11.0 11.1 11.2 4 3.93 5.04 5.76 6.29 6.71 7.05 7.35 7.60 7.83 8.03 8.21 8.37 8.52 8.66 8.79 8.91 9.03 9.13 9.23 5 3.64 4.60 5.22 5.67 6.03 6.33 6.58 6.80 6.99 7.17 7.32 7.47 7.60 7.72 7.83 7.93 8.03 8.12 8.21 6 3.46 4.34 4.90 5.30 5.63 5.90 6.12 6.32 6.49 6.65 6.79 6.92 7.03 7.14 7.24 7.34 7.43 7.51 7.59 7 3.34 4.16 4.68 5.06 5.36 5.61 5.82 6.00 6.16 6.30 6.43 6.55 6.66 6.76 6.85 6.94 7.02 7.10 7.17 8 3.26 4.04 4.53 4.89 5.17 5.40 5.60 5.77 5.92 6.05 6.18 6.29 6.39 6.48 6.57 6.65 6.73 6.80 6.87 9 3.20 3.95 4.41 4.76 5.02 5.24 5.43 5.59 5.74 5.87 5.98 6.09 6.19 6.28 6.36 6.44 6.51 6.58 6.64

10 3.15 3.88 4.33 4.65 4.91 5.12 5.30 5.46 5.60 5.72 5.83 5.93 6.03 6.11 6.19 6.27 6.34 6.40 6.47 11 3.11 3.82 4.26 4.57 4.82 5.03 5.20 5.35 5.49 5.61 5.71 5.81 5.90 5.98 6.06 6.13 6.20 6.27 6.33 12 3.08 3.77 4.20 4.51 4.75 4.95 5.12 5.27 5.39 5.51 5.61 5.71 5.80 5.88 5.95 6.02 6.09 6.15 6.21 13 3.06 3.73 4.15 4.45 4.69 4.88 5.05 5.19 5.32 5.43 5.53 5.63 5.71 5.79 5.86 5.93 5.99 6.05 6.11 14 3.03 3.70 4.11 4.41 4.64 4.83 4.99 5.13 5.25 5.36 5.46 5.55 5.64 5.71 5.79 5.85 5.91 5.97 6.03 15 3.01 3.67 4.08 4.37 4.59 4.78 4.94 5.08 5.20 5.31 5.40 5.49 5.57 5.65 5.72 5.78 5.85 5.90 5.96 16 3.00 3.65 4.05 4.33 4.56 4.74 4.90 5.03 5.15 5.26 5.35 5.44 5.52 5.59 5.66 5.73 5.79 5.84 5.90 17 2.98 3.63 4.02 4.30 4.52 4.70 4.86 4.99 5.11 5.21 5.31 5.39 5.47 5.54 5.61 5.67 5.73 5.79 5.84 18 2.97 3.61 4.00 4.28 4.49 4.67 4.82 4.96 5.07 5.17 5.27 5.35 5.43 5.50 5.57 5.63 5.69 5.74 5.79 19 2.96 3.59 3.98 4.25 4.47 4.65 4.79 4.92 5.04 5.14 5.23 5.31 5.39 5.46 5.53 5.59 5.65 5.70 5.75 20 2.95 3.58 3.96 4.23 4.45 4.62 4.77 4.90 5.01 5.11 5.20 5.28 5.36 5.43 5.49 5.55 5.61 5.66 5.71 24 2.92 3.53 3.90 4.17 4.37 4.54 4.68 4.81 4.92 5.01 5.10 5.18 5.25 5.32 5.38 5.44 5.49 5.55 5.59 30 2.89 3.49 3.85 4.10 4.30 4.46 4.60 4.72 4.82 4.92 5.00 5.08 5.15 5.21 5.27 5.33 5.38 5.43 5.47 40 2.86 3.44 3.79 4.04 4.23 4.39 4.52 4.63 4.73 4.82 4.90 4.98 5.04 5.11 5.16 5.22 5.27 5.31 5.36 60 2.83 3.40 3.74 3.98 4.16 4.31 4.44 4.55 4.65 4.73 4.81 4.88 4.94 5.00 5.06 5.11 5.15 5.20 5.24

120 2.80 3.36 3.68 3.92 4.10 4.24 4.36 4.47 4.56 4.64 4.71 4.78 4.84 4.90 4.95 5.00 5.04 5.09 5.13 ∞ 2.77 3.31 3.63 3.86 4.03 4.17 4.29 4.39 4.47 4.55 4.62 4.68 4.74 4.80 4.85 4.89 4.93 4.97 5.01

© THE UNIVERSITY OF DUBLIN 2015

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