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ST1252-1
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.
ST1252-1
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.
Page 2 of 12c© Trinity College Dublin, The University of Dublin 2016
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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.
Page 3 of 12c© Trinity College Dublin, The University of Dublin 2016
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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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Page 5 of 12c© Trinity College Dublin, The University of Dublin 2016
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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Page 6 of 12c© Trinity College Dublin, The University of Dublin 2016
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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Page 7 of 12c© Trinity College Dublin, The University of Dublin 2016
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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Page 8 of 12c© Trinity College Dublin, The University of Dublin 2016
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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Page 9 of 12c© Trinity College Dublin, The University of Dublin 2016
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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Page 10 of 12c© Trinity College Dublin, The University of Dublin 2016
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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Page 11 of 12c© Trinity College Dublin, The University of Dublin 2016
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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