BUSINESS STATICS

Can we conclude at the 0.05 level of significance that cars equipped with radial NJn-Paranrtric Jests tyres obtain better fuel economy than those equipped with regular belted tyres? NOTES Solution Let 1-11 and 1-lz represent the mean kilometres per litre for cars equipped with radial and belted tyres respectively. Self-Instructional!Yllterial 341 (i) Ho: (j:il -):i2) = 0 (ii) ~ : (j:il -):i2) > 0 (iii) a = 0.05 (iv) Test statistic: Binomial variable Xwith p= 21 (v) Computations: After replacing each positive difference by a \"+\" symbol and each negative difference by a \"-\" symbol and then discarding the two zero differences, we obtain the sequence +-++-+++++++-+ for which n= 14 and x= 11. Using the normal-curve approximation, we find Z= [I;.fz7] = 1.87 =and then P P(X\"?:. 11) A. P (Z> 1.87) =0.0307 (vi) Decision: Reject ~and conclude that, on the average, radial tyres do improve fuel economy. Remarks 1. Not only is the sign test one of our simplest non-parametric procedures to apply, it has the additional advantage ofbeing applicable to dichotomous datathat cannot be recorded on a numerical scale but can be represented by positive and negative responses. 2 Sometimes, we draw analogies or comparisons between parametric and corresponding non-parametric tests. In the case of the sign test the competition, of course, is the t-test. Ifone is sampling from a normal distribution, the use ofthe t-test will result in the larger power of the test. If the distribution is merely symmetric, though not normal, the t-test is preferred in terms ofpower unless the distribution has extremely \"heavy tails\" compared to the normal distribution. 13.5 SUMMARY I this unit, you have learned about non-parametric tests. Non-parametric tests are useful under the following conditions: (i) When the nature of the population distribution from which the samples were drawn is not known or is known to be non-normal. (ii) The variables are expressed in: • Nominal form, i.e., classified into categories and represented by frequency counts, or • Ordinal form i.e., ranked in order and expressed as 1, 2, 3, ... etc. Most non-parametric tests, thus, are applied to nominal or ordinal scaled data. You have learned about Chi-Square Test and the goodness offit. You have also learned about other non-parametric tests such as the Sign Test, the Signed-Rank Test, the Rank-Sum Test and the Kruskall-Wallis Test.

Business Statistics-D 13.6 ANSWERS TO 'CHECK YOUR PROGRESS' NOTES 1. The Chi-Square Test was discovered in 1875 by Helmert and again independently in 1900 by Karl Pearson. 342 Self-InstructionalJl.flterial 2. Two possible sources oferror in the application ofthe Chi-Square test are small theoretical frequencies and neglect offrequencies ofnon-occurrence. 3. Three non-parametric tests of significance are the Sign Test, the Signed-Rank Test and the Kruskall-Wallis Test. 4. Frank Wilcoxon devised the Signed-Rank Test. 13.7 QUESTIONS AND EXERCISES Short-Answer Questions 1. Why are non-parametric tests useful? 2. What does the Chi-Square test measure? 3. What are the conditions for the application ofthe Chi-Square Test? 4. Why is the Sign Test useful? 5. What does the Runs Test do? Long-Answer Questions 1. A set ofcoins is tossed 3200 times, and the no. ofheads appearing each time is noted. The results are given below: 1/V. of~·& I ~ I [ I 2 I~ I~ I~Frequency 570 1100 2. From the following table, test the hypothesis that flower colour is independent of flatness ofleaf Rat leaves Curved leaves Total White flowers 99 36 135 Red flowers 20 5 25 Total ll9 41 160 3. The following table shows the association among 1000 school boys between their general ability and their mathematical ability. Calculate the coefficient of contingency between the two. General Ability Good Fair Poor Total Mathematical Ability Good 44 22 4 70 Fair 265 257 178 700 Poor 41 91 98 230 Total 350 370 280 1000

4. The normal rate ofinfection for a certain disease ofcattle is known to be 50%. In NJn-Pararrrtric Tests an experiment with 7 animals injected with a new vaccine, it was found that none ofthe animals caught infection. Can the evidence be regarded as conclusive (at NOTES 1% level ofsignificance) to prove the value ofthe new vaccine. Self-Instructional Material 343 5. It is claimed that a new diet will reduce a person's weight by 4.5 kg, on average, in a period of2 weeks. The weights in kgs of 10 women who followed this diet were recorded before and after a 2-week period, yielding the following data: Ubman Vfeight Before Vfeight After I 58.5 60.0 2 60.3 54.9 3 61.7 58.1 4 69.0 62.1 5 64.0 58.5 6 62.6 59.9 7 56.7 54.4 8 63.6 60.2 9 68.2 62.3 10 59.4 58.7 Use the sign test at the 0.05 level of significance to test the hypothesis that the diet reduces the median weight by 4.5 kg against the alternative hypothesis that the median difference in weight is less than 4.5 kg. 6. The following data represent the no. ofhours that two different types ofscientific pocket calculators operate before a recharge is required. Calculator-A 5.5 5.6 6.3 4.6 5.3 5.0 6.2 5.8 5.1 Calculator-B 3.8 4.8 4.3 42 4.0 4.9 4.5 52 4.5 Use the rank-sum test with a= 0.01 to determine ifcalculator A operates longer than calculator Bon a full-battery charge. 7. In an experiment to determine which ofthree different missile systems is preferable, the propellant burning rate is measured. The data, after coding, are given in the following table. Propellant Burning Rates Mssile Mssile Mssile System-/ System-II System-m 24.0 232 18.4 16.7 19.8 19.1 22.8 18.1 17.3 19.8 17.6 17.3 18.9 20.2 19.7 17.8 18.9 18.8 19.3 Use the Kruskal-Wallis test at a significance level ofa= 0.05 to testthe hypothesis that the propellant burning rates are the same for the three missile systems. Note that c20.05 = 5.991 for 2 degress of freedom.

Business Statistics-H 8. A random sample of 15 adults living in a small town are selected to estimate the proportion ofvotes favouring a certain candidate for mayor. Each individual was NOTES also asked if he/she was a college graduate. By letting Yand Ndesignate the response ofYES and NO to the education question, the following sequence was obtained: NNNNYYNYYNYNNNN Use the runs test at the 0.1level of significance to determine ifthe sequence supports the contention that the sample was drawn at random. 9. A cigarette manufacturer claims that the tar content of brand B cigarettes is lower than that ofbrand A To test this claim, the following determinations oftask content, in milligrams were recorded: 12 9 13 10 7 Use the rank-sum test with a= 0.05 to test whether the claim is valid. 13.8 FURTHER READING Kothari, C.R. 1984. Quantitative Techniques, 3rd Edition. New Delhi: Vtkas Publishing House Pvt. Ltd. Chandan, J.S. 1998. Statistics ForBusiness andEconorrics. New Delhi: Vikas Publishing House Pvt. Ltd. Chandan, J.S., Jagjit Singh and K.K. Khanna. 1995. Business Statistics, 2nd Edition. New Delhi: Vtkas Publishing House Pvt. Ltd. Levin, Richard I. 1991. Statistics for JIJana.gerrent New Jersey: Prentice-Hall. Meyer, Paul L. 1970. Introductory Probability and Statistical Applications. Massachusetts: Addison-Wesley Publishing Co. 344 Self-In.~tructional Material

NOTES Self-Instructional Muerial 345

NOTES 346 Self-Instructional Mlterial