Prob Meth In Engr
Prob Meth In Engr CIVL 7012
University of Memphis
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Date Created: 10/23/15
Hypothesis Testing From notes by J Hurley and R Meier Hypothesis testing consists of a statistical test composed of ve parts and is based on proof by contradiction 1 De ne the null hypothesis H0 Develop the alternative hypothesis Ha Evaluate the test statistic De ne the rejection region may be one or twotailed Make a conclusion based on comparison of the value of the test statistic and the rejection region accept or reject the null hypothesis Elk59 Y acceptance rejection region region Reject H0 7 this decision means that the observed data do not support the null hypothesis that is the observed data provides evidence that H0 is not true Accept H0 7 this decision indicates that the data do not provide evidence that the null hypothesis is not true CI VL 70128012 Probabilistic Methadsfar Engineers There are two types of errors which can result from a hypothesis test Type I error 0i probability of type I error 7 reject the null hypothesis when it is actually true Type II error probability of type II error accept the null hypothesis when it is actually false Typically you de ne the amount of risk you are willing to take of making a type I error and adjust the other factors to minimize the risk of making a type II error Select level of signi cance based on how much risk you can tolerate of making a type I error 0 Levels of signi cance are typically 001 005 010 0 Level of signi cance selected depends on the severity of the consequences of a type I error Factors effecting type II errors are 0 Design of sample selection how samples are obtained 0 Sample size 0 Choice of test statistic CI VL 70128012 Probabilistic Methadsfar Engineers Single Sample Test of Hypothesis on 11 For Normal Population and Known U Ho u uo Test Statistic z x u 7 aJ Alternative Ha u 7 uo Ha u gt uo Ha u lt uo Hypothesis Rejection Region RR lzl gt za2 RR 2 gt 2a RR 2 lt 72a for level 0i test Example A company that produces biasply tires is considering a modi cation in the tread design An economic feasibility study indicates that the modi cation can be justi ed only if the true average tire life under standard test condition exceeds 40000 miles A random sample of n 16 prototype tires is manufactured and tested resulting in a sample average tire life of f 40758 miles Suppose the standard deviation for the current version of the tire is 6 1500 miles and is not expected to change Do the data suggest that the modi cation meets the condition required for changeover Test the appropriate hypothesis using signi cance level at 001 CI VL 70128012 Probabilistic Meihadsfar Engineers Large Sample Tests 0 unkown Ho H 390 Test Statistic f7 uo Z sxZ Alternative Ha u 7 uo Hypothesis Rejection Region RR lzl gt 2a2 Example Ha ugtu0 Ha ultu0 RR 2 gt 2a RR 2 lt 72a A certain type of brick is being considered for use in a particular construction project The brick will be used unless sample evidence strongly suggests that the true average compressive strength is below 3200 psi A random sample of 36 bricks is selected and each is tested to failure The sample average compressive strength is 3109 psi with a standard deviation of 156 psi At a level of signi cance ofoc 005 should the brick be used CI VL 70128012 Probabilistic Methadsfar Engineers Small Sample Tests 6 Unknown Ho u Mo Test Statistic f 7 uo t s J Alternative Ha u 7 uo Ha u gt uo Ha u lt uo Hypothesis Rejection Region RR ltl gt taLH RR tgt tow1 RR tlt 4am Example In order to test gasoline mileage performance for a new version of one of its compact cars an automobile manufacturer selected six nonprofessional drivers to drive test cars from Phoenix to Los Angeles At the conclusion of the trip the resulting gas mileage numbers for the six cars were 322 293 315 287 302 300 The manufacturer wishes to advertise that this car gets 30 mpg or better on the highway Do the sample data support the claim that the manufacturer would like to make Assume 0c 005 CI VL 70128012 Probabilistic Methadsfar Engineers TWO SAMPLE HYPOTHESIS TESTS ON u Comparison of Two Means 61 and 62 Known H0 391 7H2 A0 Test Statistic Tel f2 A0 Z 0 0i n1 quot2 Alternative Ha ul 7 pl 7 A0 Ha ul iuz gt A0 Ha ul iuz lt A0 Hypothesis Rejection Region RR lzl gt za2 RR 2 gt 2a RR 2 lt 72a Example A random sample of 20 specimens of coldrolled steel had an average yield strength of 298 ksi A second random sample of 25 galvanized steel specimens gave an average yield strength of 347 ksi Assuming that the two yield strength distributions are normal with 0391 40 and 039 50 do the data indicate that the true average yield strengths 1 and 2 are different Assume a 001 CI VL 70128012 Probabilistic Methadsfar Engineers Lar e Sam le Com arison of Two Means 6 and 6 Unknown H0 1 2 A0 Test Statistic a 7 E 7 A0 2 ii i quot1 quot2 Alternative Ha H 7p 7 A0 Ha H 7p gt A0 Ha H 7p lt A0 Hypothesis Rejection Region RR lzl gt 2042 RR 2 gt 2a RR 2 lt 72a Example In a sample of 30 women who did not live near a freeway the sample average blood lead level was 99 and the sample standard deviation was 49 while a second sample of 35 females who did live near a freeway had a sample average and sample stande deviation of 167 and 70 respectively Does proximity to heavily traveled roads result in higher blood lead levels Test at a 001 CI VL 70128012 Probabilistic Methadsfar Engineers Small Sam le Com arison of Means 6 and 6 Unknown but Egual Pooled t testg H0 ul7u2A0 Test Statistic f1 7 f2 7 A0 S ii p quot1 quot2 Where quot171512nz71s v n1 n2 2 degrees of freedom Alternative Ha I 7 pl 7 A0 Ha H1 7 pl gt A0 Ha H1 7 pl lt A0 Hypothesis Rejection RR M gt taZYV RR tgt tow RR tlt 74w Region Example A random sample of 15 ceramic insulators doped in a certain manner yielded a sample average holdolT voltage of 110 kV and a sample standard deviation of 24 kV A random sample of 76 undoped ceramic insulators produced a sample average holdoff voltage of 101 kV with a standard deviation of 22 kV If we can assume that the actual population standard deviations should be the same do the data suggest that the true average holdoff voltage for doped specimens exceeds that for plain specimens by more than 5 kV at a signi cance level of010 CI VL 70128012 Probabilistic Methadsfar Engineers n u nmmrisrm of Means 61 and 6 Unknown and Unequal the Sith 39 Procedure H0 1 7112 A0 Test Statistic a 7 E 7 A0 t Z Z L1 52 quot1 quot2 Where 2 2 2 S1 S2 J quotI quot2 C round down 2 2 2 2 s1 Hi1 s2 n2 n1 1 rt2 1 Alternative Ha ul 7 pl 7 A0 Ha ul 7 pl gt A0 Ha ul iuz lt A0 Hypothesis Rejection RR ltl gt tam RR tgt tow RR tlt itw Region Example Dextroamphetamine is a drug commonly used to treat hyperkinetic children A paper in the Journal of Nervous and Mental Disorders 1968 vol 146 pp 136 146 reported the following data on the percentage of the drug excreted within seven hours of its administration by children having organically related disorders and children with nonorganic disorders 10rganic 1753 2060 1762 2893 2710 2Nonorganic 1559 1476 1332 1245 1279 The summary values are E 2236 E 1378 s12 2863 and s22 180 The data suggest that there is much less variability in recovery percentage for children with organically related disorders Compare the means at a signi cance level of 001 CI VL 70128012 Probabilistic Methadsfar Engineers Paired Samples Baired t Test 1 H0 uD A0 Test Statistic Where Alternative Hypothesis Rejection Region Example 37A paired spJi t 2 11 11 7 11 5 D n 71 n 71 Ha MDiAO Ha uDgtA0 Ha uDltA0 RR tpaaied gt14 RR tpaaled gt raw RR tpaaled lt tav In an experiment designed to evaluate an additive to increase the strength of concrete each of ve batches of concrete was divided in half and the additive added to one half of each batch The resulting compressive strength measurements load in kips at failure were shown below Does the additive work Test at a 001 Treated Untreated 174 155 137 123 169 159 161 148 147 132 CI VL 70128012 Probabilistic Methadsfar Engineers Tests of Hypotheses on o 2 Single Sample Test of Hypothesis on U 2 From a Normal Population Test Statistic 2 n 1s2 0 Alternative Ha o2 gt 002 Ha 0392 lt 03902 Hypothesis Rejection Region RR 12 gt 1701 RR 12 lt 2141an Example 2 2 Ha 039 017 2 gt 2 or RR39 I 1H 39 2 2 I lt 17H A manufacturer of liquid detergent is interested in the uniformity of the machine used to ll bottles If the variance of ll volume exceeds 001 uid 022 an unacceptable proportion of bottles will be under lled A random sample of 20 bottles results in a sample variance of ll volume of s2 00153 uid ozz Assuming 0i 005 is there evidence in the sample data to suggest the need for replacing or overhauling the machine CI VL 70128012 Probabilistic Methadsfar Engineers Comparison of Variances may be used as a preliminary test for H0 03912 0392 comparing the means of the two samples 2 Test Statistic F where subscripts are such that S12 2 s22 4 r4 NNHN vln1 landv2nzl Alternative Ha 012 gt 0 Ha 03912 lt 03922 Ha 03912 7 03922 Hypothesis Rejection Region RR F gtFM V RR F gtFM V RR F gtFly 717 2 7271 2v1v2 note reversal of signs and subscripts Example A study of two types of materials used in electrical conduits is to be conducted The purpose of the study is to compare the strength of one to the other by measuring the load required to crush a 6 inch long piece of material to 40 of its original diameter The primary question to be answered is Is ul gt HZ However before this is done we must consider the question Is 03912 03922 If the answer to this appears to be yes then a pooled t procedure can be used Otherwise the SmithSatterthwaite procedure should be employed Material 1 Material 2 n1 25 n2 16 X1 3801b X2370 lb s12 100 sf 400 CI VL 70128012 Probabilistic Methadsfar Engineers CI VL 70128012 Probabilistic Methadsfar Engineers
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