Selecting Sample Size Larger n ® smaller confidence interval Ex. 7.17 p 315 range = 520 about equal to 4 Σ Σ = 520 4 130 [email protected] Σ Sqrt@nD == 10, nD 88n ® 649.23<< 7.18 p1 = p2 = .03; Clear@nD lect23.nb 2 Solve@Hp1 H1 - p1L n + p2 H1 - p2L nL 1.96 ^ 2 .005 ^ 2, nD 88n ® 8943.24<< Ex 8.1 p 338 Null Hypothesis : a statistical hypothesis to be tested and accepted or rejected in favor of an alternative http : www.merriam - webster.com dictionary null %20 hypothesis Null Hypothesis : p = .2 Alt Hypothesis p > .2 Test stat = ð of buyers, y rejection region y >= 4 Harbitrary L bd = BinomialDistribution @10, .2D BinomialDistribution @10, 0.2D Ex 8.2 Null hypothesis is true. Error : t = Table@8i, PDF@bd, iD<, 8i, 0, 10<D 980, 0.107374<, 81, 0.268435<, 82, 0.30199<, 83, 0.201327<, 84, 0.0880804 <, 85, 0.0264241 <, 86, 0.00550502 <, 87, 0.000786432 <, 88, 0.000073728 <, 99, 4.096 ´ 10-6 =, 910, 1.024 ´ 10-7 == lect23.nb 3 ListPlot@t, Filling ® AxisD 0.30 0.25 0.20 0.15 0.10 0.05 2 4 6 8 10 1 - CDF@bd, 3D 0.120874 Type 1 error alpha p = .6 prob of type 2 error beta. bd = BinomialDistribution @10, .6D BinomialDistribution @10, 0.6D t = Table@8i, PDF@bd, iD<, 8i, 0, 10<D 880, 0.000104858 <, 81, 0.00157286 <, 82, 0.0106168 <, 83, 0.0424673 <, 84, 0.111477<, 85, 0.200658<, 86, 0.250823<, 87, 0.214991<, 88, 0.120932<, 89, 0.0403108 <, 810, 0.00604662 << lect23.nb 4 ListPlot@t, Filling ® AxisD 0.25 0.20 0.15 0.10 0.05 2 4 6 8 CDF@bd, 3D 0.0547619 power : 1 - beta Higher the power, the greater the prob. of rejecting the null hypothesis when it ' s false. Ex 8.4 p = .3 0.3 Find power 10 lect23.nb 5 bd = BinomialDistribution @10, .3D BinomialDistribution @10, 0.3D t = Table@8i, PDF@bd, iD<, 8i, 0, 10<D 980, 0.0282475 <, 81, 0.121061<, 82, 0.233474<, 83, 0.266828<, 84, 0.200121<, 85, 0.102919<, 86, 0.0367569 <, 87, 0.00900169 <, 88, 0.0014467 <, 89, 0.000137781 <, 910, 5.9049 ´ 10-6 == ListPlot@t, Filling ® AxisD 0.25 0.20 0.15 0.10 0.05 2 4 6 8 10 lect23.nb 6 1 - CDF@bd, 3D 0.350389 low power. Null hypothesis is false but we will often make mistakes and accept it. Ex. 8.5 null hyp mu = 72 n = 50 ybar = 74.1 s = 13.3 alpha = .1 In[974]:= << HypothesisTesting` NormalCI@72, 13.3 Sqrt@50D, ConfidenceLevel -> .8D 869.5895, 74.4105< CDF@NormalDistribution @72, 13.3 Sqrt@50DD, 74.4104755144819485` D 0.9 Therefore for ybar > 74.4105 we reject the null hyp. H*or using tstat, tstat=Hybar-ΜLHsSqrt@nDL*L In[976]:= Out[976]= NormalCI@0, 1, ConfidenceLevel -> .8D 8- 1.28155, 1.28155< Therefore for tstat > 1.281 we reject the null hyp. lect23.nb In[977]:= 7 tstat = H 74.1 - 72L H13.3 Sqrt@50DL 0.2 Out[977]= 1.11648 0.15 The data does not support theory. 0.1 Ex. 8.6 0.05 Plot@PDF@NormalDistribution @78, 13.3 Sqrt@50DD, yD, 8y, 70, 90<D 75 80 85 90 Graphics CDF@NormalDistribution @78, 13.3 Sqrt@50DD, 74.4104755144819485` D 0.0281695 Type 2 error HbetaL 8.7 Null Hypothesis Μ = .5; Althernate hypotheis Μ ¹ .5;
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