Extra Homework; Statistics 301;Professor WardropFall 2007Section 2.31. Consider a balanced study with eight subjects,identified as A, B, C, D, E, G, H, and J. In theactual study,• A, B, C and D are assigned to the firsttreatment, and• There are exactly four successes, and theyare obtained by A, B, C, and H.This information is needed for parts (a)–(c) be-low.(a) Compute the observed value of the teststatistic.(b) Assume that the Skeptic is correct. Deter-mine the observed value of the test statisticfor the assignment that places A, D, E, andG on the first treatment, and the remainingsubjects on the second treatment.(c) We have obtained the sampling distribu-tion of the test statistic on the assumptionthat the Skeptic is correct. It also is possi-ble to obtain a sampling distribution of thetest statistic if the Skeptic is wrong pro-vided we specify exactly how the Skep-tic is in error. Assume that the Skepticis incorrect about subjects C, D, H, andJ, but correct about subjects A, B, E, andG. This means that for subjects C, D, H,and J, his/her/its response will change ifthe treatment changes.For the assignment that puts A, D, E, andH on the first treatment, and the other sub-jects on the second treatment, determinethe response for each of the eight subjects.2. Consider a unbalanced study with nine subjects,identified as A, B, C, D, E, G, H, J, and K. In theactual study,• A, B, C, D, and E are assigned to the firsttreatment, and• There are exactly five successes, and theyare obtained by B, C, E, H, and J.This information is needed for parts (a)–(c) be-low.(a) Compute the observed value of the teststatistic.(b) Assume that the Skeptic is correct. Deter-mine the observed value of the test statis-tic for the assignment that places A, C, D,G, and K on the first treatment, and the re-maining subjects on the second treatment.(c) We have obtained the sampling distribu-tion of the test statistic on the assumptionthat the Skeptic is correct. It also is possi-ble to obtain a sampling distribution of thetest statistic if the Skeptic is wrong pro-vided we specify exactly how the Skepticis in error. Assume that the Skeptic is in-correct about subjects C, D, E, G, H, J,and K, but correct about subjects A, andB. This means that for subjects C, D, E,G, H, J, and K, his/her/its response willchange if the treatment changes.For the assignment that puts A, E, G, H,and J on the first treatment, and the othersubjects on the second treatment, deter-mine the response for each of the nine sub-jects.3. An unbalanced CRD yields the data below.Treatment S F Total1 a b 102 c d 5Total 10 5 15On the assumption the Skeptic is correct, list allpossible values of the test statistic.14. A CRD yields the following data.Treatment S F Total1 a b 52 c d 4Total 6 3 9On the assumption the Skeptic is correct, deter-mine all possible values of the test statistic.Section 2.55. Sally performs a CRD with a dichotomous re-sponse. She obtains the sampling distributionof the test statistic for Fisher’s test for her data;it is given below.x P (X = x) P (X ≤ x) P (X ≥ x)−0.30 0.0003 0.0003 1.0000−0.25 0.0029 0.0032 0.9997−0.20 0.0151 0.0184 0.9968−0.15 0.0514 0.0697 0.9816−0.10 0.1193 0.1890 0.9303−0.05 0.1957 0.3848 0.81100.00 0.2305 0.6152 0.61520.05 0.1957 0.8110 0.38480.10 0.1193 0.9303 0.18900.15 0.0514 0.9816 0.06970.20 0.0151 0.9968 0.01840.25 0.0029 0.9997 0.00320.30 0.0003 1.0000 0.0003(a) Find the P-value for the second alternative(p1< p2) if x = −0.15.(b) Determine the P-value for the third alter-native (p16= p2) if x = −0.25.(c) Determine the value of x and the P-valuethat satisfy the following condition: Thedata are statistically significant but nothighly statistically significant for the firstalternative (p1> p2).6. Pam performs a CRD with a dichotomous re-sponse. She obtains the sampling distributionof the test statistic for Fisher’s test for her data;it is given below.x P (X = x) P (X ≤ x) P (X ≥ x)−0.289 0.0002 0.0002 1.0000−0.244 0.0016 0.0018 0.9998−0.200 0.0089 0.0107 0.9982−0.156 0.0330 0.0437 0.9893−0.111 0.0851 0.1288 0.9563−0.067 0.1576 0.2864 0.8712−0.022 0.2136 0.5000 0.71360.022 0.2136 0.7136 0.50000.067 0.1576 0.8712 0.28640.111 0.0851 0.9563 0.12880.156 0.0330 0.9893 0.04370.200 0.0089 0.9982 0.01070.244 0.0016 0.9998 0.00180.289 0.0002 1.0000 0.0002(a) Find the P-value for the second alternative(p1< p2) if x = −0.111.(b) Determine the P-value for the first alterna-tive if x = 0.289.(c) Determine all values of x and the P-valuethat satisfy the following condition. Thedata are statistically significant but nothighly statistically significant for the thirdalternative (p16= p2).7. Two comparative studies with dichotomous re-sponses, randomization, and two treatments areperformed. The first study has n = 35 and thesecond study is balanced.The observed value of the test statistic, x, is apositive number for the first study and a nega-tive number for the second study. The P-valuesfor all three alternatives are obtained for bothstudies. The six (sorted) P-values are below.0.0433, 0.0866, 0.3297, 0.5060,0.8700, and 0.9865.Match each P-value with its study and alterna-tive.AlternativeStudy > < 6=FirstSecond28. Two comparative studies, with two treatments,dichotomous responses and randomization areperformed. The first study is balanced.The observed value of the test statistic, x, is apositive number for the first study and a nega-tive number for the second study. The P-valuesfor all three alternatives are obtained for bothstudies. The six (sorted) P-values are below.0.1001, 0.1223, 0.2002, 0.2187,0.9432, and 0.9565.Match each P-value with its study and alterna-tive. Hint: For the second study, for the ac-tual x,P (X = x) = 0.0655.AlternativeStudy > < 6=FirstSecondSection 2.49. (Extra Credit.) A balanced CRD with n = 60subjects is performed. The study yields a to-tal of 20 successes. Write an expression forP (X = 0.00) using binomial coefficients; donot compute the answer.10. (Extra Credit.) A balanced CRD with n = 80subjects is performed. The study yields a to-tal of 30 successes. Write an expression forP (X = 0.25) using binomial coefficients; donot compute the answer.Section 3.111. Below is the sampling distribution of the teststatistic for Fisher’s Test for an unbalancedCRD.x P (X = x)x P (X = x)−0.8 0.0070 0.1 0.3916−0.5 0.09320.4 0.1632−0.2 0.32630.7 0.0186Draw the probability histogram for this distri-bution.12. Below is the probability