1 ) I need help in working out these problems. Thank you in advance.

In this exercise involving paired differences, consider that it is reasonable to assume the populations being compared have approximately the same shape and that the distribution of paired differences is approximately symmetric.Two fuel additives are tested to determine their effect on miles per gallon for passenger cars. Suppose test results for 12 cars follow; each car was tested with both fuel additives. (d_i = additive 1 - additive 2)

	Additive
Car	1	2
1	20.12	17.05
2	23.56	22.77
3	21.03	23.57
4	18.15	17.06
5	21.23	20.22
6	23.77	22.80
7	17.16	17.20
8	19.55	15.98
9	22.87	21.03
10	23.23	20.15
11	22.21	21.78
12	25.02	23.70

Use?= 0.05 and the Wilcoxon signed-rank test to see whether there is a significant difference between the median miles per gallon for the additives.

State the null and alternative hypotheses.

H₀: Median for additive 1?Median for additive 2?0 H_a: Median for additive 1?Median for additive 2 = 0H₀: Median for additive 1?Median for additive 2?0 H_a: Median for additive 1?Median for additive 2 > 0 H₀: Median for additive 1?Median for additive 2?0 H_a: Median for additive 1?Median for additive 2 H₀: Median for additive 1?Median for additive 2 > 0 H_a: Median for additive 1?Median for additive 2 = 0H₀: Median for additive 1?Median for additive 2 = 0 H_a: Median for additive 1?Median for additive 2?0

Find the value of T+. - X Find the value of T-. - X Find the critical value for the test. critical value = * State your conclusion. Reject Hy- There is sufficient evidence to conclude that there is a significant difference between the median miles per gallon for the additives. O Reject Hg. There is not sufficient evidence to conclude that there is a significant difference between the median miles per gallon for the additives. O Do not reject Hy- There is sufficient evidence to conclude that there is a significant difference between the median miles per gallon for the additives. O Do not reject Hy. There is not sufficient evidence to conclude that there is a significant difference between the median miles per gallon for the additives. Appendix A Tables 803 TABLE A. 14 1-SIDED 2-SIDED n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 Critical Values of T for the a =.05 Wilcoxon Matched-Pairs x =.10 o 11 Signed Rank a =.025 a=.05 ONA A Samples) a =.01 a =.02 N U a =.005 x = .01 2 3 1-SIDED 2-SIDED n = 11 n = 12 n = 13 n = 14 n = 15 n = 16 a = .05 x = .10 14 21 26 30 36 x = .025 =.05 14 17 21 25 30 a =.01 x = .02 C 3 20 a =.005 x =.01 7 13 16 10 1-SIDED 2-SIDED n = 17 n = 18 n = 19 n = 20 n = 21 n = 22 a =.05 a =.10 41 4 6 75 =.025 a =.05 35 52 59 66 a=.01 x =.02 28 33 38 43 19 56 a =.005 x = .01 23 8 32 37 13 19 1-SIDED 2-SIDED n = 23 n = 24 n = 25 n = 26 n = 27 n = 28 a =.05 = .10 33 101 110 120 130 a =.025 x = .05 73 31 90 98 107 117 a =.01 x = .02 52 59 77 35 93 102 a =.005 x = .01 51 68 76 92 1-SIDED 2-SIDED n = 29 n = 30 n = 31 n = 32 n = 33 n = 34 a =.05 x = .10 152 163 175 188 201 a =.025 x = .05 127 137 148 159 171 183 a=.01 a =.02 111 20 130 141 51 162 a =.005 a =.01 100 109 118 128 138 149 1-SIDED 2-SIDED n = 35 n = 36 n = 37 n = 38 n = 39 a =.05 x = .10 214 242 256 71 x =.025 = .05 195 208 222 235 250 a =.01 =.02 174 186 198 21 1 224 a =.005 x =.01 160 171 183 195 208 1-SIDED 2-SIDED n = 40 n = 41 n = 42 n = 43 n = 44 n = 45 a =.05 = .10 287 303 $19 336 353 371 a =.025 x = .05 264 279 295 311 327 344 a =.01 x =.02 238 252 267 281 297 313 a = .005 a =.01 221 234 248 262 277 292 1-SIDED 2-SIDED n = 46 n = 47 n = 48 n = 49 n = 50 a =.05 X = .10 389 427 446 466 a =.025 x = .05 361 179 397 415 434 a =.01 x =.02 329 345 362 380 398 =.005 a =.01 307 323 339 356 373 From E. Wilcoxon and R. A. Wilcox, "Some Rapid Approximate Statistical Procedures," 1964. Reprinted by permission of Lederle Labs, a division of the American Cyanamid Co.Before After 5.6 5.4 1.5 1.5 4.7 1.6 3.8 4.3 2.4 2.1 5.5 5.0 5.1 5.2 4.6 4.5 3.7 4.5 What is the value of T-? D x What is the value of T+? What is the test statistic, T? Using the table of Critical Values, what is the critical value for this study? X Brand A Brand B 24 27 15 24 22 26 27 25 22 26 27 22 27 25 27 265 27 23 What is the value of T? |:| What is the critical value#|| [ "HE R R T = AL a 5% significance level, what can you conclude from this study? (O Brand A is rated significantly higher than Brand B. (O There is a significant difference in the median ratings between the two coffee brands. There is no significant difference in the median ratings between the two coffee brands. O Brand A is rated significantly lower than Brand B