Question

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Minitab Problem 3 Imagine choosing n = 16 women at random from a large population and measuring their heights. Assume that the heights of the women in this population are normal, with u = 64 inches and o= 3 inches. Suppose you then test the null hypothesis Ho: u=64 versus the alternative that Hi: u #64, using a=0.10. Assume o is known. Simulate the results of doing this test 20 times by generating 20 columns of 16 numbers selected randomly from a normal population with a mean of 64 and a standard deviation of 3. Conduct a test of hypothesis on the 20 samples and answer the following questions regarding the results: a) In how many tests did you fail to reject Hi? That is, how many times did you make the correct" decision? b) How many times did you make an “incorrect" decision that is, reject H.)? On the average, how many times out of 20 would you expect to make the wrong decision? c) What is the attained significance level (p-value) of each test? Are they all the same? d) Suppose you used a=0.05 instead of a=0.10. Does this change any of your decisions to reject or not? Should it in some cases? Since you are generating random numbers to construct your data sets, it is necessary for you to submit a copy of your MINITAB results with your solution.

Answer 1

(a) Answer: 19 (Because we see that P-value of 19 out of 20 tests>0.10 and data corresponding C5 does not support the claim and its P-value=0.072<0.10).

Session Variable C1 C2 C3 C4 C5 C6 C7 Се c9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 N Mean St Dev 16 63.278 2.155 16 63.407 2.569 16 64.579 3.431 16 64.411 2.552 16 62.650 3.356 16 63.520 2.867 16 64.608 3.023 16 63.349 3.419 16 64.572 3.113 16 64.034 2.985 16 63.030 2.922 16 63.434 3.528 16 64.160 2.700 16 63.109 3.564 16 64.467 2.431 16 64.520 2.320 16 64.514 2.834 16 63.946 3.000 16 64.510 3.417 16 64.443 2.693 SE Mean 90% CI 0.750 (62.044, 64.512) 0.750 (62.174, 64.641) 0.750 (63.345, 65.813) 0.750 (63.178, 65.645) 0.750 (61.416, 63.884) 0.750 (62.286, 64.753) 0.750 (63.375, 65.842) 0.750 (62.115, 64.582) 0.750 (63.339, 65.806) 0.750 (62.800, 65.267) 0.750 (61.797, 64.264) 0.750 (62.200, 64.667) 0.750 (62.927, 65.394) 0.750 (61.875, 64.343) 0.750 (63.233, 65.700) 0.750 (63.287, 65.754) 0.750 (63.281, 65.748) 0.750 (62.712, 65.179) 0.750 (63.277, 65.744) 0.750 (63.209, 65.676) Z P -0.96 0.336 -0.79 0.430 0.77 0.440 0.55 0.583 -1.80 0.072 -0.64 0.522 0.81 0.417 -0.87 0.385 0.76 0.445 0.04 0.964 -1.29 0.196 -0.75 0.450 0.21 0.831 -1.19 0.235 0.62 0.534 0.69 0.488 0.69 0.493 -0.07 0.942 0.68 0.496 0.59 0.555 < AL Worksheet 1 *** C13 C14 + C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C15 C16 C17 C18 C19 C20 C21 1 2 63.6107 3 4 5 6 7 8 64.4952 60.6923 72.5810 64.8689 58.9771 59.3022 69.4634 62.3817 57.9549 68.8865 63.6120 59.0484 62.8039 66.3487 61.0484 62.1682 64.0093 65.3965 65.2722 63.3660 65.2566 65.8389 62.5378 65.4844 60.7238 65.0704 63.5746 58.9126 67.0209 64.0859 63.6405 65.4344 65.2524 68.4136 62.6965 67.4849 61.7828 60.0772 63.4615 61.4784 61.0161 60.6732 65.8283 60.8384 63.8829 60.5224 63.2701 60.6881 62.5814 57.3187 64.6324 63.6464 63.9187 63.6662 61.4881 63.0437 64.3831 66.4413 65.3021 62.3684 61.6015 64.7928 69.2212 65.6005 64.7946 63.2535 60.5489 65.2493 62.4924 63.8059 65.7197 62.2453 66.4209 64.9927 57.9655 61.6280 65.1159 65.5527 66.2823 69.5123 60.8814 64.5707 64.2020 68.1965 64.4969 64.8358 60.9365 63.1625 63.5061 61.7991 69.8491 59.3477 67.1102 62 9222 60.3505 69.3865 63.0519 60.7179 58.8202 63.2939 61.5428 62.0396 59.7477 67.7331 64.0947 66.2350 64.2075 64 0328 67.8335 66.9105 68.2538 61.4017 62.3920 67.2357 59.8542 64.8994 63.5347 61.3241 63.4584 64.1572 60.4134 63.2819 63.6878 60.6692 59.3277 65.4003 64.9017 60.1109 63.7444 64.5424 64.6036 60.3760 63.9985 57.1519 60.4778 60.9793 66.1654 64.5052 66.5433 67.3952 64.6786 71.5095 65.8482 62.7616 67.4836 64.2835 62.9939 62.5345 59.0493 65,0716 65.5321 66.4291 64.3387 67.0013 59.7986 60.9339 60.1094 61.2573 58.1082 63.5257 65.2706 66.1373 61.5441 65.9161 69.9436 66.2868 67.2573 68.9701 65.3986 67.2623 71.6427 70.8077 61.2446 63.4895 65.7498 59.3755 60.6593 63.1710 65.0711 58.7716 61.5606 57.9957 61.7898 64.4408 64.1529 60.7659 59.8986 64.0459 66.2458 66.6658 65.7561 61.2031 67.8507 61.8510 65.6007 67.5314 65.7626 55.7608 62.3719 62.2219 60.9536 62.3574 64.6564 68,2726 64.9383 61.1954 69.0413 61.6017 63.2803 68.5989 62.1970 66.2530 59.3090 64.3908 65.4412 64.6976 67.8602 63.1939 68.3137 61.5178 64.9325 67.7479 65.7026 66.2783 60.2025 59.9779 62.6319 60.1295 64.3069 64.9353 61.3040 64.5850 62.9177 66.5220 61.6687 65.9785 63.8345 66.8352 64.8151 68.0488 66.1399 60.2157 67.5082 65.8969 65.9272 69.3260 61.9940 63.0958 65.9412 63.5795 64.7869 64.8006 68.6246 68.6246 60.2962 66.1684 69.3239 61.8373 60.5609 65.5198 64.8740 64.5407 61.0282 61.3276 62.4382 65.3924 64.5786 65.4137 60.6672 60.9714 63.2086 64.7635 63.5102 65.0509 60.8135 63.9746 60.2873 67.9333 66.1380 71.7311 61.9083 63.1232 63.4375 67.8627 62.3446 63.3276 65.2495 60.5069 57.4967 63.2944 64.4156 61.6160 65.8279 63.0364 63.2242 57.2765 67.8939 65.6046 67.5546 64.4281 64.2269 68.2811 68.8043 65.1181 64.1176 58.6009 66,2547 63.2076 65.9572 65.0835 60.8234 62.3103 63.6623 68.9262 66.2348 66.1913 61.8086 67.0013 67.3041 63.5975 9 10 11 12 13 14 15 16

(b) Answer 1 ( the data corresponding C5 does not support the claim and its P-value=0.072<0.10).

(c)

Variable   P-value
C1 0.336
C2 0.430
C3 0.440
C4    0.583
C5    0.072
C6 0.522
C7    0.417
C8 0.385
C9 0.445
C10 0.964
C11    0.196
C12 0.450
C13 0.831
C14 0.235
C15 0.534
C16 0.488
C17 0.493
C18 0.942
C19 0.496
C20 0.555

We see that P-values are not same.

(d) Since P-value for C5>0.05, all P-value>0.05, hence at $\alpha$ =0.05, we fail to reject all 20 tests. However for $\alpha$ =0.10, we fail to reject 19 out of 20 tests.