Question

2. [-/10 Points] DETAILS DEVORESTAT9 9.E.046. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER As part of a larger project to study the behavior of stressed-skin panels, a structural component being used extensively in North America, an article reported on various mechanical properties of Scotch pine lumber specimens. Data on the modulus of elasticity (MPa) obtained 1 minute after loading in a certain configuration and 4 weeks after loading for the same lumber specimens is presented here. Observation 1 min 1 2 3 4 5 10,420 16,620 17,100 15,480 12,920 17,260 13,400 13,900 13,630 13,280 14,370 11,700 15,470 17,840 14,010 14,760 6 7 8 9 10 4 weeks 9,180 13,250 14,720 12,740 10,120 14,570 11,220 11,100 11,420 10,930 12,110 8,650 12,590 15,090 10,550 12,230 Difference 1240 3370 2380 2740 2800 2690 2180 2800 2210 2350 2260 11 12 3050 13 2880 14 15 16 2750 3460 2530 Calculate an upper confidence bound for the true average difference between 1-minute modulus and 4-week modulus; first check the plausibility of any necessary assumptions. (Use a = 0.05. Round your answer to two decimal places.) MPa Interpret your upper confidence bound. We are 95% confident that the true mean difference between 1-minute modulus and 4-week modulus is at least the upper bound. O We are 95% confident that the true mean difference between 1-minute modulus and 4-week modulus is never the upper bound. We are 95% confident that the true mean difference between 1-minute modulus and 4-week modulus is at most the upper bound. O We are 95% confident that the true mean difference between 1-minute modulus and 4-week modulus is exactly the upper bound. You may need to use the appropriate table in the Appendix of Tables to answer this question. Need Help? Read It Talk to a Tutor

Answer 1

Since the 16 observations belongs to same lumber specimen of 1 min and 4 weeks. So, we use Paired t - test.

Let X = 1 Min

Y = 4 Weeks

X	Y	d = X - Y	$(d - \bar{d})$	(P-P)
10420	9180	1240	-1365.63	1864931.6406
16620	13250	3370	3370	11356900.0000
17100	14720	2380	2380	5664400.0000
15480	12740	2740	2740	7507600.0000
12920	10120	2800	2800	7840000.0000
17260	14570	2690	2690	7236100.0000
13400	11220	2180	2180	4752400.0000
13900	11100	2800	2800	7840000.0000
13630	11420	2210	2210	4884100.0000
13280	10930	2350	2350	5522500.0000
14370	12110	2260	2260	5107600.0000
11700	8650	3050	3050	9302500.0000
15470	12590	2880	2880	8294400.0000
17840	15090	2750	2750	7562500.0000
14010	10550	3460	3460	11971600.0000
14760	12230	2530	2530	6400900.0000
	TOTAL	41690		113108431.6406
	MEAN	2605.6250

$\bar{d} = 2605.6250$

(d - d)2 = 113108431.6406

$S^{2} = \frac{1}{n-1}*\Sigma (d - \bar{d})^{2} = \frac{1}{16-1}*113108431.6406$

$\Rightarrow S^{2} = 7540562.109$

Therefore

$S.E(d) = \sqrt{\frac{S^{2}}{n}} = \sqrt{\frac{ 7540562.109}{16}} = \sqrt{471285.1318} = 686.5021$

Since Confidence Interva = 95%. So,the level of significance = $\alpha$ = 5% = 0.05

$t_{cri} = t_{(n-1). \alpha } = t_{(16-1),5\%} = t_{15,0.05} = 2.1314$

EXCEL COMMAND FOR $t_{cri}$ is

=T.INV.2T(0.05,15)

Margin Error:

$E_{\alpha } = S.E(d)*t_{cri}$

$E_{\alpha } =686.5021*2.1314=1463.2106$

Therefore the 95% Confidence Interval of $\mu _{d}$ is

$\bar{d} - E_{\alpha } \leq \mu _{d} \leq \bar{d} + E_{\alpha }$

$\Rightarrow 2605.6250 - 1463.2106 \leq \mu _{d} \leq 2605.6250 + 1463.2106$

$\Rightarrow 1142.4144 \leq \mu _{d} \leq 4068.8356$

$\Rightarrow 1142.41 \leq \mu _{d} \leq 4068.84$

The Upper Confidence Bound for the true average difference between 1 Min modulus and 4 Week modulus is

4068.84

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Therefore we are 95% Confident that the true mean difference between 1 Min modulus and 4 Week modulus is atmost the Upper bound.

OPTION (C)

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NOTE: We can also extract the $t_{cri}$ value from the tabulated value which is posted below.

t 999 t Table cum. prob. one-tail two-tails df t.50 0.50 1.00 0.25 0.50 7.80 0.20 0.40 1.8 0.15 0.30 t90 0.10 0.20 4.95 0.05 0.10 t 975 0.025 0.05 t 99 0.01 0.02 t 995 t .9995 0.005 0.001 0.0005 0.01 0.002 0.001 1.376 1.061 0.978 0.941 0.920 5.893 6.869 2 31 41 5 6 71 81 9 100 11 121 1.356 1.782 3.930 13 14 15 16 171 18 19 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0% 1.000 0.816 0.765 0.741 0.727 0.718 0.711 0.706 0.703 0.700 0.697 0.695 0.694 0.692 0.691 0.690 0.689 0.688 0.688 0.687 0.686 0.686 0.685 0.685 0.684 0.684 0.684 0.683 0.683 0.683 0.681 0.679 0.678 0.677 0.675 0.674 1.963 1.386 1.250 1.190 1.156 1.134 1.119 1.108 1.100 1.093 1.088 1.083 1.079 1.076 1.074 1.071 1.069 1.067 1.066 1.064 1.063 1.061 1.060 1.059 1.058 1.058 1.057 1.056 1.055 1.055 1.050 1.045 1.043 1.042 1.037 0.906 0.896 0.889 0.883 0.879 0.876 0.873 0.870 0.868 0.866 0.865 0.863 0.862 0.861 0.860 0.859 0.858 0.858 0.857 0.856 0.856 0.855 0.855 0.854 0.854 0.851 0.848 0.846 0.845 0.842 0.842 3.078 6.314 12.71 1.886 2.920 4.303 1.638 2.353 3.182 1.533 2.132 2.776 1.476 2.015 2.571 1.440 1.943 2.447 1.415 1.895 2.365 1.397 1.860 2.306 1.383 1.833 2.262 1.372 1.812 2.228 1.363 1.796 2.201 2.179 1.350 1.771 2.160 1.345 1.761 2.145 1.341 1.753 2.131 1.337 1.746 2.120 1.333 1.740 2.110 1.330 1.734 2.101 1.328 1.729 2.093 1.325 1.725 2.086 1.323 1.721 2.080 1.321 1.717 2.074 1.319 1.714 2.069 1.318 1.711 2.064 1.316 1.708 2.060 1.315 1.706 2.056 1.314 1.703 2.052 1.313 1.701 2.048 1.311 1.699 2.045 1.310 1.697 2.042 1.303 1.684 2.021 1.296 1.671 2.000 1.292 1.664 1.990 1.290 1.660 1.984 1.282 1.646 1.962 1.282 1.645 1.960 80% 90% 95% Confidence Level 31.82 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.423 2.390 2.374 2.364 2.330 2.326 98% 20 63.66 318.31 636.62 9.925 22.327 31.599 5.841 10.215 12.924 4.604 7.173 8.610 4.032 3.707 5.208 5.959 3.499 4.785 5.408 3.355 4.501 5.041 3.250 4.297 4.781 3.169 4.144 4.587 3.106 4.025 4.437 3.055 4.318 3.012 3.852 4.221 2.977 3.787 4.140 2.947 3.733 4.073 2.921 3.686 4.015 2.898 3.646 3.965 2.878 3.610 3.922 2.861 3.579 3.883 2.845 3.552 3.850 2.831 3.527 3.819 2.819 3.792 2.807 3.485 3.768 2.797 3.467 3.745 2.787 3.450 3.725 2.779 3.435 3.707 2.771 3.421 3.690 2.763 3.408 3.674 2.756 3.396 3.659 2.750 3.385 3.646 2.704 3.307 3.551 2.660 3.232 3.460 2.639 3.195 3.416 2.626 3.174 3.390 2.581 3.098 3.300 2.576 3.090 3.291 99% 99.8% 99.9% 21 22 23 24 25 26 27 28 29 30 40 601 80 100 3.505 1000 Z 1.036 70% 50% 60% t-table.xls 7/14/2007