Question

During the off-season (January – October), animal scientists at the North Pole train the reindeer to improve their take-off speed. During the most recent off-season, a new training device was used to improve the average time required for reindeer to sprint 200 yards. Each of the current reindeer was timed in the 200-yard dash before and after using the device. The results (in sec) are given in the table below.

Reindeer	Time Before Training	Time After Training
1	8.19	7.83
2	8.62	8.42
3	7.98	7.65
4	8.41	7.89
5	9.15	9.01
6	7.45	6.69
7	8.55	7.98
8	7.11	6.88
9	9.02	8.56

Using this data, the animal scientists wish to test the following hypotheses using a level of significance of 0.05.

Note that mu1 denotes the average time required for an untrained reindeer to sprint 200 yards and mu2 denotes the average time required for a trained reindeer to sprint 200 yards.

Compute the test statistic. (

Answer 1

We frame the Null Hypothesis

: There is no Significant difference between the two Sample i.e before and after.

он

To test the we use the Paired t - test

он

10 SE

X	Y	d = X - Y	$(d - \bar{d})$	(P-P)
8.19	7.83	0.36	-0.0367	0.0013
8.62	8.42	0.2	-0.1967	0.0387
7.98	7.65	0.33	-0.0667	0.0044
8.41	7.89	0.52	0.1233	0.0152
9.15	9.01	0.14	-0.2567	0.0659
7.45	6.69	0.76	0.3633	0.1320
8.55	7.98	0.57	0.1733	0.0300
7.11	6.88	0.23	-0.1667	0.0278
9.02	8.56	0.46	0.0633	0.0040
	TOTAL			0.3194
	MEAN	0.3967

$\Sigma (d - \bar{d})^{2}$ = 0.3194

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

$S^{2}= \frac{1}{9-1}*0.3194$

$\Rightarrow S^{2}= \frac{0.3194}{8}$

$\Rightarrow S^{2}= 0.0399$

Therefore

$S.E = \sqrt{\frac{S^{2}}{n}}$

$\Rightarrow S.E = \sqrt{\frac{0.0399}{9}}$

$\Rightarrow S.E = \sqrt{0.0044}$

$\therefore \, \, \, S.E = 0.0666$

Therefore

10 SE

$\Rightarrow t = \frac{\left | 0.3967 \right |}{0.0666}$

Given level of significance = 0.05.= 5%

$t_{cri} = t_{(n-1),\alpha } = t_{(9-1),0.05} = t_{8,0.05} = 2.306$

Since So we Reject
он
at 5% Level of Significant.

NOTE: The Critical value of t has been extracted from the tabulated value of t; 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