Critical Values of the Chi-Squared Distribution U 0.995 0.99 0.98 0.975 0.95 0.90 0.80 0.75 0.70 0.50 0.04393 0.03157 0.03628 0.03982 0.00393 0.0158 0.0642 0.102 0.148 0.455 0.0100 0.0201 0.0404 0.0506 0.103 0.211 0.446 0.575 0.713 1.386 3 0.0717 0.115 0.185 0.216 0.352 0.584 1.005 1.213 1.424 2.366 4 0.207 0.297 0.429 0.484 0.711 1.064 1.649 1.923 2.195 3.357 5 0.412 0.554 0.752 0.831 1.145 1.610 2.343 2.675 3.000 4.351 6 0.676 0.872 1.134 1.237 1.635 2.204 3.070 3.455 3.828 5.348 7 0.989 1.239 1.564 1.690 2.167 2.833 3.822 4.255 4.671 6.346 8 1.344 1.647 2.032 2.180 2.733 3.490 4.594 5.071 5.527 7.344 9 1.735 2.088 2.532 2.700 3.325 4.168 5.380 5.899 6.393 8.343 10 2.156 2.558 3.059 3.247 3.940 4.865 6.179 6.737 7.267 9.342 11 2.603 3.053 3.609 3.816 4.575 5.578 6.989 7.584 8.148 10.341 12 3.074 3.571 4.178 4.404 5.226 6.304 7.807 8.438 9.034 11.340 13 3.565 4.107 4.765 5.009 5.892 7.041 8.634 9.299 9.926 12.340 14 4.075 4.660 5.368 5.629 6.571 7.790 9.467 10.165 10.821 13.339 15 4.601 5.229 5.985 6.262 7.261 8.547 10.307 11.037 11.721 14.339 16 5.142 5.812 6.614 6.908 7.962 9.312 11.152 11.912 12.624 15.338 17 5.697 6.408 7.255 7.564 8.672 10.085 12.002 12.792 13.531 16.338 18 6.265 7.015 7.906 8.231 9.390 10.865 12.857 13.675 14.440 17.338 19 6.844 7.633 8.567 8.907 10.117 11.651 13.716 14.562 15.352 18.338 20 7.434 8.260 9.237 9.591 10.851 12.443 14.578 15.452 16.266 19.337 21 8.034 8.897 9.915 10.283 11.591 13.240 15.445 16.344 17.182 20.337 22 8.643 9.542 10.600 10.982 12.338 14.041 16.314 17.240 18.101 21.337 23 9.260 10.196 11.293 11.689 13.091 14.848 17.187 18.137 19.021 22.337 24 9.886 10.856 11.992 12.401 13.848 15.659 18.062 19.037 19.943 23.337 25 10.520 11.524 12.697 13.120 14.611 16.473 18.940 19.939 20.867 24.337 26 11.160 12.198 13.409 13.844 15.379 17.292 19.820 20.843 21.792 25.336 27 11.808 12.878 14.125 14.573 16.151 18.114 20.703 21.749 22.719 26.336 28 12.461 13.565 14.847 15.308 16.928 18.939 21.588 22.657 23.647 27.336 29 13.121 14.256 15.574 16.047 17.708 19.768 22.475 23.567 24.577 28.336 30 13.787 14.953 16.306 16.791 18.493 20.599 23.364 24.478 25.508 29.336 40 20.707 22.164 23.838 24.433 26.509 29.051 32.345 33.66 34.872 39.335 50 27.991 29.707 31.664 32.357 34.764 37.689 41.449 42.942 44.313 49.335 60 35.534 37.485 39.699 40.482 43.188 46.459 50.641 52.294 53.809 59.335Critical Values of the Chi-Squared Distribution U 0.30 0.25 0.20 0.10 0.05 0.025 0.02 0.01 0.005 0.001 1.074 1.323 1.642 2.706 3.841 5.024 5.412 6.635 7.879 10.827 2.408 2.773 3.219 4.605 5.991 7.378 7.824 9.210 10.597 13.815 3.665 4.108 4.642 6.251 7.815 9.348 9.837 11.345 12.838 16.266 4.878 5.385 5.989 7.779 9.488 11.143 11.668 13.277 14.860 18.466 6.064 6.626 7.289 9.236 11.070 12.832 13.388 15.086 16.750 20.515 7.231 7.841 8.558 10.645 12.592 14.449 15.033 16.812 18.548 22.457 8.383 9.037 9.803 12.017 14.067 16.013 16.622 18.475 20.278 24.321 9.524 10.219 11.030 13.362 15.507 17.535 18.168 20.090 21.955 26.124 10.656 11.389 12.242 14.684 16.919 19.023 19.679 21.666 23.589 27.877 10 11.781 12.549 13.442 15.987 18.307 20.483 21.161 23.209 25.188 29.588 11 12.899 13.701 14.631 17.275 19.675 21.920 22.618 24.725 26.757 31.264 12 14.011 14.845 15.812 18.549 21.026 23.337 24.054 26.217 28.300 32.909 13 15.119 15.984 16.985 19.812 22.362 24.736 25.471 27.688 29.819 34.527 14 16.222 17.117 18.151 21.064 23.685 26.119 26.873 29.141 31.319 36.124 15 17.322 18.245 19.311 22.307 24.996 27.488 28.259 30.578 32.801 37.698 16 18.418 19.369 20.465 23.542 26.296 28.845 29.633 32.000 34.267 39.252 17 19.511 20.489 21.615 24.769 27.587 30.191 30.995 33.409 35.718 40.791 18 20.601 21.605 22.760 25.989 28.869 31.526 32.346 34.805 37.156 42.312 19 21.689 22.718 23.900 27.204 30.144 32.852 33.687 36.191 38.582 43.819 20 22.775 23.828 25.038 28.412 31.410 34.170 35.020 37.566 39.997 45.314 21 23.858 24.935 26.171 29.615 32.671 35.479 36.343 38.932 41.401 46.796 22 24.939 26.039 27.301 30.813 33.924 36.781 37.659 40.289 42.796 48.268 23 26.018 27.141 28.429 32.007 35.172 38.076 38.968 41.638 44.181 49.728 24 27.096 28.241 29.553 33.196 36.415 39.364 40.270 42.980 45.558 51.179 25 28.172 29.339 30.675 34.382 37.652 40.646 41.566 44.314 46.928 52.619 26 29.246 30.435 31.795 35.563 38.885 41.923 42.856 45.642 48.290 54.051 27 30.319 31.528 32.912 36.741 40.113 43.195 44.140 46.963 49.645 55.475 28 31.391 32.620 34.027 37.916 41.337 44.461 45.419 48.278 50.994 56.892 29 32.461 33.711 35.139 39.087 42.557 45.722 46.693 49.588 52.335 58.301 30 33.530 34.800 36.250 40.256 43.773 46.979 47.962 50.892 53.672 59.702 40 44.165 45.616 47.269 51.805 55.758 59.342 60.436 63.691 66.766 73.403 50 54.723 56.334 58.164 63.167 67.505 71.420 72.613 76.154 79.490 86.660 60 65.226 66.981 68.972 74.397 79.082 83.298 84.58 88.379 91.952 99.608Past experience indicates that the time required for high school seniors to complete a standardized test is a normal random variable with a standard deviation of 8 minutes. Test the hypothesis that o = 8 against the alternative that o