Question

Conduct the hypothesis test and provide the test statistic and the critical​ value, and state the conclusion.

A person drilled a hole in a die and filled it with a lead​ weight, then proceeded to roll it

200

times. Here are the observed frequencies for the outcomes of​ 1, 2,​ 3, 4,​ 5, and​ 6, respectively:

29,32,43,39,29,28.

Use a 0.10 significance level to test the claim that the outcomes are not equally likely. Does it appear that the loaded die behaves differently than a fair​ die?

The test statistic is

?

The critical value is:

?

State the conclusion.

Answer 1

1) = 200 k = 6= No.of categories 2 04 = 39; 0,= 29 ; O2 =32 ; 02= 43; 04 05 = 29 ; 06 = 28. Oi ith observed frequency To test; Ho : P = P₂ → under Ho 6 ( Ti ithe proportion of population Hi The vs Ho is not true. Expected frequencies (Ei) : Ei = npsiz 1,2,3, 4, 5, 6 Ei- nh= 2 60 x - 2 50 - 100 x 33.3333 l= (tt) 4 Test Statistic : x?- 5 (O.- Ec)? Ei (29-33-3 333) (32-33-3333)+ (43-33.3333)? Et 33.3333 33-3333 33.3333 (39-33.3333) (29-33.33330? (28–33.3333)? 33.3333 33.3333 33.3333 (Aus of Test = 5.8000058 Statistic) df z k-1=6-1=5 at x = 0.10;since the test is 2 sided find the coloumn l-a in table for upper fould coitical values and reject to if test & tabulated statistie & value

- Q = 10.05 = 0.95 For upper feul critical values; for k-1= 5 (dif) and x=0.95 we get 11.070 Here; Test statistic ( 11.070 Hence, we fail to reject Hoon & Critical value in 11.070 71 using appar fail bitial үл хо, Conclusion) - sesele There is not support the outcomes are The loaded as the fair sufficient evidence to claim that tooded not equally likely same die behaves die . s

Upper-tail critical values of chi- square distribution with v degrees of freedom Probability less than the critical value 0.90 0.95 0.975 0.99 0.999 2.706 4.605 6.251 7.779 9.236 10.645 12.017 13.362 14.684 15.987 17.275 18.549 19.812 21.064 22.307 23.542 24.769 25.989 27.204 28.412 29.615 30.813 32.007 33. 196 34.382 35.563 36.741 37.916 39.087 40.256 41.422 42.585 43.745 44.903 46.059 47.212 48.363 49.513 50.660 51.805 52.949 54.090 55.230 56.369 57.505 58.641 59.774 60.907 62.038 63.167 64.295 65.422 66.548 67.673 68.796 69.919 71.040 72.160 73.279 74.397 75.514 76.630 77.745 78.860 79.973 81.085 82.197 83.308 84.418 85.527 86.635 87.743 88.850 89.956 91.061 92.166 93.270 94.374 95.476 96.578 97.680 98.780 99.880 100.980 102.079 103. 177 3.841 5.024 6.635 5.991 7.378 9.210 7.815 9.348 11.345 9.488 11.143 13.277 11.070 12.833 15.086 12.592 14.449 16.812 14.067 16.013 18.475 15.507 17.535 20.090 16.919 19.023 21.666 18.307 20.483 23. 209 19.675 21.920 24.725 21.026 23.337 26.217 22.362 24.736 27.688 23.685 26.119 29.141 24.996 27.488 30.578 26.296 28.845 32.000 27.587 30.191 33.409 28.869 31.526 34.805 30. 144 32.852 36.191 31.410 34. 170 37.566 32.671 35.479 38.932 33.924 36.781 40.289 35. 172 38.076 41.638 36.415 39.364 42.980 37.652 40.646 44.314 38.885 41.923 45.642 40.113 43. 195 46.963 41.337 44.461 48.278 42.557 45.722 49.588 43.773 46.979 50.892 44.985 48.232 52. 191 46. 194 49.480 53.486 47.400 50.725 54.776 48.602 51.966 56.061 49.802 53.203 57.342 50.998 54.437 58.619 52. 192 55.668 59.893 53.384 56.896 61.162 54.572 58. 120 62.428 55.758 59.342 63.691 56.942 60.561 64.950 58. 124 61.777 66.206 59.304 62.990 67.459 60.481 64.201 68.710 61.656 65.410 69.957 62.830 66.617 71.201 64.001 67.821 72.443 65. 171 69.023 73.683 66.339 70.222 74.919 67.505 71.420 76.154 68.669 72.616 77.386 69.832 73.810 78.616 70.993 75.002 79.843 72.153 76.192 81.069 73.311 77.380 82.292 74.468 78.567 83.513 75.624 79.752 84.733 76.778 80.936 85.950 77.931 82.117 87.166 79.082 83.298 88.379 80.232 84.476 89.591 81.381 85.654 90.802 82.529 86.830 92.010 83.675 88.004 93.217 84.821 89.177 94.422 85.965 90.349 95.626 87.108 91.519 96.828 88.250 92.689 98.028 89.391 93.856 99.228 90.531 95.023 100.425 91.670 96.189 101.621 92.808 97.353 102.816 93.945 98.516 104.010 95.081 99.678 105.202 96.217 100.839 106.393 97.351 101.999107.583 98.484 103.158 108.771 99.617 104.316 109.958 100.749 105.473 111.144 101.879106.629 112.329 103.010 107.783 113.512 104.139 108.937 114.695 105.267 110.090 115.876 106.395 111.242 117.057 107.522 112.393118.236 108.648 113.544 119.414 10.828 13.816 16.266 18.467 20.515 22.458 24.322 26.125 27.877 29.588 31.264 32.910 34.528 36. 123 37.697 39.252 40.790 42.312 43.820 45.315 46.797 48.268 49.728 51. 179 52.620 54.052 55.476 56.892 58.301 59.703 61.098 62.487 63.870 65.247 66.619 67.985 69.347 70.703 72.055 73.402 74.745 76.084 77.419 78.750 80.077 81.400 82.720 84.037 85.351 86.661 87.968 89.272 90.573 91.872 93.168 94.461 95.751 97.039 98.324 99.607 100.888 102.166 103.442 104.716 105.988 107.258 108.526 109.791 111.055 112.317 113.577 114.835 116.092 117.346 118.599 119.850 121.100 122.348 123.594 124.839 126.083 127.324 128.565 129.804 131.041 132.277