University of Wollongong School of Mathematics and Applied Statistics STAT101 Introduction to Statistics Assignment 3 Due: 5pm Friday, 6th November, 2020 (end of Week 12) This Assignment is worth 5% of the final mark in STAT101 and must be carried out individually. Lodge your assignment as a single PDF document (with each page num- bered) in the drop box for Assignment 3 on the Moodle eLearning site for this subject. It must include your name and student ID number at the top of the page. Except in extenuating circumstances (e.g. illness or bereavement), all assignments will have 10% of the maximum mark deducted for each day (or part thereof) that they are late. Submissions received more than 7 days after the due date will receive no marks. All working must be shown. Question 1 [15 marks] Let x1, . . . , xn be a random sample of size n from a continuous distribution with pdf: fX(x, θ) = θx −θ−1, x ≥ 1, θ > 1 (a) (i) Determine a formula for the cumulative density function, F (x). (ii) Use the cdf from part (i) to find P (2 ≤ X < 4) when θ = 2. Leave your answer correct to 4 decimal places. (b) (i) Determine a formula for the mean, E(X). (ii) Hence, find the Method of Moments estimator of θ. (c) Determine the likelihood function L(θ;x1, . . . , xn) and hence find the Maximum Likelihood Estimator of θ. Question 2 [5 marks] Let y1, . . . , y25 be a simple random sample of size n = 25 from a normally-distributed population with unknown mean and variance. The sample statistics ∑25 i=1 yi = 64.528 and ∑25 i=1(yi − y¯)2 = 151.947. (a) Using the probability tables provided at the end of this document, construct a 90% confidence interval for the population mean. Round your answer to 3 decimal places. (b) Using the probability tables, construct a 98% confidence interval for the population standard deviation. Round your answer to 3 decimal places. Total Marks = 20 Percentage points of the t distribution 0 t α α df α 0.10 0.05 0.025 0.01 0.005 1 3.078 6.314 12.706 31.821 63.657 2 1.886 2.920 4.303 6.965 9.925 3 1.638 2.353 3.182 4.541 5.841 4 1.533 2.132 2.776 3.747 4.604 5 1.476 2.015 2.571 3.365 4.032 6 1.440 1.943 2.447 3.143 3.707 7 1.415 1.895 2.365 2.998 3.499 8 1.397 1.860 2.306 2.896 3.355 9 1.383 1.833 2.262 2.821 3.250 10 1.372 1.812 2.228 2.764 3.169 11 1.363 1.796 2.201 2.718 3.106 12 1.356 1.782 2.179 2.681 3.055 13 1.350 1.771 2.160 2.650 3.012 14 1.345 1.761 2.145 2.624 2.977 15 1.341 1.753 2.131 2.602 2.947 16 1.337 1.746 2.120 2.583 2.921 17 1.333 1.740 2.110 2.567 2.898 18 1.330 1.734 2.101 2.552 2.878 19 1.328 1.729 2.093 2.539 2.861 20 1.325 1.725 2.086 2.528 2.845 21 1.323 1.721 2.080 2.518 2.831 22 1.321 1.717 2.074 2.508 2.819 23 1.319 1.714 2.069 2.500 2.807 24 1.318 1.711 2.064 2.492 2.797 25 1.316 1.708 2.060 2.485 2.787 26 1.315 1.706 2.056 2.479 2.779 27 1.314 1.703 2.052 2.473 2.771 28 1.313 1.701 2.048 2.467 2.763 29 1.311 1.699 2.045 2.462 2.756 30 1.310 1.697 2.042 2.457 2.750 40 1.303 1.684 2.021 2.423 2.704 50 1.299 1.676 2.009 2.403 2.678 60 1.296 1.671 2.000 2.390 2.660 70 1.294 1.667 1.994 2.381 2.648 80 1.292 1.664 1.990 2.374 2.639 90 1.291 1.662 1.987 2.368 2.632 100 1.290 1.660 1.984 2.364 2.626 ∞ 1.282 1.645 1.960 2.326 2.576 Percentage Points of the χ2 distribution α 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0.005 1-α 0.005 0.01 0.025 0.05 0.10 0.90 0.95 0.975 0.99 0.995 df 1 0.000 0.000 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879 2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597 3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838 4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860 5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.833 15.086 16.750 6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548 7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278 8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955 9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589 10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188 11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757 12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300 13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819 14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319 15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801 16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267 17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718 18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156 19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582 20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997 21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401 22 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.796 23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181 24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559 25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928 26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290 27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645 28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993 29 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.336 30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672 31 14.458 15.655 17.539 19.281 21.434 41.422 44.985 48.232 52.191 55.003 32 15.134 16.362 18.291 20.072 22.271 42.585 46.194 49.480 53.486 56.328 33 15.815 17.074 19.047 20.867 23.110 43.745 47.400 50.725 54.776 57.648 34 16.501 17.789 19.806 21.664 23.952 44.903 48.602 51.966 56.061 58.964 35 17.192 18.509 20.569 22.465 24.797 46.059 49.802 53.203 57.342 60.275 36 17.887 19.233 21.336 23.269 25.643 47.212 50.998 54.437 58.619 61.581 37 18.586 19.960 22.106 24.075 26.492 48.363 52.192 55.668 59.893 62.883 38 19.289 20.691 22.878 24.884 27.343 49.513 53.384 56.896 61.162 64.181 39 19.996 21.426 23.654 25.695 28.196 50.660 54.572 58.120 62.428 65.476 40 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766
欢迎咨询51作业君