MA2604 Exam
1. Let X1, . . . , Xn be independent identically distributed random variables with
probability density function
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f(x|θ) = θxθ−1, 0 ≤ x ≤ 1, 0 ≤ θ ≤ ∞
a. Find the maximum likelihood estimator or θ. [7 marks]
b. Find the method of moments estimator of θ. [3 marks]
2. A study at Uxbridge University is made to determine the percentage of stu-
dents at the university (which has many thousands of students) in favour of
instituting a new “upgrade” feature in lecture halls, such that students can pay
to reserve their preferred seats and receive a selection of beverages.
a. In a random sample of 400 students, 230 are in favour. Find a 95%
confidence interval for the percentage in favour in the whole population
of students, stating any assumption that you make. [6 marks]
b. Test at the 5% significance level whether more than 50% of the population
is in favour. Define the null and alternative hypotheses, the test statistic
used and its distribution under the null hypothesis. [6 marks]
c. Do your conclusions in (a) agree with the conclusions in (b)? [3 marks]
3. a. Define the p-value in statistical hypothesis testing. [2 marks]
b. A survey found that the average hotel room rate in London is £127 and
the average room rate in Edinburgh is £115. The data were obtained
from two samples of 50 hotels each and the sample standard deviations
were £23.62 and £19.83, respectively. At a 5% significant level, can it be
concluded that there is a significant difference in the rates? Clearly state
the null and alternative hypotheses, the test statistic and the distribution
of the test statistic, together with the assumptions that you make. [9 marks]
c. Compute the p-value of the test conducted in part (b) and discuss its
interpretation. [4 marks]
Page 1 of 8
4. A study was conducted at the Uxbridge University Leisure Centre concerning
how the amount of gas used to heat the buildings depends on the temperature
outside. The amount of gas used each day in thousands of litres (Y ) and the
average temperature outside the factory (X) was recorded; n measurements
(xi, yi) were made on random days during a period. The data is analysed in R
and the output provided.
> summary(lm(y~x))
lm(formula = y ~ x)
Residuals:
Min 1Q Median 3Q Max
-6.9340 -2.0017 0.1318 1.5643 6.8753
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 44.49860 1.26086 35.29 <2e-16 *** x -1.62458 0.07491 -21.69 <2e-16 *** Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 3.612 on 28 degrees of freedom Multiple R-squared: 0.9438,Adjusted R-squared: 0.9418 F-statistic: 470.4 on 1 and 28 DF, p-value: < 2.2e-16 A linear regression model is used to study the relationship of Y on x: Y = α + βx+ ϵ. a. Write down the assumptions about ϵ for a linear regression model. [2 marks] Page 2 of 8 b. Write down estimates for α and β and interpret these in the context of the study conducted. [2 marks] c. Without counting the points in the plot, how could we tell that n = 30 from this output? [1 mark] d. Calculate a 95% confidence interval for β. [3 marks] e. A residual plot for the model is shown below. What does this plot tell you about the fit of the linear model? How could the model be improved? Interpret in the context of these particular data. [2 marks] Page 3 of 8 STATISTICAL Cumulative normal distribution Critical values of the t distribution Critical values of the F distribution Critical values of the chi-squared distribution © C. Dougherty 2001, 2002 These tables have been computed to accompany the text C. to Econometrics (second edition 2002, Oxford University Press, Oxford), They may be reproduced freely provided that this attribution is retained. Page 4 of 8 STATISTICAL TABLES Cumulative Standardized Normal Distribution A(z) is the integral of the standardized normal distribution from ∞− to z (in other words, the area under the curve to the left of z). It gives the probability of a normal random variable not being more than z standard deviations above its mean. Values of z of particular importance: 1.645 0.9500 Lower limit of right 5% tail 1.960 0.9750 Lower limit of right 2.5% tail 2.326 0.9900 Lower limit of right 1% tail 2.576 0.9950 Lower limit of right 0.5% tail 3.090 0.9990 Lower limit of right 0.1% tail 3.291 0.9995 Lower limit of right 0.05% tail -4 -3 -2 -1 0 41 32 z z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 3.6 0.9998 0.9998 0.9999 Page 5 of 8 STATISTICAL TABLES t Distribution: Critical Values of t Significance level Degrees of Two-tailed test: 10% 5% 2% 1% 0.2% 0.1% freedom One-tailed test: 5% 2.5% 1% 0.5% 0.1% 0.05% 1 6.314 12.706 31.821 63.657 318.309 636.619 2 2.920 4.303 6.965 9.925 22.327 31.599 3 2.353 3.182 4.541 5.841 10.215 12.924 4 2.132 2.776 3.747 4.604 7.173 8.610 5 2.015 2.571 3.365 4.032 5.893 6.869 6 1.943 2.447 3.143 3.707 5.208 5.959 7 1.894 2.365 2.998 3.499 4.785 5.408 8 1.860 2.306 2.896 3.355 4.501 5.041 9 1.833 2.262 2.821 3.250 4.297 4.781 10 1.812 2.228 2.764 3.169 4.144 4.587 11 1.796 2.201 2.718 3.106 4.025 4.437 12 1.782 2.179 2.681 3.055 3.930 4.318 13 1.771 2.160 2.650 3.012 3.852 4.221 14 1.761 2.145 2.624 2.977 3.787 4.140 15 1.753 2.131 2.602 2.947 3.733 4.073 16 1.746 2.120 2.583 2.921 3.686 4.015 17 1.740 2.110 2.567 2.898 3.646 3.965 18 1.734 2.101 2.552 2.878 3.610 3.922 19 1.729 2.093 2.539 2.861 3.579 3.883 20 1.725 2.086 2.528 2.845 3.552 3.850 21 1.721 2.080 2.518 2.831 3.527 3.819 22 1.717 2.074 2.508 2.819 3.505 3.792 23 1.714 2.069 2.500 2.807 3.485 3.768 24 1.711 2.064 2.492 2.797 3.467 3.745 25 1.708 2.060 2.485 2.787 3.450 3.725 26 1.706 2.056 2.479 2.779 3.435 3.707 27 1.703 2.052 2.473 2.771 3.421 3.690 28 1.701 2.048 2.467 2.763 3.408 3.674 29 1.699 2.045 2.462 2.756 3.396 3.659 30 1.697 2.042 2.457 2.750 3.385 3.646 32 1.694 2.037 2.449 2.738 3.365 3.622 34 1.691 2.032 2.441 2.728 3.348 3.601 36 1.688 2.028 2.434 2.719 3.333 3.582 38 1.686 2.024 2.429 2.712 3.319 3.566 40 1.684 2.021 2.423 2.704 3.307 3.551 42 1.682 2.018 2.418 2.698 3.296 3.538 44 1.680 2.015 2.414 2.692 3.286 3.526 46 1.679 2.013 2.410 2.687 3.277 3.515 48 1.677 2.011 2.407 2.682 3.269 3.505 50 1.676 2.009 2.403 2.678 3.261 3.496 60 1.671 2.000 2.390 2.660 3.232 3.460 70 1.667 1.994 2.381 2.648 3.211 3.435 80 1.664 1.990 2.374 2.639 3.195 3.416 90 1.662 1.987 2.368 2.632 3.183 3.402 100 1.660 1.984 2.364 2.626 3.174 3.390 120 1.658 1.980 2.358 2.617 3.160 3.373 150 1.655 1.976 2.351 2.609 3.145 3.357 200 1.653 1.972 2.345 2.601 3.131 3.340 300 1.650 1.968 2.339 2.592 3.118 3.323 400 1.649 1.966 2.336 2.588 3.111 3.315 500 1.648 1.965 2.334 2.586 3.107 3.310 600 1.647 1.964 2.333 2.584 3.104 3.307 1.645 1.960 2.326 2.576 3.090 3.291 ∞ Page 6 of 8 STATISTICAL TABLES F Distribution: Critical Values of F (5% significance level) v1 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 1 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54 241.88 243.91 245.36 246.46 247.32 248.01 2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.42 19.43 19.44 19.45 3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.71 8.69 8.67 8.66 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.87 5.84 5.82 5.80 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.64 4.60 4.58 4.56 6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.00 3.96 3.92 3.90 3.87 7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.53 3.49 3.47 3.44 8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.24 3.20 3.17 3.15 9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.03 2.99 2.96 2.94 10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.91 2.86 2.83 2.80 2.77 11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.79 2.74 2.70 2.67 2.65 12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.69 2.64 2.60 2.57 2.54 13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 2.60 2.55 2.51 2.48 2.46 14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 2.53 2.48 2.44 2.41 2.39 15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.48 2.42 2.38 2.35 2.33 16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.42 2.37 2.33 2.30 2.28 17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 2.38 2.33 2.29 2.26 2.23 18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.34 2.29 2.25 2.22 2.19 19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.31 2.26 2.21 2.18 2.16 20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.28 2.22 2.18 2.15 2.12 21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.25 2.20 2.16 2.12 2.10 22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30 2.23 2.17 2.13 2.10 2.07 23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 2.20 2.15 2.11 2.08 2.05 24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 2.18 2.13 2.09 2.05 2.03 25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 2.16 2.11 2.07 2.04 2.01 26 4.22 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22 2.15 2.09 2.05 2.02 1.99 27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25 2.20 2.13 2.08 2.04 2.00 1.97 28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19 2.12 2.06 2.02 1.99 1.96 29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22 2.18 2.10 2.05 2.01 1.97 1.94 30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.09 2.04 1.99 1.96 1.93 35 4.12 3.27 2.87 2.64 2.49 2.37 2.29 2.22 2.16 2.11 2.04 1.99 1.94 1.91 1.88 40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.00 1.95 1.90 1.87 1.84 50 4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07 2.03 1.95 1.89 1.85 1.81 1.78 60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 1.92 1.86 1.82 1.78 1.75 70 3.98 3.13 2.74 2.50 2.35 2.23 2.14 2.07 2.02 1.97 1.89 1.84 1.79 1.75 1.72 80 3.96 3.11 2.72 2.49 2.33 2.21 2.13 2.06 2.00 1.95 1.88 1.82 1.77 1.73 1.70 90 3.95 3.10 2.71 2.47 2.32 2.20 2.11 2.04 1.99 1.94 1.86 1.80 1.76 1.72 1.69 100 3.94 3.09 2.70 2.46 2.31 2.19 2.10 2.03 1.97 1.93 1.85 1.79 1.75 1.71 1.68 120 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.96 1.91 1.83 1.78 1.73 1.69 1.66 150 3.90 3.06 2.66 2.43 2.27 2.16 2.07 2.00 1.94 1.89 1.82 1.76 1.71 1.67 1.64 200 3.89 3.04 2.65 2.42 2.26 2.14 2.06 1.98 1.93 1.88 1.80 1.74 1.69 1.66 1.62 250 3.88 3.03 2.64 2.41 2.25 2.13 2.05 1.98 1.92 1.87 1.79 1.73 1.68 1.65 1.61 300 3.87 3.03 2.63 2.40 2.24 2.13 2.04 1.97 1.91 1.86 1.78 1.72 1.68 1.64 1.61 400 3.86 3.02 2.63 2.39 2.24 2.12 2.03 1.96 1.90 1.85 1.78 1.72 1.67 1.63 1.60 500 3.86 3.01 2.62 2.39 2.23 2.12 2.03 1.96 1.90 1.85 1.77 1.71 1.66 1.62 1.59 600 3.86 3.01 2.62 2.39 2.23 2.11 2.02 1.95 1.90 1.85 1.77 1.71 1.66 1.62 1.59 750 3.85 3.01 2.62 2.38 2.23 2.11 2.02 1.95 1.89 1.84 1.77 1.70 1.66 1.62 1.58 1000 3.85 3.00 2.61 2.38 2.22 2.11 2.02 1.95 1.89 1.84 1.76 1.70 1.65 1.61 1.58 Page 7 of 8 STATISTICAL TABLES χ2 (Chi-Squared) Distribution: Critical Values of χ2 Significance level Degrees of 5% 1% 0.1% 1 3.841 6.635 10.828 2 5.991 9.210 13.816 3 7.815 11.345 16.266 4 9.488 13.277 18.467 5 11.070 15.086 20.515 6 12.592 16.812 22.458 7 14.067 18.475 24.322 8 15.507 20.090 26.124 9 16.919 21.666 27.877 10 18.307 23.209 29.588 Page 8 of 8 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com