In an online survey of a random sample of 248 residents of the City of Waterloo, 89 reported
that they felt the city has inadequate municipal parking facilities, compared to 147 of the 328
residents randomly sampled from the City of Kitchener.
a. In your own words and using the Waterloo and Kitchener case above as an example, write
one sentence to illustrate the trade-off between confidence and precision. [2 marks]
b. Draw a second sketch demonstrating how increasing the sample size (n) affects the shape
of the sampling distribution for one sample proportion (p-hat Waterloo). Label the axes but
note that the sketch doesn't have to be to scale. In your own words, explain why smaller
samples affect the sampling distribution in the way they do. [5 marks]
c. Why is it useful to use confidence intervals to communicate our uncertainty, rather than
just providing the estimate alone? [3 marks]
please show all steps
C-2 APPENDIX C Tables Table Z Second decimal place in z Areas under the standard Normal curve 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.0000T 0.0001 0.0001 0.0001 -3.9 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 -3.8 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 -3.7 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0002 -3.6 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 -3.5 0.0002 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 -3.4 0.0003 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0005 0.0005 0.0005 -3.3 0.0005 0.0005 0.0005 0.0006 0.0006 0.0006 0.0006 0.0006 0.0007 0.0007 -3.2 0.0007 0.0007 0.0008 0.0008 0.0008 0.0008 0.0009 0.0009 0.0009 0.0010 -3.1 0.0010 0.0010 0.0011 0.0011 0.0011 0.0012 0.0012 0.0013 0.0013 0.0013 -3.0 0.0014 0.0014 0.0015 0.0015 0.0016 0.0016 0.0017 0.0018 0.0018 0.0019 -2.9 0.0019 0.0020 0.0021 0.0021 0.0022 0.0023 0.0023 0.0024 0.0025 0.0026 -2.8 0.0026 0.0027 0.0028 0.0029 0.0030 0.0031 0.0032 0.0033 0.0034 0.0035 -2.7 0.0036 0.0037 0.0038 0.0039 0.0040 0.0041 0.0043 0.0044 0.0045 0.0047 -2.6 0.0048 0.0049 0.0051 0.0052 0.0054 0.0055 0.0057 0.0059 0.0060 0.0062 -2.5 0.0064 0.0066 0.0068 0.0069 0.0071 0.0073 0.0075 0.0078 0.0080 0.0082 -2.4 0.0084 0.0087 0.0089 0.0091 0.0094 0.0096 0.0099 0.0102 0.0104 0.0107 -2.3 0.0110 0.0113 0.0116 0.0119 0.0122 0.0125 0.0129 0.0132 0.0136 0.0139 -2.2 0.0143 0.0146 0.0150 0.0154 0.0158 0.0162 0.0166 0.0170 0.0174 0.0179 -2.1 0.0183 0.0188 0.0192 0.0197 0.0202 0.0207 0.0212 0.0217 0.0222 0.0228 -2.0 0.0233 0.0239 0.0244 0.0250 0.0256 0.0262 0.0268 0.0274 0.0281 0.0287 -1.9 0.0294 0.0301 0.0307 0.0314 0.0322 0.0329 0.0336 0.0344 0.0351 0.0359 1.8 0.0367 0.0375 0.0384 0.0392 0.0401 0.0409 0.0418 0.0427 0.0436 0.0446 -1.7 0.0455 0.0465 0.0475 0.0485 0.0495 0.0505 0.0516 0.0526 0.0537 0.0548 -1.6 0.0559 0.0571 0.0582 0.0594 0.0606 0.0618 0.0630 0.0643 0.0655 0.0668 -1.5 0.0681 0.0694 0.0708 0.0721 0.0735 0.0749 0.0764 0.0778 0.0793 0.0808 -1.4 0.0823 0.0838 0.0853 0.0869 0.0885 0.0901 0.0918 0.0934 0.0951 0.0968 -1.3 0.0985 0.1003 0.1020 0 0.1038 0.1056 0.1075 0.1093 0.1112 0.1131 0.1151 -1.2 0.1170 0.1190 0.1210 0.1230 0.1251 0.1271 0.1292 0.1314 0.1335 0.1357 -1.1 0.1379 0.1401 0.1423 0.1446 0.1469 0.1492 0.1515 0.1539 0.1562 0.1587 -1.0 0.1611 0.1635 0.1660 0.1685 0.1711 0.1736 0.1762 0.1788 0.1814 0.1841 -0.9 -0.8 0.1867 0.1894 0.1922 0.1949 0.1977 0.2005 0.2033 0.2061 0.2090 0.2119 0.2148 0.2177 0.2206 0.2236 0.2266 0.2296 0.2327 0.2358 0.2389 0.2420 -0.7 0.2451 0.2483 0.2514 0.2546 0.2578 0.2611 0.2643 0.2676 0.2709 0.2743 -0.6 -0.5 0.2776 0.2810 0.2843 0.2877 0.2912 0.2946 0.2981 0.3015 0.3050 0.3085 0.3121 0.3156 0.3192 0.3228 0.3264 0.3300 0.3336 0.3372 0.3409 0.3446 -0.4 -0.3 0.3483 0.3520 0.3557 0.3594 0.3632 0.3669 0.3707 0.3745 0.3783 0.3821 0.3859 0.3897 0.3936 0.3974 0.4013 0.4052 0.4090 0.4129 0.4168 0.4207 -0.2 0.4247 0.4286 0.4325 0.4364 0.4404 0.4443 0.4483 0.4522 0.4562 0.4602 -0.1 0.4641 0.4681 0.4721 0.4761 0.4801 0.4840 0.4880 0.4920 0.4960 0.5000 -0.0 For z S -3.90, the areas are 0.0000 to four decimal places.APPENDIX C Tables C-5 Right-tail probability 0.10 0.05 0.025 0.01 0.005 Table X if Values of Xa 2.706 3.841 5.024 6.635 7.879 4.605 5.991 7.378 9.210 10.597 W N 6.251 7.815 9.348 11.345 12.838 A 7.779 9.488 11.143 13.277 14.860 9.236 11.070 12.833 15.086 16.750 10.645 12.592 14.449 16.812 18.548 12.017 14.067 16.013 18.475 20.278 6 0 Va 13.362 15.507 17.535 20.090 21.955 0 14.684 16.919 19.023 21.666 23.589 10 15.987 18.307 20.483 23.209 25.188 11 17.275 19.675 21.920 24.725 26.757 12 18.549 21.026 23.337 26.217 28.300 13 19.812 22.362 24.736 27.688 29.819 14 21.064 23.685 26.119 29.141 31.319 15 22.307 24.996 27.488 30.578 32.801 16 23.542 26.296 28.845 32.000 34.267 17 24.769 27.587 30.191 33.409 35.718 18 25.989 28.869 31.526 34.805 37.156 19 27.204 30.143 32.852 36.191 38.582 20 28.412 31.410 34.170 37.566 39.997 21 29.615 32.671 35.479 38.932 41.401 22 30.813 33.924 36.781 40.290 42.796 23 32.007 35.172 38.076 41.638 44.181 24 33.196 36.415 39.364 42.980 45.559 25 34.382 37.653 40.647 44.314 46.928 26 35.563 38.885 41.923 45.642 48.290 27 36.741 40.113 43.195 46.963 49.645 28 37.916 41.337 44.461 48.278 50.994 29 39.087 42.557 45.722 59.588 52.336 30 40.256 43.773 46.979 50.892 53.672 40 51.805 55.759 59.342 63.691 66.767 50 63.167 67.505 71.420 76.154 79.490 60 74.397 79.082 83.298 88.381 91.955 70 85.527 90.531 95.023 100.424 104.213 80 96.578 101.879 106.628 112.328 90 116.320 107.565 113.145 118.135 124.115 100 128.296 118.499 124.343 129.563 135.811 140.177APPENDIX C Tables C-3 Table Z (cont.) Areas under the 0.00 0.01 Second decimal place in z standard Normal curve 0.0 0.5000 0.02 0.5040 0.03 0.5080 0.04 0.5120 0.05 0.1 0.5398 0.5160 0.06 0.5438 0.5199 0.07 0.08 0.09 0.2 0.5478 0.5793 0.5517 0.5239 0.5832 0.5557 0.5279 0.5319 0.5359 0.3 0.5871 0.5596 0.5636 0.6179 0.5910 0.5675 0.6217 0.5948 0.5987 0.5714 0.5753 0.6026 0 0.4 0.6255 0.6554 0.6293 0.6064 0.6103 0.6591 0.6331 0.6628 0.6368 0.6141 0.6664 0.6406 0.6443 0.6700 0.6480 0.6736 0.6517 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.6 0.7257 0.7019 0.7054 0.7291 0.7088 0.7324 0.7123 0. 7357 0.7157 0.7 0.7580 0.7389 0.7190 0.7224 0.7611 0.7422 0.7642 0.7454 0.7486 0.7673 0.7517 0.8 0.7881 0.7704 0.7549 0.7910 0.7734 0.7764 0.7939 0.7794 0.7967 0.7823 0.7852 0.9 0.8159 0.7995 0.8023 0.8186 0.8051 0.8078 0.8212 0.8238 0.8106 0.8133 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 1.1 0.8554 0.8577 0.8643 0.8599 0.8665 0.8621 0.8686 0.8708 0.8729 0.8749 1.2 0.8770 0.8790 0.8810 0.8849 0.8830 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 1.3 0.8997 0.9015 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 0.9793 0.9798 0.9808 0.9772 0.9778 0.9783 0.9788 0.9803 0.9812 0.9817 2.0 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.1 0.9821 0.9826 0.9830 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.2 0.9861 0.9864 0.9868 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.3 0.9893 0.9896 0.9898 0.9925 0.9927 0.9929 0.9931 0.9932 0.9922 0.9934 0.9936 2.4 0.9918 0.9920 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.5 0.9938 0.9940 0.9941 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 0.9956 2.6 0.9953 0.9955 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.7 0.9965 0.9966 0.9967 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 0.9976 2.8 0.9974 0.9975 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 29 0.9982 0.9982 0.9983 0.9981 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 0.9987 0,9987 0.9988 0.9992 0.9992 0.9992 0.9993 0.9987 0.9992 0.9993 3.0 0.9991 0.9991 0.9991 0.9994 0.9994 0.9994 0,9995 0.9995 0.9995 3.1 0.9990 0.9994 0.9994 0,9996 0.9996 0.9996 0.9997 3.2 0.9993 0.9993 0.9996 0.9996 0.9995 0.9995 0.9996 0,9997 0.9997 0.9997 0.9997 0.9997 0.9998 3.3 0.9995 0.9997 0.9997 0.9997 0.9997 3.4 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9999 0.9999 0.9999 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 3.5 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.6 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.7 0.9999 0.9999 3.8 0.9999 0.9999 0,9999 3.9 1.0000* "For z 2 3.90, the areas are 1.0000 to four decimal places.C-4 APPENDIX C Tables Two-tail probability 0.20 0.10 0.05 0.02 0.01 One-tail probability 0.10 0.05 0.025 0.01 0.005 Table T df Value of to 3.078 6.314 12.706 31.821 63.657 AWNNE 1.886 2.920 4.303 6.965 9.925 1.638 2.353 3.182 WN 4.541 5.841 1.533 2.132 2.776 3.747 4.604 1.476 2.015 2.571 3.365 4.032 1.440 1.943 2.447 3.143 3.707 -tal2 0 tal2 1.415 1.895 2.365 2.998 3.499 Two tails 1.397 1.860 2.306 2.896 3.355 1.383 1.833 2.262 2.821 3.250 10 1.372 1.812 2.228 2.764 3.169 10 11 1.363 1.796 2.201 2.718 3.106 11 12 1.356 1.782 2.179 2.681 3.055 12 0 13 1.350 1.771 2.160 2.650 3.012 13 14 One tail 1.345 1.761 2.145 2.624 2.977 14 Are 15 1.341 1.753 2.131 2.602 2.947 15 16 1.337 1.746 2.120 2.583 2.921 16 17 1.333 1.740 2.110 2.567 2.898 17 18 1.330 1.734 2.101 2.552 2.878 18 19 1.328 1.729 2.093 2.539 2.861 19 20 1.325 1.725 2.086 2.528 2.845 20 21 1.323 1.721 2.080 2.518 2.831 21 22 1.321 1.717 2.074 2.508 2.819 22 23 1.319 1.714 2.069 2.500 2.807 23 24 1.318 1.711 2.064 2.492 2.797 24 25 1.316 1.708 2.060 2.485 2.787 25 26 1.315 1.706 2.056 2.479 2.779 26 27 1.314 1.703 2.052 2.473 2.771 27 28 1.313 1.70 2.048 2.467 2.763 28 29 1.311 1.699 2.045 2.462 2.756 29 30 1.310 1.697 2.042 2.457 2.750 30 32 1.309 1.694 2.03 2.449 2.738 32 35 1.306 1.690 2.030 2.438 2.725 35 40 1.303 1.684 2.02 2.423 2.704 40 45 1.301 1.679 2.014 2.412 2.690 45 50 1.299 1.676 2.009 2.403 2.678 50 60 1.296 1.671 2.000 2.390 2.660 60 75 1.293 1.665 1.992 2.377 2.643 75 100 1.290 1.660 1.984 2.364 100 120 2.626 1.289 1.658 1.980 2.358 2.617 120 140 1.288 1.656 1.977 2.353 180 2.611 140 1.286 1.653 1.973 2.347 2.603 180 250 1.285 1.651 1.969 2.341 2.596 250 400 1.284 .649 1.966 2.336 2.588 400 1000 1.282 1.646 1.962 2.330 2.581 1000 1.282 1.645 1.960 2.326 2.576 Confidence levels 80% 90% 95% 98 99%CONFIDENCE INTERVAL ON THE DIFFERENCE IN PROPORTIONS (p1 p2) Population 1 ( Waterloo ) Population 2 ( Kitchener ) m = 248 m = 328 (Sample size) 931 = 89 332 = 147 (Number of successes) Sample 1 proportion A a: P1 = \"1 A _ 4 p1 _ 248 p1 = 0.3589 Sample 2 proportion A $2 P2 = n \"'2 _ 328 p3 = 0.4482 Estimator of the parameter p A2501 + 932 + n , 819 + 1247 p: 243 + 328 p _ 576 :0: 0.4097 Verify the requirements of the confidence interval Calculus of n1 P1, n1 (1 -PI), n2 p2 and n2 (1 -p2). n1 P1 = 248 . 0.3589 n1 (1 - p1) = 248 . (1 - 0.3589) n1 P1 = 89.0072 n1 (1 - p1) = 248 . 0.6411 n1 (1 -p1) = 158.9928 n2 P2 = 328 . 0.4482 n2 (1 -p2) = 328 . (1 - 0.4482) n2 P2 = 147.0096 n2 (1 -p2) = 328 . 0.5518 n2 (1 -p2) = 180.9904 Since n1 p1 > 5, ni (1 -p1) > 5, n2 p2 > 5 and n2 (1 -p2) > 5, then the requirements of the confidence interval are verified. The conditions are satisfied. This is important because it allows us to say that p is approximately normal.Standard error of the difference in sample proportions pi . (1 - p1) p2 . (1 -p2) Sp1 -P2 = n1 n2 0.3589 . (1 - 0.3589) 0.4482 . (1 - 0.4482 Sp1 -P2 = + 248 328 0.3589 . 0.6411 0.4482 . 0.5518 Sp1 -P2 = 1 + 248 328 0.23009079 0.24731676 Spi -P2 = + 248 328 Sp1-p2 = V0.000927785 + 0.000754015 Spi-p2 = V0.0016818 Spi-p2 = 0.041We have a bell shaped distribution with mean 0.4097 and standard error 0.041. Difference -0.255 -0.205 -0.155 -0.105 -0.055 -0.005 0.045 191.192 Confidence interval on the difference in proportions If pi and p2 are the sample proportions of observation in two independent random samples of sizes n, and n2 that belong to a class of interest, an approximate two-sided 100 (1 - a) % confidence interval on the difference in the true proportions p1 - P2 is: P1 - P2 - Za/2 . 1 pi (1 - P1) P2(1 -P2)
