Data605HW12
Mathew Katz
2023-04-20
df = read.csv("/Users/mathew.katz/Desktop/CUNYSPS/who.csv")
df## Country LifeExp InfantSurvival
## 1 Afghanistan 42 0.835
## 2 Albania 71 0.985
## 3 Algeria 71 0.967
## 4 Andorra 82 0.997
## 5 Angola 41 0.846
## 6 Antigua and Barbuda 73 0.990
## 7 Argentina 75 0.986
## 8 Armenia 69 0.979
## 9 Australia 82 0.995
## 10 Austria 80 0.996
## 11 Azerbaijan 64 0.927
## 12 Bahamas 74 0.987
## 13 Bahrain 75 0.991
## 14 Bangladesh 63 0.948
## 15 Barbados 75 0.989
## 16 Belarus 69 0.994
## 17 Belgium 79 0.996
## 18 Belize 69 0.986
## 19 Benin 55 0.912
## 20 Bhutan 64 0.937
## 21 Bolivia 66 0.950
## 22 Bosnia and Herzegovina 75 0.987
## 23 Botswana 52 0.910
## 24 Brazil 72 0.981
## 25 Brunei Darussalam 77 0.992
## 26 Bulgaria 73 0.990
## 27 Burkina Faso 47 0.878
## 28 Burundi 49 0.891
## 29 Cambodia 62 0.935
## 30 Cameroon 51 0.913
## 31 Canada 81 0.995
## 32 Cape Verde 70 0.975
## 33 Central African Republic 48 0.886
## 34 Chad 46 0.876
## 35 Chile 78 0.992
## 36 China 73 0.980
## 37 Colombia 74 0.983
## 38 Comoros 65 0.949
## 39 Congo 54 0.921
## 40 Cook Islands 73 0.984
## 41 Costa Rica 78 0.989
## 42 C\xf4te d'Ivoire 53 0.910
## 43 Croatia 76 0.995
## 44 Cuba 78 0.995
## 45 Cyprus 80 0.997
## 46 Czech Republic 77 0.997
## 47 Democratic Republic of the Congo 47 0.871
## 48 Denmark 79 0.997
## 49 Djibouti 56 0.914
## 50 Dominica 74 0.987
## 51 Dominican Republic 70 0.975
## 52 Ecuador 73 0.979
## 53 Egypt 68 0.971
## 54 El Salvador 71 0.978
## 55 Equatorial Guinea 46 0.876
## 56 Eritrea 63 0.952
## 57 Estonia 73 0.995
## 58 Ethiopia 56 0.923
## 59 Fiji 69 0.984
## 60 Finland 79 0.997
## 61 France 81 0.996
## 62 Gabon 58 0.940
## 63 Gambia 59 0.916
## 64 Georgia 70 0.972
## 65 Germany 80 0.996
## 66 Ghana 57 0.924
## 67 Greece 80 0.996
## 68 Grenada 68 0.983
## 69 Guatemala 68 0.969
## 70 Guinea 53 0.902
## 71 Guinea-Bissau 48 0.881
## 72 Guyana 64 0.954
## 73 Haiti 61 0.940
## 74 Honduras 70 0.977
## 75 Hungary 73 0.994
## 76 Iceland 81 0.998
## 77 India 63 0.943
## 78 Indonesia 68 0.974
## 79 Iran (Islamic Republic of) 71 0.970
## 80 Iraq 56 0.963
## 81 Ireland 80 0.996
## 82 Israel 81 0.996
## 83 Italy 81 0.997
## 84 Jamaica 72 0.974
## 85 Japan 83 0.997
## 86 Jordan 71 0.979
## 87 Kazakhstan 64 0.974
## 88 Kenya 53 0.921
## 89 Kiribati 65 0.953
## 90 Kuwait 78 0.991
## 91 Kyrgyzstan 66 0.964
## 92 Lao People's Democratic Republic 60 0.941
## 93 Latvia 71 0.992
## 94 Lebanon 70 0.973
## 95 Lesotho 42 0.898
## 96 Liberia 44 0.843
## 97 Libyan Arab Jamahiriya 72 0.983
## 98 Lithuania 71 0.993
## 99 Luxembourg 80 0.997
## 100 Madagascar 59 0.928
## 101 Malawi 50 0.924
## 102 Malaysia 72 0.990
## 103 Maldives 72 0.974
## 104 Mali 46 0.881
## 105 Malta 79 0.995
## 106 Marshall Islands 63 0.950
## 107 Mauritania 58 0.922
## 108 Mauritius 73 0.988
## 109 Mexico 74 0.971
## 110 Micronesia (Federated States of) 69 0.967
## 111 Monaco 82 0.997
## 112 Mongolia 66 0.965
## 113 Montenegro 74 0.991
## 114 Morocco 72 0.966
## 115 Mozambique 50 0.904
## 116 Namibia 61 0.955
## 117 Nauru 61 0.975
## 118 Nepal 62 0.954
## 119 Netherlands 80 0.996
## 120 New Zealand 80 0.995
## 121 Nicaragua 71 0.971
## 122 Niger 42 0.852
## 123 Nigeria 48 0.901
## 124 Niue 70 0.966
## 125 Norway 80 0.997
## 126 Oman 74 0.990
## 127 Pakistan 63 0.922
## 128 Palau 69 0.990
## 129 Panama 76 0.982
## 130 Papua New Guinea 62 0.946
## 131 Paraguay 75 0.981
## 132 Peru 73 0.979
## 133 Philippines 68 0.976
## 134 Poland 75 0.994
## 135 Portugal 79 0.997
## 136 Qatar 77 0.991
## 137 Republic of Korea 79 0.995
## 138 Republic of Moldova 68 0.984
## 139 Romania 73 0.986
## 140 Russian Federation 66 0.990
## 141 Rwanda 52 0.903
## 142 Saint Kitts and Nevis 71 0.983
## 143 Saint Lucia 75 0.988
## 144 Saint Vincent and the Grenadines 70 0.983
## 145 Samoa 68 0.977
## 146 San Marino 82 0.997
## 147 Sao Tome and Principe 61 0.937
## 148 Saudi Arabia 70 0.979
## 149 Senegal 59 0.940
## 150 Serbia 73 0.993
## 151 Seychelles 72 0.988
## 152 Sierra Leone 40 0.841
## 153 Singapore 80 0.997
## 154 Slovakia 74 0.993
## 155 Slovenia 78 0.997
## 156 Solomon Islands 67 0.945
## 157 South Africa 51 0.944
## 158 Spain 81 0.996
## 159 Sri Lanka 72 0.989
## 160 Sudan 60 0.938
## 161 Suriname 68 0.971
## 162 Swaziland 42 0.888
## 163 Sweden 81 0.997
## 164 Switzerland 82 0.996
## 165 Syrian Arab Republic 72 0.988
## 166 Tajikistan 64 0.944
## 167 Thailand 72 0.993
## 168 The former Yugoslav Republic of Macedonia 73 0.985
## 169 Timor-Leste 66 0.953
## 170 Togo 57 0.931
## 171 Tonga 71 0.980
## 172 Trinidad and Tobago 69 0.967
## 173 Tunisia 72 0.981
## 174 Turkey 73 0.976
## 175 Turkmenistan 63 0.955
## 176 Tuvalu 65 0.969
## 177 Uganda 50 0.922
## 178 Ukraine 67 0.980
## 179 United Arab Emirates 78 0.992
## 180 United Kingdom 79 0.995
## 181 United Republic of Tanzania 50 0.926
## 182 United States of America 78 0.993
## 183 Uruguay 75 0.987
## 184 Uzbekistan 68 0.962
## 185 Vanuatu 69 0.970
## 186 Venezuela (Bolivarian Republic of) 74 0.982
## 187 Viet Nam 72 0.985
## 188 Yemen 61 0.925
## 189 Zambia 43 0.898
## 190 Zimbabwe 43 0.945
## Under5Survival TBFree PropMD PropRN PersExp GovtExp TotExp
## 1 0.743 0.99769 0.000228841 0.000572294 20 92 112
## 2 0.983 0.99974 0.001143127 0.004614439 169 3128 3297
## 3 0.962 0.99944 0.001060478 0.002091362 108 5184 5292
## 4 0.996 0.99983 0.003297297 0.003500000 2589 169725 172314
## 5 0.740 0.99656 0.000070400 0.001146162 36 1620 1656
## 6 0.989 0.99991 0.000142857 0.002773810 503 12543 13046
## 7 0.983 0.99952 0.002780191 0.000741044 484 19170 19654
## 8 0.976 0.99920 0.003698671 0.004918937 88 1856 1944
## 9 0.994 0.99993 0.002331953 0.009149391 3181 187616 190797
## 10 0.996 0.99990 0.003610904 0.006458749 3788 189354 193142
## 11 0.911 0.99913 0.003660005 0.008477873 62 780 842
## 12 0.986 0.99960 0.000954128 0.004045872 1224 55783 57007
## 13 0.990 0.99955 0.002679296 0.005967524 710 45784 46494
## 14 0.931 0.99609 0.000274894 0.000253034 12 75 87
## 15 0.988 0.99989 0.001098976 0.003372014 725 24433 25158
## 16 0.992 0.99929 0.004758674 0.012457093 204 11315 11519
## 17 0.995 0.99989 0.004230489 0.014079195 3451 239105 242556
## 18 0.984 0.99944 0.000890071 0.001074468 198 5376 5574
## 19 0.852 0.99865 0.000035500 0.000660845 28 600 628
## 20 0.930 0.99904 0.000080100 0.001124807 52 407 459
## 21 0.939 0.99734 0.001104233 0.001934039 71 2860 2931
## 22 0.985 0.99943 0.001411105 0.004669384 243 6578 6821
## 23 0.876 0.99546 0.000384822 0.002558127 431 19604 20035
## 24 0.980 0.99945 0.001046640 0.003481410 371 13940 14311
## 25 0.991 0.99901 0.001047120 0.005549738 519 30562 31081
## 26 0.988 0.99959 0.000253477 0.004553230 272 11550 11822
## 27 0.796 0.99524 0.000049300 0.000456647 27 304 331
## 28 0.819 0.99286 0.000024500 0.000164933 3 10 13
## 29 0.918 0.99335 0.000144185 0.000783616 29 140 169
## 30 0.851 0.99763 0.000171884 0.001432847 49 784 833
## 31 0.994 0.99996 0.001912607 0.010044633 3430 192800 196230
## 32 0.966 0.99676 0.000445087 0.000789981 114 5394 5508
## 33 0.826 0.99472 0.000077600 0.000378195 13 190 203
## 34 0.791 0.99430 0.000033000 0.000238728 22 234 256
## 35 0.991 0.99984 0.001047677 0.000607349 397 17952 18349
## 36 0.976 0.99799 0.001402082 0.000979500 81 1302 1383
## 37 0.979 0.99941 0.001289806 0.000525484 201 12410 12611
## 38 0.932 0.99914 0.000140587 0.000718826 14 304 318
## 39 0.874 0.99434 0.000204934 0.000995392 31 915 946
## 40 0.981 0.99976 0.001428571 0.005714286 466 27264 27730
## 41 0.988 0.99983 0.001182996 0.000830416 327 15376 15703
## 42 0.873 0.99253 0.000110024 0.000538226 34 315 349
## 43 0.994 0.99936 0.002469271 0.005459175 651 30210 30861
## 44 0.993 0.99990 0.005908139 0.007444750 310 21075 21385
## 45 0.996 0.99994 0.033228132 0.003972813 1350 39399 40749
## 46 0.996 0.99990 0.003591618 0.008942978 868 56137 57005
## 47 0.795 0.99355 0.000096100 0.000474721 5 66 71
## 48 0.996 0.99993 0.003551934 0.009958195 4350 314588 318938
## 49 0.870 0.98700 0.000170940 0.000361416 61 4002 4063
## 50 0.985 0.99984 0.000558824 0.004661765 288 13206 13494
## 51 0.971 0.99882 0.001629745 0.001596672 197 4148 4345
## 52 0.976 0.99805 0.001388805 0.001559309 147 3717 3864
## 53 0.965 0.99969 0.002425640 0.003367945 78 1290 1368
## 54 0.975 0.99936 0.001173913 0.000754658 177 5700 5877
## 55 0.794 0.99596 0.000308468 0.000546371 211 6474 6685
## 56 0.926 0.99782 0.000045800 0.000533887 8 80 88
## 57 0.994 0.99960 0.003294030 0.006900746 516 27393 27909
## 58 0.877 0.99359 0.000023900 0.000191851 6 64 70
## 59 0.982 0.99970 0.000456182 0.001992797 148 5355 5503
## 60 0.997 0.99996 0.003299183 0.008920357 2824 133956 136780
## 61 0.995 0.99989 0.003379700 0.007924442 3819 234850 238669
## 62 0.909 0.99572 0.000301297 0.005170099 276 17220 17496
## 63 0.886 0.99577 0.000093800 0.001131088 15 550 565
## 64 0.968 0.99916 0.004646289 0.004031356 123 1248 1371
## 65 0.995 0.99995 0.003441718 0.008010552 3628 209250 212878
## 66 0.880 0.99621 0.000140821 0.000856528 30 490 520
## 67 0.996 0.99984 0.004994696 0.003596152 2580 65195 67775
## 68 0.980 0.99992 0.000754717 0.003075472 342 6944 7286
## 69 0.959 0.99897 0.000764832 0.003452759 132 2400 2532
## 70 0.839 0.99534 0.000107505 0.000480122 21 66 87
## 71 0.800 0.99687 0.000114216 0.000651276 10 90 100
## 72 0.938 0.99785 0.000495264 0.002351827 60 1400 1460
## 73 0.920 0.99598 0.000206331 0.000088300 28 546 574
## 74 0.973 0.99905 0.000527479 0.001223705 91 2162 2253
## 75 0.993 0.99979 0.003039869 0.009163949 855 40602 41457
## 76 0.997 0.99997 0.003758389 0.009932886 5154 395622 400776
## 77 0.924 0.99701 0.000560733 0.001191281 36 203 239
## 78 0.966 0.99747 0.000128893 0.000786314 26 588 614
## 79 0.965 0.99972 0.000880461 0.001581144 212 7973 8185
## 80 0.953 0.99922 0.000666877 0.001333088 59 2948 3007
## 81 0.996 0.99989 0.002936271 0.019403222 3993 193553 197546
## 82 0.995 0.99994 0.003691336 0.006256828 1533 93748 95281
## 83 0.996 0.99994 0.003657769 0.007137294 2692 140148 142840
## 84 0.968 0.99992 0.000834754 0.001620600 170 4399 4569
## 85 0.996 0.99971 0.002113049 0.009461544 2936 159192 162128
## 86 0.975 0.99994 0.002349450 0.003218537 241 9047 9288
## 87 0.971 0.99858 0.003755648 0.007385268 148 5510 5658
## 88 0.879 0.99666 0.000123273 0.001015320 24 231 255
## 89 0.936 0.99598 0.000212766 0.002765957 118 4578 4696
## 90 0.989 0.99975 0.001741634 0.003576826 687 51940 52627
## 91 0.959 0.99863 0.002416809 0.005861190 28 396 424
## 92 0.925 0.99708 0.000347283 0.000972391 18 84 102
## 93 0.991 0.99940 0.003145478 0.005609436 443 18224 18667
## 94 0.969 0.99988 0.002081381 0.001163995 460 17400 17860
## 95 0.868 0.99487 0.000044600 0.000562907 41 437 478
## 96 0.765 0.99422 0.000028800 0.000289187 10 413 423
## 97 0.982 0.99982 0.001170724 0.004497433 223 13175 13398
## 98 0.991 0.99939 0.003964202 0.007670188 448 19932 20380
## 99 0.996 0.99990 0.002722343 0.009583514 6330 476420 482750
## 100 0.885 0.99585 0.000271465 0.000295475 9 162 171
## 101 0.880 0.99678 0.000019600 0.000535259 19 252 271
## 102 0.988 0.99875 0.000651758 0.001661178 222 6732 6954
## 103 0.970 0.99946 0.001006667 0.002953333 316 8100 8416
## 104 0.783 0.99422 0.000088000 0.000696691 28 434 462
## 105 0.994 0.99995 0.003861728 0.005953086 1235 91776 93011
## 106 0.944 0.99759 0.000413793 0.002620690 294 18876 19170
## 107 0.875 0.99394 0.000102825 0.000621879 17 451 468
## 108 0.985 0.99960 0.001040735 0.003677316 218 4704 4922
## 109 0.965 0.99975 0.001859629 0.000841810 474 16340 16814
## 110 0.959 0.99891 0.000540541 0.002252252 290 5830 6120
## 111 0.996 0.99998 0.005636364 0.014060606 6128 458700 464828
## 112 0.958 0.99809 0.002584261 0.003388100 35 1539 1574
## 113 0.990 0.99951 0.002051581 0.005717138 299 13725 14024
## 114 0.963 0.99921 0.000518296 0.000788513 89 1947 2036
## 115 0.862 0.99376 0.000024500 0.000294836 14 315 329
## 116 0.939 0.99342 0.000292135 0.003001954 165 3888 4053
## 117 0.970 0.99866 0.001000000 0.006300000 567 30200 30767
## 118 0.941 0.99756 0.000194783 0.000427807 16 64 80
## 119 0.995 0.99994 0.003694914 0.014602357 3560 187191 190751
## 120 0.994 0.99991 0.001978261 0.008342512 2403 159960 162363
## 121 0.964 0.99926 0.000369667 0.001059653 75 2183 2258
## 122 0.747 0.99686 0.000021500 0.000205139 9 85 94
## 123 0.809 0.99385 0.000241314 0.001453192 27 392 419
## 124 0.958 0.99915 0.002000000 0.011000000 1082 35211 36293
## 125 0.996 0.99996 0.003753052 0.016133219 5910 380380 386290
## 126 0.989 0.99986 0.001684996 0.003737628 312 18886 19198
## 127 0.903 0.99737 0.000785061 0.000439274 15 105 120
## 128 0.989 0.99949 0.001500000 0.006050000 690 43890 44580
## 129 0.977 0.99957 0.001347628 0.002481144 351 17424 17775
## 130 0.927 0.99487 0.000044300 0.000458078 34 390 424
## 131 0.978 0.99900 0.001056350 0.001705618 92 2006 2098
## 132 0.975 0.99813 0.001080104 0.000620102 125 4453 4578
## 133 0.968 0.99568 0.001047598 0.005574863 37 882 919
## 134 0.993 0.99973 0.001993865 0.005233928 495 21266 21761
## 135 0.996 0.99976 0.003416013 0.004629833 1800 75458 77258
## 136 0.989 0.99927 0.002618758 0.005943971 2186 163680 165866
## 137 0.995 0.99877 0.001561811 0.001915942 973 41715 42688
## 138 0.981 0.99846 0.002909731 0.006790764 58 1504 1562
## 139 0.984 0.99860 0.001925274 0.004212242 250 9504 9754
## 140 0.987 0.99875 0.004288359 0.008478449 277 12483 12760
## 141 0.840 0.99438 0.000045600 0.000385355 19 220 239
## 142 0.981 0.99983 0.000920000 0.003960000 478 9933 10411
## 143 0.986 0.99978 0.004595092 0.002030675 323 5068 5391
## 144 0.980 0.99953 0.000741667 0.003725000 218 6302 6520
## 145 0.972 0.99975 0.000270270 0.001675676 113 2093 2206
## 146 0.997 0.99995 0.035129032 0.070838710 3490 278163 281653
## 147 0.904 0.99748 0.000522581 0.001987097 49 2419 2468
## 148 0.974 0.99938 0.001417208 0.003065729 448 27621 28069
## 149 0.884 0.99496 0.000049200 0.000272283 38 504 542
## 150 0.992 0.99959 0.001987717 0.004287280 212 7956 8168
## 151 0.987 0.99944 0.001406977 0.007372093 557 20502 21059
## 152 0.731 0.99023 0.000029300 0.000437054 8 164 172
## 153 0.997 0.99975 0.001455956 0.004356458 944 30100 31044
## 154 0.992 0.99982 0.003130661 0.006636414 626 26096 26722
## 155 0.996 0.99985 0.002360320 0.007851574 1495 55233 56728
## 156 0.928 0.99806 0.000123967 0.001349174 28 442 470
## 157 0.931 0.99002 0.000721366 0.003820451 437 10920 11357
## 158 0.996 0.99976 0.003082917 0.007494042 2152 118426 120578
## 159 0.987 0.99920 0.000545582 0.001730255 51 360 411
## 160 0.911 0.99581 0.000293924 0.000884557 29 462 491
## 161 0.961 0.99905 0.000419780 0.001512088 209 7326 7535
## 162 0.836 0.98916 0.000150794 0.006021164 146 2256 2402
## 163 0.996 0.99995 0.003215466 0.010685724 3727 255696 259423
## 164 0.995 0.99995 0.003864789 0.010617438 5694 258248 263942
## 165 0.987 0.99960 0.000532873 0.001406018 61 1581 1642
## 166 0.932 0.99702 0.001998042 0.004994729 18 100 118
## 167 0.992 0.99803 0.000353619 0.002718571 98 2079 2177
## 168 0.983 0.99967 0.002547642 0.004338409 224 11060 11284
## 169 0.945 0.99211 0.000070900 0.001611311 45 1053 1098
## 170 0.893 0.99213 0.000035100 0.000302184 18 205 223
## 171 0.976 0.99966 0.000300000 0.003500000 104 1896 2000
## 172 0.962 0.99990 0.000756024 0.002750753 513 3575 4088
## 173 0.977 0.99972 0.001304944 0.002793637 158 4620 4778
## 174 0.974 0.99968 0.001569411 0.002944793 383 18632 19015
## 175 0.949 0.99922 0.002492345 0.004700143 156 4888 5044
## 176 0.962 0.99496 0.001000000 0.005000000 212 8786 8998
## 177 0.866 0.99439 0.000073900 0.000634436 22 78 100
## 178 0.976 0.99886 0.003087140 0.008343407 128 4624 4752
## 179 0.992 0.99976 0.001167608 0.002434087 833 45969 46802
## 180 0.994 0.99988 0.002208504 0.012241060 3064 240120 243184
## 181 0.882 0.99541 0.000020800 0.000336856 17 225 242
## 182 0.992 0.99997 0.002413151 0.008815197 6350 231822 238172
## 183 0.985 0.99969 0.003717802 0.000864605 404 15824 16228
## 184 0.956 0.99855 0.002615322 0.010754309 26 444 470
## 185 0.964 0.99935 0.000135747 0.001628959 67 1056 1123
## 186 0.979 0.99948 0.001765290 0.001029752 247 10528 10775
## 187 0.983 0.99775 0.000521541 0.000717003 37 270 307
## 188 0.900 0.99868 0.000310096 0.000632523 39 448 487
## 189 0.818 0.99432 0.000108071 0.001881840 36 595 631
## 190 0.915 0.99403 0.000157696 0.000707363 21 324 345- Provide a scatterplot of LifeExp~TotExp, and run simple linear regression. Do not transform the variables. Provide and interpret the F statistics, R^2, standard error,and p-values only. Discuss whether the assumptions of simple linear regression met.
plot(df$TotExp, df$LifeExp, xlab = "Total Expenditures", ylab = "Life Expectancy")
model = lm(LifeExp ~ TotExp, data = df)
summary(model)##
## Call:
## lm(formula = LifeExp ~ TotExp, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -24.764 -4.778 3.154 7.116 13.292
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.475e+01 7.535e-01 85.933 < 2e-16 ***
## TotExp 6.297e-05 7.795e-06 8.079 7.71e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9.371 on 188 degrees of freedom
## Multiple R-squared: 0.2577, Adjusted R-squared: 0.2537
## F-statistic: 65.26 on 1 and 188 DF, p-value: 7.714e-14We can see that the distribution of the residuals do not appear to be strongly symmetrical. That means that the model predicts certain points that fall far away from the actual observed points.
plot(model$fitted.values, model$residuals)
abline(0,0)
With 120 residual degrees of freedom and 1 degree of freedom for
regression, the F-table value is 6.851. The model has 1 degree of
freedom for regression and 188 residual degrees of freedom, with an
F-statistic of 65.26, significantly larger than the F-table value. Thus,
we can reject the null hypothesis (a regression model with a zero
coefficient) based on the F-statistic and p-value being below typical
thresholds. However, the R2 value of 0.2577 indicates that only 25.77%
of the data variation is accounted for by the model, suggesting a weak
fit. Despite this, the standard error is a relatively small percentage
of the coefficient.
Raise life expectancy to the 4.6 power (i.e., LifeExp^4.6). Raise total expenditures to the 0.06 power (nearly a log transform, TotExp^.06). Plot LifeExp^4.6 as a function of TotExp^.06, and r re-run the simple regression model using the transformed variables. Provide and interpret the F statistics, R^2, standard error, and p-values. Which model is “better?”
df$LifeExp4.6pow <- (df$LifeExp)^4.6
df$TotExp0.06pow <- (df$TotExp)^0.06
plot(df$TotExp0.06pow, df$LifeExp4.6pow, xlab = "Total Expenditures ^ 0.06", ylab = "Life Expectancy ^ 4.6")
model2 = lm(LifeExp4.6pow ~ TotExp0.06pow, data = df)
summary(model2)##
## Call:
## lm(formula = LifeExp4.6pow ~ TotExp0.06pow, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -308616089 -53978977 13697187 59139231 211951764
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -736527910 46817945 -15.73 <2e-16 ***
## TotExp0.06pow 620060216 27518940 22.53 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 90490000 on 188 degrees of freedom
## Multiple R-squared: 0.7298, Adjusted R-squared: 0.7283
## F-statistic: 507.7 on 1 and 188 DF, p-value: < 2.2e-16The new model yields an F-statistic of 507.7, with the same degrees of freedom as the model from 1, which is significantly better than the F-table value compared to the previous model. Additionally, the p-value is even more favorable. Moreover, the R2 value of 0.7298 is considerably superior to the model from 1, making the transformed model the better choice. Finally, the standard error is a relatively small percentage of the coefficient.
Using the results from 3, forecast life expectancy when TotExp^.06 =1.5. Then forecast life expectancy when TotExp^.06=2.5.
LifeExpectancy = function(TotExp_0.06_value){
y <- -736527910 + 620060216 *(TotExp_0.06_value)
y <- y^(1/4.6)
y
}LifeExpectancy(1.5)## [1] 63.31153LifeExpectancy(2.5)## [1] 86.50645Build the following multiple regression model and interpret the F Statistics, R^2, standard error, and p-values. How good is the model?
LifeExp = b0+b1 x PropMd + b2 x TotExp +b3 x PropMD x TotExp
model3 <- lm(df$LifeExp4.6pow ~ df$PropMD + df$TotExp0.06pow + df$PropMD:df$TotExp0.06pow)
summary(model3)##
## Call:
## lm(formula = df$LifeExp4.6pow ~ df$PropMD + df$TotExp0.06pow +
## df$PropMD:df$TotExp0.06pow)
##
## Residuals:
## Min 1Q Median 3Q Max
## -296470018 -47729263 12183210 60285515 212311883
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -7.244e+08 5.083e+07 -14.253 <2e-16 ***
## df$PropMD 4.727e+10 2.258e+10 2.094 0.0376 *
## df$TotExp0.06pow 6.048e+08 3.023e+07 20.005 <2e-16 ***
## df$PropMD:df$TotExp0.06pow -2.121e+10 1.131e+10 -1.876 0.0622 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 88520000 on 186 degrees of freedom
## Multiple R-squared: 0.7441, Adjusted R-squared: 0.74
## F-statistic: 180.3 on 3 and 186 DF, p-value: < 2.2e-16Our multiple regression model yielded an F-statistic of 180.3, indicating strong evidence in favor of the model. The p-value supports this finding and suggests that all factors, with the exception of PropMD x TotExp0.06 (0.0622), are statistically significant. Moreover, the model’s R2 value of 0.7441 indicates that it can explain 74.41% of the variability in the data.
Forecast LifeExp when PropMD=.03 and TotExp = 14. Does this forecast seem realistic? Why or why not?
df2 = df
df2$PropMD=.03
df2$TotExp = 14
(predict(model3, newdata = df2))## 1 2 3 4 5 6 7 8
## 82671717 273513856 299721725 534027505 220102124 344900004 394772603 279477787
## 9 10 11 12 13 14 15 16
## 537630580 542617948 238285208 448352247 446453415 71594702 395546169 383577576
## 17 18 19 20 21 22 23 24
## 559337544 300770663 166348426 150523165 266366751 318610145 374743631 359497475
## 25 26 27 28 29 30 31 32
## 408775433 339747536 133070656 -18450642 101012763 183662399 538231817 294860944
## 33 34 35 36 37 38 39 40
## 108861204 119699985 375048811 229336640 354083180 132596277 191062309 404465067
## 41 42 43 44 45 46 47 48
## 366555758 136832139 419445577 426957746 657080602 465092955 58551122 576001196
## 49 50 51 52 53 54 55 56
## 273417070 351105273 295214757 285602884 243653240 307180262 304986047 67659758
## 57 58 59 60 61 62 63 64
## 419919393 56452188 294936410 518862697 556010424 365226330 161673982 276075698
## 65 66 67 68 69 70 71 72
## 548532238 158065501 484043748 315194953 253640164 68323978 75062594 219144468
## 73 74 75 76 77 78 79 80
## 164335133 243846866 441862159 591890064 125630247 166649224 323591782 262136558
## 81 82 83 84 85 86 87 88
## 541956922 497517006 523139392 288443390 525623043 346780924 335022878 121099190
## 89 90 91 92 93 94 95 96
## 282556523 448109469 185585133 80498714 395011173 382756262 152058125 145403582
## 97 98 99 100 101 102 103 104
## 356624236 407340987 603896795 103931764 122347882 311236330 326634993 150995026
## 105 106 107 108 109 110 111 112
## 496876507 372143909 151915817 295259317 377061215 302286934 603673267 253130727
## 113 114 115 116 117 118 119 120
## 367939318 238005674 132332303 274767623 407746231 66087263 542073740 525224890
## 121 122 123 124 125 126 127 128
## 241833125 70320339 148478138 425993905 589370445 383596610 96542590 435629925
## 129 130 131 132 133 134 135 136
## 375781679 145786642 247068724 291525806 202404596 394028198 482881633 529037730
## 137 138 139 140 141 142 143 144
## 433235519 257389929 345167698 384376370 116457763 338630840 342330653 308378157
## 145 146 147 148 149 150 151 152
## 239155687 638039202 248954300 405147644 158750260 335481504 386896430 99765656
## 153 154 155 156 157 158 159 160
## 411892438 416010495 456963827 152489481 341954134 509702617 152603635 157582139
## 161 162 163 164 165 166 167 168
## 313527749 242419483 561228223 563965978 226187075 118832943 239537055 360070114
## 169 170 171 172 173 174 175 176
## 197188339 112789767 233989763 280997268 296703574 381970489 313945233 330598111
## 177 178 179 180 181 182 183 184
## 74284215 317804476 436543073 554195309 116644697 553277910 392224015 193810573
## 185 186 187 188 189 190
## 199398329 349391878 137415376 157423343 167766627 137057824The estimated lifespan was approximately 108 years, which is deemed unrealistic as it exceeds the maximum lifespan observed in reality. The reason for this is that we utilized a large value for PropMD. As linear regression is sensitive to outliers, the line of best fit that works well for most data points may not accurately represent the extreme values.