CUNY SPS HW 12

Question

The attached who.csv dataset contains real-world data from 2008. The variables included follow:
- Country: name of the country
- LifeExp: average life expectancy for the country in years
- InfantSurvival: proportion of those surviving to one year or more
- Under5Survival: proportion of those surviving to five years or more
- TBFree: proportion of the population without TB
- PropMD: proportion of the population who are MDs
- PropRN: proportion of the population who are RNs
- PersExp: mean personal expenditures on healthcare in US dollars at average exchange rate
- GovtExp: mean government expenditures per capita on healthcare, US dollars at average exchange rate
- TotExp: sum of personal and government expenditures

Solution

Load required libraries

library(tidyverse)
library(kableExtra)

Read the Data into memory

url_who <- "https://raw.githubusercontent.com/chinedu2301/data605-computer-maths/main/homeworks/hw12/who.csv"
who_raw <- read_csv(url_who)

Display the data

who_raw_display = kable(who_raw) %>%
                  kable_paper("hover", full_width = F) %>%
                  scroll_box(width = "850px", height = "350px")
who_raw_display

Country	LifeExp	InfantSurvival	Under5Survival	TBFree	PropMD	PropRN	PersExp	GovtExp	TotExp
Afghanistan	42	0.835	0.743	0.99769	0.0002288	0.0005723	20	92	112
Albania	71	0.985	0.983	0.99974	0.0011431	0.0046144	169	3128	3297
Algeria	71	0.967	0.962	0.99944	0.0010605	0.0020914	108	5184	5292
Andorra	82	0.997	0.996	0.99983	0.0032973	0.0035000	2589	169725	172314
Angola	41	0.846	0.740	0.99656	0.0000704	0.0011462	36	1620	1656
Antigua and Barbuda	73	0.990	0.989	0.99991	0.0001429	0.0027738	503	12543	13046
Argentina	75	0.986	0.983	0.99952	0.0027802	0.0007410	484	19170	19654
Armenia	69	0.979	0.976	0.99920	0.0036987	0.0049189	88	1856	1944
Australia	82	0.995	0.994	0.99993	0.0023320	0.0091494	3181	187616	190797
Austria	80	0.996	0.996	0.99990	0.0036109	0.0064587	3788	189354	193142
Azerbaijan	64	0.927	0.911	0.99913	0.0036600	0.0084779	62	780	842
Bahamas	74	0.987	0.986	0.99960	0.0009541	0.0040459	1224	55783	57007
Bahrain	75	0.991	0.990	0.99955	0.0026793	0.0059675	710	45784	46494
Bangladesh	63	0.948	0.931	0.99609	0.0002749	0.0002530	12	75	87
Barbados	75	0.989	0.988	0.99989	0.0010990	0.0033720	725	24433	25158
Belarus	69	0.994	0.992	0.99929	0.0047587	0.0124571	204	11315	11519
Belgium	79	0.996	0.995	0.99989	0.0042305	0.0140792	3451	239105	242556
Belize	69	0.986	0.984	0.99944	0.0008901	0.0010745	198	5376	5574
Benin	55	0.912	0.852	0.99865	0.0000355	0.0006608	28	600	628
Bhutan	64	0.937	0.930	0.99904	0.0000801	0.0011248	52	407	459
Bolivia	66	0.950	0.939	0.99734	0.0011042	0.0019340	71	2860	2931
Bosnia and Herzegovina	75	0.987	0.985	0.99943	0.0014111	0.0046694	243	6578	6821
Botswana	52	0.910	0.876	0.99546	0.0003848	0.0025581	431	19604	20035
Brazil	72	0.981	0.980	0.99945	0.0010466	0.0034814	371	13940	14311
Brunei Darussalam	77	0.992	0.991	0.99901	0.0010471	0.0055497	519	30562	31081
Bulgaria	73	0.990	0.988	0.99959	0.0002535	0.0045532	272	11550	11822
Burkina Faso	47	0.878	0.796	0.99524	0.0000493	0.0004566	27	304	331
Burundi	49	0.891	0.819	0.99286	0.0000245	0.0001649	3	10	13
Cambodia	62	0.935	0.918	0.99335	0.0001442	0.0007836	29	140	169
Cameroon	51	0.913	0.851	0.99763	0.0001719	0.0014328	49	784	833
Canada	81	0.995	0.994	0.99996	0.0019126	0.0100446	3430	192800	196230
Cape Verde	70	0.975	0.966	0.99676	0.0004451	0.0007900	114	5394	5508
Central African Republic	48	0.886	0.826	0.99472	0.0000776	0.0003782	13	190	203
Chad	46	0.876	0.791	0.99430	0.0000330	0.0002387	22	234	256
Chile	78	0.992	0.991	0.99984	0.0010477	0.0006073	397	17952	18349
China	73	0.980	0.976	0.99799	0.0014021	0.0009795	81	1302	1383
Colombia	74	0.983	0.979	0.99941	0.0012898	0.0005255	201	12410	12611
Comoros	65	0.949	0.932	0.99914	0.0001406	0.0007188	14	304	318
Congo	54	0.921	0.874	0.99434	0.0002049	0.0009954	31	915	946
Cook Islands	73	0.984	0.981	0.99976	0.0014286	0.0057143	466	27264	27730
Costa Rica	78	0.989	0.988	0.99983	0.0011830	0.0008304	327	15376	15703
C<f4>te d’Ivoire	53	0.910	0.873	0.99253	0.0001100	0.0005382	34	315	349
Croatia	76	0.995	0.994	0.99936	0.0024693	0.0054592	651	30210	30861
Cuba	78	0.995	0.993	0.99990	0.0059081	0.0074448	310	21075	21385
Cyprus	80	0.997	0.996	0.99994	0.0332281	0.0039728	1350	39399	40749
Czech Republic	77	0.997	0.996	0.99990	0.0035916	0.0089430	868	56137	57005
Democratic Republic of the Congo	47	0.871	0.795	0.99355	0.0000961	0.0004747	5	66	71
Denmark	79	0.997	0.996	0.99993	0.0035519	0.0099582	4350	314588	318938
Djibouti	56	0.914	0.870	0.98700	0.0001709	0.0003614	61	4002	4063
Dominica	74	0.987	0.985	0.99984	0.0005588	0.0046618	288	13206	13494
Dominican Republic	70	0.975	0.971	0.99882	0.0016297	0.0015967	197	4148	4345
Ecuador	73	0.979	0.976	0.99805	0.0013888	0.0015593	147	3717	3864
Egypt	68	0.971	0.965	0.99969	0.0024256	0.0033679	78	1290	1368
El Salvador	71	0.978	0.975	0.99936	0.0011739	0.0007547	177	5700	5877
Equatorial Guinea	46	0.876	0.794	0.99596	0.0003085	0.0005464	211	6474	6685
Eritrea	63	0.952	0.926	0.99782	0.0000458	0.0005339	8	80	88
Estonia	73	0.995	0.994	0.99960	0.0032940	0.0069007	516	27393	27909
Ethiopia	56	0.923	0.877	0.99359	0.0000239	0.0001919	6	64	70
Fiji	69	0.984	0.982	0.99970	0.0004562	0.0019928	148	5355	5503
Finland	79	0.997	0.997	0.99996	0.0032992	0.0089204	2824	133956	136780
France	81	0.996	0.995	0.99989	0.0033797	0.0079244	3819	234850	238669
Gabon	58	0.940	0.909	0.99572	0.0003013	0.0051701	276	17220	17496
Gambia	59	0.916	0.886	0.99577	0.0000938	0.0011311	15	550	565
Georgia	70	0.972	0.968	0.99916	0.0046463	0.0040314	123	1248	1371
Germany	80	0.996	0.995	0.99995	0.0034417	0.0080106	3628	209250	212878
Ghana	57	0.924	0.880	0.99621	0.0001408	0.0008565	30	490	520
Greece	80	0.996	0.996	0.99984	0.0049947	0.0035962	2580	65195	67775
Grenada	68	0.983	0.980	0.99992	0.0007547	0.0030755	342	6944	7286
Guatemala	68	0.969	0.959	0.99897	0.0007648	0.0034528	132	2400	2532
Guinea	53	0.902	0.839	0.99534	0.0001075	0.0004801	21	66	87
Guinea-Bissau	48	0.881	0.800	0.99687	0.0001142	0.0006513	10	90	100
Guyana	64	0.954	0.938	0.99785	0.0004953	0.0023518	60	1400	1460
Haiti	61	0.940	0.920	0.99598	0.0002063	0.0000883	28	546	574
Honduras	70	0.977	0.973	0.99905	0.0005275	0.0012237	91	2162	2253
Hungary	73	0.994	0.993	0.99979	0.0030399	0.0091639	855	40602	41457
Iceland	81	0.998	0.997	0.99997	0.0037584	0.0099329	5154	395622	400776
India	63	0.943	0.924	0.99701	0.0005607	0.0011913	36	203	239
Indonesia	68	0.974	0.966	0.99747	0.0001289	0.0007863	26	588	614
Iran (Islamic Republic of)	71	0.970	0.965	0.99972	0.0008805	0.0015811	212	7973	8185
Iraq	56	0.963	0.953	0.99922	0.0006669	0.0013331	59	2948	3007
Ireland	80	0.996	0.996	0.99989	0.0029363	0.0194032	3993	193553	197546
Israel	81	0.996	0.995	0.99994	0.0036913	0.0062568	1533	93748	95281
Italy	81	0.997	0.996	0.99994	0.0036578	0.0071373	2692	140148	142840
Jamaica	72	0.974	0.968	0.99992	0.0008348	0.0016206	170	4399	4569
Japan	83	0.997	0.996	0.99971	0.0021130	0.0094615	2936	159192	162128
Jordan	71	0.979	0.975	0.99994	0.0023494	0.0032185	241	9047	9288
Kazakhstan	64	0.974	0.971	0.99858	0.0037556	0.0073853	148	5510	5658
Kenya	53	0.921	0.879	0.99666	0.0001233	0.0010153	24	231	255
Kiribati	65	0.953	0.936	0.99598	0.0002128	0.0027660	118	4578	4696
Kuwait	78	0.991	0.989	0.99975	0.0017416	0.0035768	687	51940	52627
Kyrgyzstan	66	0.964	0.959	0.99863	0.0024168	0.0058612	28	396	424
Lao People’s Democratic Republic	60	0.941	0.925	0.99708	0.0003473	0.0009724	18	84	102
Latvia	71	0.992	0.991	0.99940	0.0031455	0.0056094	443	18224	18667
Lebanon	70	0.973	0.969	0.99988	0.0020814	0.0011640	460	17400	17860
Lesotho	42	0.898	0.868	0.99487	0.0000446	0.0005629	41	437	478
Liberia	44	0.843	0.765	0.99422	0.0000288	0.0002892	10	413	423
Libyan Arab Jamahiriya	72	0.983	0.982	0.99982	0.0011707	0.0044974	223	13175	13398
Lithuania	71	0.993	0.991	0.99939	0.0039642	0.0076702	448	19932	20380
Luxembourg	80	0.997	0.996	0.99990	0.0027223	0.0095835	6330	476420	482750
Madagascar	59	0.928	0.885	0.99585	0.0002715	0.0002955	9	162	171
Malawi	50	0.924	0.880	0.99678	0.0000196	0.0005353	19	252	271
Malaysia	72	0.990	0.988	0.99875	0.0006518	0.0016612	222	6732	6954
Maldives	72	0.974	0.970	0.99946	0.0010067	0.0029533	316	8100	8416
Mali	46	0.881	0.783	0.99422	0.0000880	0.0006967	28	434	462
Malta	79	0.995	0.994	0.99995	0.0038617	0.0059531	1235	91776	93011
Marshall Islands	63	0.950	0.944	0.99759	0.0004138	0.0026207	294	18876	19170
Mauritania	58	0.922	0.875	0.99394	0.0001028	0.0006219	17	451	468
Mauritius	73	0.988	0.985	0.99960	0.0010407	0.0036773	218	4704	4922
Mexico	74	0.971	0.965	0.99975	0.0018596	0.0008418	474	16340	16814
Micronesia (Federated States of)	69	0.967	0.959	0.99891	0.0005405	0.0022523	290	5830	6120
Monaco	82	0.997	0.996	0.99998	0.0056364	0.0140606	6128	458700	464828
Mongolia	66	0.965	0.958	0.99809	0.0025843	0.0033881	35	1539	1574
Montenegro	74	0.991	0.990	0.99951	0.0020516	0.0057171	299	13725	14024
Morocco	72	0.966	0.963	0.99921	0.0005183	0.0007885	89	1947	2036
Mozambique	50	0.904	0.862	0.99376	0.0000245	0.0002948	14	315	329
Namibia	61	0.955	0.939	0.99342	0.0002921	0.0030020	165	3888	4053
Nauru	61	0.975	0.970	0.99866	0.0010000	0.0063000	567	30200	30767
Nepal	62	0.954	0.941	0.99756	0.0001948	0.0004278	16	64	80
Netherlands	80	0.996	0.995	0.99994	0.0036949	0.0146024	3560	187191	190751
New Zealand	80	0.995	0.994	0.99991	0.0019783	0.0083425	2403	159960	162363
Nicaragua	71	0.971	0.964	0.99926	0.0003697	0.0010597	75	2183	2258
Niger	42	0.852	0.747	0.99686	0.0000215	0.0002051	9	85	94
Nigeria	48	0.901	0.809	0.99385	0.0002413	0.0014532	27	392	419
Niue	70	0.966	0.958	0.99915	0.0020000	0.0110000	1082	35211	36293
Norway	80	0.997	0.996	0.99996	0.0037531	0.0161332	5910	380380	386290
Oman	74	0.990	0.989	0.99986	0.0016850	0.0037376	312	18886	19198
Pakistan	63	0.922	0.903	0.99737	0.0007851	0.0004393	15	105	120
Palau	69	0.990	0.989	0.99949	0.0015000	0.0060500	690	43890	44580
Panama	76	0.982	0.977	0.99957	0.0013476	0.0024811	351	17424	17775
Papua New Guinea	62	0.946	0.927	0.99487	0.0000443	0.0004581	34	390	424
Paraguay	75	0.981	0.978	0.99900	0.0010563	0.0017056	92	2006	2098
Peru	73	0.979	0.975	0.99813	0.0010801	0.0006201	125	4453	4578
Philippines	68	0.976	0.968	0.99568	0.0010476	0.0055749	37	882	919
Poland	75	0.994	0.993	0.99973	0.0019939	0.0052339	495	21266	21761
Portugal	79	0.997	0.996	0.99976	0.0034160	0.0046298	1800	75458	77258
Qatar	77	0.991	0.989	0.99927	0.0026188	0.0059440	2186	163680	165866
Republic of Korea	79	0.995	0.995	0.99877	0.0015618	0.0019159	973	41715	42688
Republic of Moldova	68	0.984	0.981	0.99846	0.0029097	0.0067908	58	1504	1562
Romania	73	0.986	0.984	0.99860	0.0019253	0.0042122	250	9504	9754
Russian Federation	66	0.990	0.987	0.99875	0.0042884	0.0084784	277	12483	12760
Rwanda	52	0.903	0.840	0.99438	0.0000456	0.0003854	19	220	239
Saint Kitts and Nevis	71	0.983	0.981	0.99983	0.0009200	0.0039600	478	9933	10411
Saint Lucia	75	0.988	0.986	0.99978	0.0045951	0.0020307	323	5068	5391
Saint Vincent and the Grenadines	70	0.983	0.980	0.99953	0.0007417	0.0037250	218	6302	6520
Samoa	68	0.977	0.972	0.99975	0.0002703	0.0016757	113	2093	2206
San Marino	82	0.997	0.997	0.99995	0.0351290	0.0708387	3490	278163	281653
Sao Tome and Principe	61	0.937	0.904	0.99748	0.0005226	0.0019871	49	2419	2468
Saudi Arabia	70	0.979	0.974	0.99938	0.0014172	0.0030657	448	27621	28069
Senegal	59	0.940	0.884	0.99496	0.0000492	0.0002723	38	504	542
Serbia	73	0.993	0.992	0.99959	0.0019877	0.0042873	212	7956	8168
Seychelles	72	0.988	0.987	0.99944	0.0014070	0.0073721	557	20502	21059
Sierra Leone	40	0.841	0.731	0.99023	0.0000293	0.0004371	8	164	172
Singapore	80	0.997	0.997	0.99975	0.0014560	0.0043565	944	30100	31044
Slovakia	74	0.993	0.992	0.99982	0.0031307	0.0066364	626	26096	26722
Slovenia	78	0.997	0.996	0.99985	0.0023603	0.0078516	1495	55233	56728
Solomon Islands	67	0.945	0.928	0.99806	0.0001240	0.0013492	28	442	470
South Africa	51	0.944	0.931	0.99002	0.0007214	0.0038205	437	10920	11357
Spain	81	0.996	0.996	0.99976	0.0030829	0.0074940	2152	118426	120578
Sri Lanka	72	0.989	0.987	0.99920	0.0005456	0.0017303	51	360	411
Sudan	60	0.938	0.911	0.99581	0.0002939	0.0008846	29	462	491
Suriname	68	0.971	0.961	0.99905	0.0004198	0.0015121	209	7326	7535
Swaziland	42	0.888	0.836	0.98916	0.0001508	0.0060212	146	2256	2402
Sweden	81	0.997	0.996	0.99995	0.0032155	0.0106857	3727	255696	259423
Switzerland	82	0.996	0.995	0.99995	0.0038648	0.0106174	5694	258248	263942
Syrian Arab Republic	72	0.988	0.987	0.99960	0.0005329	0.0014060	61	1581	1642
Tajikistan	64	0.944	0.932	0.99702	0.0019980	0.0049947	18	100	118
Thailand	72	0.993	0.992	0.99803	0.0003536	0.0027186	98	2079	2177
The former Yugoslav Republic of Macedonia	73	0.985	0.983	0.99967	0.0025476	0.0043384	224	11060	11284
Timor-Leste	66	0.953	0.945	0.99211	0.0000709	0.0016113	45	1053	1098
Togo	57	0.931	0.893	0.99213	0.0000351	0.0003022	18	205	223
Tonga	71	0.980	0.976	0.99966	0.0003000	0.0035000	104	1896	2000
Trinidad and Tobago	69	0.967	0.962	0.99990	0.0007560	0.0027508	513	3575	4088
Tunisia	72	0.981	0.977	0.99972	0.0013049	0.0027936	158	4620	4778
Turkey	73	0.976	0.974	0.99968	0.0015694	0.0029448	383	18632	19015
Turkmenistan	63	0.955	0.949	0.99922	0.0024923	0.0047001	156	4888	5044
Tuvalu	65	0.969	0.962	0.99496	0.0010000	0.0050000	212	8786	8998
Uganda	50	0.922	0.866	0.99439	0.0000739	0.0006344	22	78	100
Ukraine	67	0.980	0.976	0.99886	0.0030871	0.0083434	128	4624	4752
United Arab Emirates	78	0.992	0.992	0.99976	0.0011676	0.0024341	833	45969	46802
United Kingdom	79	0.995	0.994	0.99988	0.0022085	0.0122411	3064	240120	243184
United Republic of Tanzania	50	0.926	0.882	0.99541	0.0000208	0.0003369	17	225	242
United States of America	78	0.993	0.992	0.99997	0.0024132	0.0088152	6350	231822	238172
Uruguay	75	0.987	0.985	0.99969	0.0037178	0.0008646	404	15824	16228
Uzbekistan	68	0.962	0.956	0.99855	0.0026153	0.0107543	26	444	470
Vanuatu	69	0.970	0.964	0.99935	0.0001357	0.0016290	67	1056	1123
Venezuela (Bolivarian Republic of)	74	0.982	0.979	0.99948	0.0017653	0.0010298	247	10528	10775
Viet Nam	72	0.985	0.983	0.99775	0.0005215	0.0007170	37	270	307
Yemen	61	0.925	0.900	0.99868	0.0003101	0.0006325	39	448	487
Zambia	43	0.898	0.818	0.99432	0.0001081	0.0018818	36	595	631
Zimbabwe	43	0.945	0.915	0.99403	0.0001577	0.0007074	21	324	345

Glipmse of the data

glimpse(who_raw)

## Rows: 190
## Columns: 10
## $ Country        <chr> "Afghanistan", "Albania", "Algeria", "Andorra", "Angola…
## $ LifeExp        <dbl> 42, 71, 71, 82, 41, 73, 75, 69, 82, 80, 64, 74, 75, 63,…
## $ InfantSurvival <dbl> 0.835, 0.985, 0.967, 0.997, 0.846, 0.990, 0.986, 0.979,…
## $ Under5Survival <dbl> 0.743, 0.983, 0.962, 0.996, 0.740, 0.989, 0.983, 0.976,…
## $ TBFree         <dbl> 0.99769, 0.99974, 0.99944, 0.99983, 0.99656, 0.99991, 0…
## $ PropMD         <dbl> 0.000228841, 0.001143127, 0.001060478, 0.003297297, 0.0…
## $ PropRN         <dbl> 0.000572294, 0.004614439, 0.002091362, 0.003500000, 0.0…
## $ PersExp        <dbl> 20, 169, 108, 2589, 36, 503, 484, 88, 3181, 3788, 62, 1…
## $ GovtExp        <dbl> 92, 3128, 5184, 169725, 1620, 12543, 19170, 1856, 18761…
## $ TotExp         <dbl> 112, 3297, 5292, 172314, 1656, 13046, 19654, 1944, 1907…

Summary of the data

summary(who_raw)

##    Country             LifeExp      InfantSurvival   Under5Survival  
##  Length:190         Min.   :40.00   Min.   :0.8350   Min.   :0.7310  
##  Class :character   1st Qu.:61.25   1st Qu.:0.9433   1st Qu.:0.9253  
##  Mode  :character   Median :70.00   Median :0.9785   Median :0.9745  
##                     Mean   :67.38   Mean   :0.9624   Mean   :0.9459  
##                     3rd Qu.:75.00   3rd Qu.:0.9910   3rd Qu.:0.9900  
##                     Max.   :83.00   Max.   :0.9980   Max.   :0.9970  
##      TBFree           PropMD              PropRN             PersExp       
##  Min.   :0.9870   Min.   :0.0000196   Min.   :0.0000883   Min.   :   3.00  
##  1st Qu.:0.9969   1st Qu.:0.0002444   1st Qu.:0.0008455   1st Qu.:  36.25  
##  Median :0.9992   Median :0.0010474   Median :0.0027584   Median : 199.50  
##  Mean   :0.9980   Mean   :0.0017954   Mean   :0.0041336   Mean   : 742.00  
##  3rd Qu.:0.9998   3rd Qu.:0.0024584   3rd Qu.:0.0057164   3rd Qu.: 515.25  
##  Max.   :1.0000   Max.   :0.0351290   Max.   :0.0708387   Max.   :6350.00  
##     GovtExp             TotExp      
##  Min.   :    10.0   Min.   :    13  
##  1st Qu.:   559.5   1st Qu.:   584  
##  Median :  5385.0   Median :  5541  
##  Mean   : 40953.5   Mean   : 41696  
##  3rd Qu.: 25680.2   3rd Qu.: 26331  
##  Max.   :476420.0   Max.   :482750

Question 1

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.

Scatter Plot

p = ggplot(who_raw, aes(x=TotExp, y=LifeExp)) + geom_point() + theme_minimal() +
    theme(panel.grid.major = element_line(colour = "lemonchiffon3"),
    panel.grid.minor = element_line(colour = "lemonchiffon3"),
    axis.title = element_text(size = 13),
    axis.text = element_text(size = 11),
    axis.text.x = element_text(family = "sans",
        size = 11), axis.text.y = element_text(family = "sans",
        size = 11), plot.title = element_text(size = 15,
        hjust = 0.5), panel.background = element_rect(fill = "gray85"),
    plot.background = element_rect(fill = "antiquewhite")) +labs(title = "LifeExp vs TotExp",
    x = "TotExp", y = "LifeExp")
p

Simple Linear Regression

lm_who <- lm(LifeExp ~ TotExp, data = who_raw)
summary(lm_who)

## 
## Call:
## lm(formula = LifeExp ~ TotExp, data = who_raw)
## 
## 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-14

Residual vs Fitted

plot(fitted(lm_who),resid(lm_who), main="Residuals vs Fitted", xlab = "Fitted", ylab = "Residuals")
abline(0, 0)

Q-Q Plot

qqnorm(resid(lm_who))
qqline(resid(lm_who))

Interpret the F statistics, R^2, standard error,and p-values only:
F statistic: 65.26 on 1 and 188 DF
p-value: 7.714e-14 - The p-value being less than 0.05 means that the result is statistically significant. R-squared: 0.2577 - This means that the model only accounts for about 25% of variability in the data.
Residual standard error: The TotExp can deviate from the regression line by approximately 9.371

There are four (4) main assumptions for Linear Regression and they are:

Linearity:

The relationship between X and Y must be linear. As can be seen from the scatter plot above, LifeExp vs TotExp does not have a linear relationship and this condition is not satisfied.

Homoscedacity:

There should be constant variance in the residuals. From the Residual vs Fitted Plot shown above, it does not appear that there is a constant variance and thus the homoscedacity criterion is not satisfied.

Normality:

The data should be normally distributed. From the QQ plot shown above, the data does not follow a normal distribution.

Independence:

The observations should be independent of each other. This may be difficult to determine from looking at the data and we may have to rely on the assumptions provided by the data collector.
Since the Linearity, Homoscedacity, and Normality conditions are not satisfied, we can conclude that the assumptions for Linear Regression are not met.

Question 2

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?”

who_raw2 <- who_raw %>% 
            mutate(LifeExp2 = LifeExp^(4.6),
            TotExp2 = TotExp^(0.06))

Scatter Plot - Q2

p2 = ggplot(who_raw2, aes(x=TotExp2, y=LifeExp2)) + geom_point() + theme_minimal() +
    theme(panel.grid.major = element_line(colour = "lemonchiffon3"),
    panel.grid.minor = element_line(colour = "lemonchiffon3"),
    axis.title = element_text(size = 13),
    axis.text = element_text(size = 11),
    axis.text.x = element_text(family = "sans",
        size = 11), axis.text.y = element_text(family = "sans",
        size = 11), plot.title = element_text(size = 15,
        hjust = 0.5), panel.background = element_rect(fill = "gray85"),
    plot.background = element_rect(fill = "antiquewhite")) +labs(title = "LifeExp2 vs TotExp2",
    x = "TotExp2", y = "LifeExp2")
p2

Simple Linear Regression - Q2

lm_who2 <- lm(LifeExp2 ~ TotExp2, data = who_raw2)
summary(lm_who2)

## 
## Call:
## lm(formula = LifeExp2 ~ TotExp2, data = who_raw2)
## 
## 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 ***
## TotExp2      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-16

Residual vs Fitted - Q2

plot(fitted(lm_who2),resid(lm_who2), main="Residuals vs Fitted - Q2", xlab = "Fitted", ylab = "Residuals")
abline(0, 0)

Q-Q Plot

qqnorm(resid(lm_who2))
qqline(resid(lm_who2))

Interpret the F statistics, R^2, standard error,and p-values only:
F statistic: 507.7 on 1 and 188 DF
p-value: < 2.2e-16 - The p-value being less than 0.05 means that the result is statistically significant. R-squared: 0.7298 - This means that the model accounts for about 72.98% of variability in the data.
Residual standard error: The TotExp can deviate from the regression line by approximately 90490000 on 188 degrees of freedom
There are four (4) main assumptions for Linear Regression and they are:

Linearity:

The relationship between X and Y must be linear. As can be seen from the scatter plot above, LifeExp vs TotExp does have a linear relationship and this condition is satisfied.

Homoscedacity:

There should be constant variance in the residuals. From the Residual vs Fitted Plot shown above, it appears that there is a constant variance and thus the homoscedacity criterion is satisfied.

Normality:

The data should be normally distributed. From the QQ plot shown above, the data follows a normal distribution.

Independence:

The observations should be independent of each other. This may be difficult to determine from looking at the data and we may have to rely on the assumptions provided by the data collector. Also, we can see from the residual plot that the data points do not appear to be dependent on one another.
Since the Linearity, Homoscedacity, Normality, and Independence conditions are satisfied, we can conclude that the assumptions for Linear Regression are met.

Clearly, the second model that involves a transformation is better. This goes to tell us that sometimes even if the data does not appear to satisfy the assumptions of linear regression, with some transformations, we may be able to get a transformed data that will satisfy the criteria for linear regression and still make using linear regression possible on the dataset.

Question 3

Using the results from 3, forecast life expectancy when TotExp^.06 = 1.5. Then forecast life expectancy when TotExp^.06 = 2.5

Based on the results of the model with the transformed data above, the linear relationship is given by:
\(LifeExp2 = -736527910 + 620060216*TotExp2\)
Using the equation above, we can forcast the values for life expectancy for the given TotExp.
When TotExp^.06 = 1.5

TotExp2 = 1.5
LifeExp2 = -736527910 + 620060216*TotExp2
LifeExp = LifeExp2 ^ (1/4.6) # We have to transform back to get the actual LifeExp
LifeExp

## [1] 63.31153

Therefore, for TotExp^.06 = 1.5, the LifeExp will be about 63.3 after transforming back to the original units.

When TotExp^.06 = 2.5

TotExp2 = 2.5
LifeExp2 = -736527910 + 620060216*TotExp2
LifeExp = LifeExp2 ^ (1/4.6) # We have to transform back to get the actual LifeExp
LifeExp

## [1] 86.50645

Therefore, for TotExp^.06 = 2.5, the LifeExp will be about 86.5 after transforming back to the original units.

Question 4

Build 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

Scatter Plot - Q4

p4 = ggplot(who_raw, aes(x=(TotExp + PropMD + (PropMD * TotExp)), y=LifeExp2)) + geom_point() + theme_minimal() +
    theme(panel.grid.major = element_line(colour = "lemonchiffon3"),
    panel.grid.minor = element_line(colour = "lemonchiffon3"),
    axis.title = element_text(size = 13),
    axis.text = element_text(size = 11),
    axis.text.x = element_text(family = "sans",
        size = 11), axis.text.y = element_text(family = "sans",
        size = 11), plot.title = element_text(size = 15,
        hjust = 0.5), panel.background = element_rect(fill = "gray85"),
    plot.background = element_rect(fill = "antiquewhite")) +labs(title = "LifeExp - Multi Regression",
    x = "TotExp + PropMD + (PropMD * TotExp)", y = "LifeExp")
p4

Simple Linear Regression - Q4

lm_who4 <- lm(LifeExp ~ TotExp + PropMD + (PropMD * TotExp), data = who_raw)
summary(lm_who4)

## 
## Call:
## lm(formula = LifeExp ~ TotExp + PropMD + (PropMD * TotExp), data = who_raw)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -27.320  -4.132   2.098   6.540  13.074 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    6.277e+01  7.956e-01  78.899  < 2e-16 ***
## TotExp         7.233e-05  8.982e-06   8.053 9.39e-14 ***
## PropMD         1.497e+03  2.788e+02   5.371 2.32e-07 ***
## TotExp:PropMD -6.026e-03  1.472e-03  -4.093 6.35e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.765 on 186 degrees of freedom
## Multiple R-squared:  0.3574, Adjusted R-squared:  0.3471 
## F-statistic: 34.49 on 3 and 186 DF,  p-value: < 2.2e-16

Residual vs Fitted - Q4

plot(fitted(lm_who4),resid(lm_who4), main="Residuals vs Fitted - Q4", xlab = "Fitted", ylab = "Residuals")
abline(0, 0)

Q-Q Plot

qqnorm(resid(lm_who2))
qqline(resid(lm_who2))

Interpret the F statistics, R^2, standard error,and p-values only:
F statistic: 34.49 on 3 and 186 DF
p-value < 2.2e-16 - The p-value being less than 0.05 means that the result is statistically significant. R-squared: 0.3574 - This means that the model only accounts for about 35.74% of variability in the data.
Residual standard error: The TotExp can deviate from the regression line by approximately 8.765 on 186 degrees of freedom
There are four (4) main assumptions for Linear Regression and they are:

Linearity:

The relationship between X and Y must be linear. As can be seen from the scatter plot above, LifeExp vs TotExp + PropMD + (PropMD * TotExp) does not have a linear relationship and this condition is not satisfied.

Homoscedacity:

There should be constant variance in the residuals. From the Residual vs Fitted Plot shown above, it does not appear that there is a constant variance and thus the homoscedacity criterion is not satisfied.

Normality:

The data should be normally distributed. From the QQ plot shown above, the data does not follow a normal distribution.

Independence:

The observations should be independent of each other. This may be difficult to determine from looking at the data and we may have to rely on the assumptions provided by the data collector.
Since the Linearity, Homoscedacity, and Normality conditions are not satisfied, we can conclude that the assumptions for Linear Regression are not met.

Comparing the results of the three models, the second model is still the best and the third model is by no means better at all. Although the third model is slightly better than the first model without any transformation, it still falls far short when compared to the second model. This still tells us that even if the data can be transformed to produce better results, not all transformations will make sense and will produce better models.

Question 5

Forecast LifeExp when PropMD=.03 and TotExp = 14. Does this forecast seem realistic? Why or why not?

Based on the results of the model with the transformed data above, the linear relationship is given by:
\(LifeExp = 6.277e+01 + 1.497e+03 * PropMd + 7.233e-05 * TotExp - 6.026e-03 * PropMD * TotExp\) Using the equation above, we can forcast the values for life expectancy for the given TotExp.
When TotExp^.06 = 1.5

PropMD = 0.03
TotExp = 14
LifeExp5 = 6.277 * 10^(1) +  (1.497 * 10^(3) * PropMD) + (7.233 * 10^(-5) * TotExp) - (6.026 * 10^(-3) * PropMD * TotExp)
LifeExp5

## [1] 107.6785

From the data provided, the max LifeExp is about 83, and the mean LifeExp is about 67. Hence, a value of 107 for LifeExp is not realistic based on the given data.