5 Assignment: Linear Regression and Time Studies

1

Linear Regression and Time Studies 2

Linear Regression and Time Studies

Linear Regression and Time Studies

In this assignment, we are going to perform linear regression on the COVID tracking data for the state of my residence, i.e., Maryland, for the following variables:

1. “%tested”

2. “%positive” and

3. “deaths”.

In linear regression for the data, we will first be choosing an independent variable x and a dependent variable y and try to find a best possible linear relationship between the two. We will be trying to find coefficients a and b such that y = bx+c. Here b is the slope of the line and c is the y intercept.

1. LINEAR REGRESSION FOR THE VARIABLE “% TESTED”

For the variable % tested we choose the independent variable, x to be the “date” and the dependent variable y to be “% tested”. We formulate the null and alternate hypotheses as follows:

Null Hypothesis: “ The slope which means that y is independent of x. ”

Alternate Hypothesis: “ The slope ≠ 0 which means that y is dependent on x.”

Output:

First, we note that the equation of the straight line we obtained is y = 0.0038x - 165.24. Hence the value of slope b = 0.0038 and c =- 165.24.

In the following two tables, we summarise the output data we got by performing the linear regression:

Regression Statistics
Multiple R	0.9624
R Square	0.926214
Adjusted R Square	0.926014
Standard Error	0.114243
Observations	372
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-165.242	2.431183	-67.9679	4.1E-211	-170.023	-160.462	-170.023	-160.462
% tested	0.003759	5.52E-05	68.15053	1.6E-211	0.003651	0.003868	0.003651	0.003868

We observe the following from the output data:

1. r-squared value: The r-squared value, 0.926214 is very close to 1. This means that there is a strong positive linear relationship between the two variables x and y.

2. p-value: We find that the p-value is 1.6E-211which is way less than 0.05. Hence we reject the null hypothesis and we conclude that the alternate hypothesis is significant in the 0.05 significance level.

Graph of the data and the linear fit:

Using the p-value, we concluded that this model is significant. Now, we use this model predicted outcomes for the next 7 days after the end of the workshop and put it in the following table:

DATE	% tested
08-03-2021	1.144655
09-03-2021	1.148415
10-03-2021	1.152174
11-03-2021	1.155933
12-03-2021	1.159692
13-03-2021	1.163451
14-03-2021	1.16721

2. LINEAR REGRESSION FOR THE VARIABLE “% POSITIVE”

For the variable % positive, we choose the independent variable, x to be the “date” and the dependent variable y to be “% positive”. We formulate the null and alternate hypotheses as follows:

Null Hypothesis: “ The slope which means that y is independent of x. ”

Alternate Hypothesis: “ The slope ≠ 0 which means that y is dependent on x.”

Output:

First, we note that the equation of the straight line we obtained is y = -0.0004x + 19.331. Hence the value of slope b = -0.0004 and c = 19.331.

In the following two tables, we summarise the output data we got by performing the linear regression:

Regression Statistics
Multiple R	0.802194
R Square	0.643515
Adjusted R Square	0.642488
Standard Error	0.032809
Observations	349

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	19.3307	0.768533	25.15271	3.32E-80	17.81913	20.84227	17.81913	20.84227
% positive	-0.00044	1.74E-05	-25.0278	1.01E-79	-0.00047	-0.0004	-0.00047	-0.0004

We observe the following from the output data:

1. r-squared value: The r-squared value, 0.643515 is in the middle of 0 and 1. This means the linear relationship between the two variables x and y is not very weak and at the same time not very strong as well.

2. p-value: We find that the p-value is 1.01E-79 is way less than 0.05. Hence we reject the null hypothesis and we conclude that the alternate hypothesis is significant in the 0.05 significance level.

Graph of the data and the linear fit:

Using the p-value, we concluded that this model is significant. Now, we use this model predicted outcomes for the next 7 days after the end of the workshop and put it in the following table:

DATE	% positive
08-03-2021	0.019669
09-03-2021	0.019233
10-03-2021	0.018797
11-03-2021	0.018361
12-03-2021	0.017924
13-03-2021	0.017488
14-03-2021	0.017052

3. LINEAR REGRESSION FOR THE VARIABLE “DEATHS”

For the variable deaths, we choose the independent variable, x to be the “date” and the dependent variable y to be “deaths”. We formulate the null and alternate hypotheses as follows:

Null Hypothesis: “ The slope which means that y is independent of x. ”

Alternate Hypothesis: “ The slope ≠ 0 which means that y is dependent on x.”

Output:

First, we note that the equation of the straight line we obtained is y = 19.335x - 848470. Hence the value of slope b = 19.335 and c = -848470.

In the following two tables, we summarise the output data we got by performing the linear regression:

Regression Statistics
Multiple R	0.965519
R Square	0.932227
Adjusted R Square	0.932036
Standard Error	538.7689
Observations	357

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-848470	12197.61	-69.5603	6.5E-209	-872458	-824481	-872458	-824481
deaths	19.33474	0.276689	69.87885	1.4E-209	18.79058	19.87889	18.79058	19.87889

We observe the following from the output data:

1. r-squared value: The r-squared value, 0.932227 is very close to 1. This means that there is a strong positive linear relationship between the two variables x and y.

2. p-value: We find that the p-value is 1.4E-209 and it is way less than 0.05. Hence we reject the null hypothesis and we conclude that the alternate hypothesis is significant in the 0.05 significance level.

Graph of the data and the linear fit:

Using the p-value, we concluded that this model is significant. Now, we use this model predicted outcomes for the next 7 days after the end of the workshop and put it in the following table:

DATE	deaths
08-03-2021	7343.826
09-03-2021	7363.16
10-03-2021	7382.495
11-03-2021	7401.83
12-03-2021	7421.165
13-03-2021	7440.499
14-03-2021	7459.834

% tested 44262 44261 44260 44259 44258 44257 44256 44255 44254 44253 44252 44251 44250 44249 44248 44247 44246 44245 44244 44243 44242 44241 44240 44239 44238 44237 44236 44235 44234 44233 44232 44231 44230 44229 44228 44227 44226 44225 44224 44223 44222 44221 44220 44219 44218 44217 44216 44215 44214 44213 44212 44211 44210 44209 44208 44207 44206 44205 44204 44203 44202 44201 44200 44199 44198 44197 44196 44195 44194 44193 44192 44191 44190 44189 44188 44187 44186 44185 44184 44183 44182 44181 44180 44179 44178 44177 44176 44175 44174 44173 44172 44171 44170 44169 44168 44167 44166 44165 44164 44163 44162 44161 44160 44159 44158 44157 44156 44155 44154 44153 44152 44151 44150 44149 44148 44147 44146 44145 44144 44143 44142 44141 44140 44139 44138 44137 44136 44135 44134 44133 44132 44131 44130 44129 44128 44127 44126 44125 44124 44123 44122 44121 44120 44119 44118 44117 44116 44115 44114 44113 44112 44111 44110 44109 44108 44107 44106 44105 44104 44103 44102 44101 44100 44099 44098 44097 44096 44095 44094 44093 44092 44091 44090 44089 44088 44087 44086 44085 44084 44083 44082 44081 44080 44079 44078 44077 44076 44075 44074 44073 44072 44071 44070 44069 44068 44067 44066 44065 44064 44063 44062 44061 44060 44059 44058 44057 44056 44055 44054 44053 44052 44051 44050 44049 44048 44047 44046 44045 44044 44043 44042 44041 44040 44039 44038 44037 44036 44035 44034 44033 44032 44031 44030 44029 44028 44027 44026 44025 44024 44023 44022 44021 44020 44019 44018 44017 44016 44015 44014 44013 44012 44011 44010 44009 44008 44007 44006 44005 44004 44003 44002 44001 44000 43999 43998 43997 43996 43995 43994 43993 43992 43991 43990 43989 43988 43987 43986 43985 43984 43983 43982 43981 43980 43979 43978 43977 43976 43975 43974 43973 43972 43971 43970 43969 43968 43967 43966 43965 43964 43963 43962 43961 43960 43959 43958 43957 43956 43955 43954 43953 43952 43951 43950 43949 43948 43947 43946 43945 43944 43943 43942 43941 43940 43939 43938 43937 43936 43935 43934 43933 43932 43931 43930 43929 43928 43927 43926 43925 43924 43923 43922 43921 43920 43919 43918 43917 43916 43915 43914 43913 43912 43911 43910 43909 43908 43907 43906 43905 43904 43903 43902 43901 43900 43899 43898 43897 43896 43895 43894 43893 43892 43891 1.3350384044939225 1.3295766371947539 1.3229185832642534 1.3160389459224366 1.3104790818005498 1.3053193537941872 1.3034992043440901 1.3003233403171677 1.2939292410306531 1.2866502919163603 1.2787166495533049 1.2725861751735572 1.2677639332110668 1.2656973051896021 1.2620392004795697 1.2567363665200655 1.2525430983032382 1.2455844888974181 1.2396574623819294 1.2358399297264038 1.2341966842944183 1.2309387156999101 1.2249312333029316 1.2181505171268809 1.2107963879266057 1.2043350222473701 1.1993286220479451 1.1969611088139418 1.193194850296005 1.1851807520514601 1.1779232358564167 1.1709558884142872 1.1651051960650478 1.162696960284471 1.1604379305955912 1.1562489489626138 1.1486270731403316 1.1406118208155194 1.1319095609944612 1.1242471274942147 1.1184953563107418 1.1142209727379861 1.10 84944264517835 1.1010346164727483 1.0931718676118254 1.0846005134667978 1.078414148628392 1.0733203680658736 1.0684781770016203 1.0642347887274715 1.0565313029434322 1.0473237208339186 1.0383718499379104 1.031146483121741 1.0254858842793824 1.0203330807546234 1.0131579988643851 1.0032367005438685 0.99407231401007279 0.98466655983180762 0.97812918972354168 0.97181785447905145 0.96771559373472904 0.96186424191105135 0.95750181850076399 0.94989609980222367 0.94107496971363647 0.9352468643639138 0.92860612163742229 0.92519169273239388 0.91877368749748578 0.91436707929982275 0.90808772856559694 0.90115005747319732 0.8935876992189844 0.88702427987040011 0.87784142805232801 0.87119392571284238 0.86416854452013014 0.85816650278726869 0.85088293735190679 0.84601172941236213 0.83993500219934725 0.83446383738943086 0.82786744431892445 0.81939435186522458 0.80854072155736212 0.80059454917997652 0.79249785176201681 0.78656043852412261 0.77989974669586815 0.77248791348222945 0.76451684594479274 0.7556304278867999 0.7464122941862712 0.7412510823624221 0.73538736539302363 0.73032655855242723 0.72581740207958667 0.71934647402099372 0.71477829458591269 0.70697671197915535 0.70008998528712529 0.69258599052071446 0.68777809212726015 0.68267590326565142 0.67444978398914768 0.66595740190812336 0.65915624202448098 0.65190812334018522 0.64666513668596948 0.64209168145538098 0.6363488131768269 0.63163785752582335 0.62543434635201822 0.62015541834090737 0.61494590001444249 0.6104520763222957 0.60640471682497354 0.60250557420769091 0.59751566086922692 0.59149861609289089 0.58566787284541455 0.58106078441846554 0.57689455465361439 0.57361581261429517 0.56919502571620573 0.56428952510586217 0.55885149888647745 0.55416972497937489 0.54959973198958822 0.54590354263073582 0.54208073417970282 0.53848544440993196 0.53323190616470106 0.52787202766627161 0.52251841417500733 0.51750195699039603 0.5146866606126913 0.51172463117243339 0.50765715770473874 0.50278891740016707 0.49676676169693323 0.49220863924703845 0.48748746174223911 0.48387832300926098 0.48079313671762425 0.47684370917441055 0.47203251340876401 0.46648303600928276 0.46201179931665259 0.45722681765993411 0.45380843190827502 0.45088267356213141 0.44765619487205865 0.44330959884829385 0.43868239645097235 0.43358762667679618 0.42963325307529415 0.42679372101197671 0.42430700777322522 0.4213555299239824 0.416401722811023 0.41118544487156405 0.40638908728078244 0.40248137149580016 0.39932281867288683 0.3969406321326282 0.39408032662449988 0.38931793196729797 0.38319619562385954 0.37913317360862436 0.3746251712160511 0.37023983106902786 0.36710551393172725 0.36394053123303916 0.3600633161408347 0.35457269683498432 0.35084435809725795 0.34778588052037807 0.3461518677305308 0.34401632463024917 0.34032821383326772 0.33757111607475537 0.33292314023262304 0.32807056244596433 0.32395544195009229 0.32190612513263678 0.31965484426840873 0.31646776917603286 0.31232874273176736 0.30812739595306915 0.30359317945156788 0.30005262606018762 0.29801204727904146 0.29589018827335745 0.2922553300372801 0.28876984276150963 0.28406152500825 993 0.27921669604625288 0.27624230146027423 0.27396233345797399 0.27131470845624289 0.26564306341704041 0.26142720820069654 0.25588762291779188 0.2519076946818003 0.24698768563381099 0.24401757763168222 0.24119090531991436 0.23632200554090424 0.22961646945083586 0.22618604829067523 0.22147360882218525 0.21872046791030356 0.21513259722796516 0.21243254400837797 0.20839590097068042 0.20447450109110046 0.19936291471874404 0.19464239668838315 0.19040708697610526 0.18781090098057254 0.18386328699206456 0.18000618587023257 0.17637231684581289 0.1706226889542648 0.16662957122950436 0.1639844192569174 0.16051871621430017 0.15713066628680938 0.15469802995201004 0.14993349200288322 0.14695711899358924 0.14297207983069973 0.14018728414577286 0.13663683863781598 0.13409538902067386 0.13227078811811716 0.12878546571095631 0.1267857743449935 0.1245933515743963 0.12210070306569883 0.12013316107861001 0.11850442408427028 0.11670125610096289 0.11520490860014021 0.11306903576263932 0.11050302072266528 0.10825108038399878 0.10667312292141901 0.10524173365278275 0.10317494076270857 0.10192787459961658 0.10021093289913537 9.857642550345927E-2 9.6657849493424708E-2 9.519266215981835E-2 9.3484623364256086E-2 9.2259154989023054E-2 9.0705433212055986E-2 8.8759653881435724E-2 8.7337167517718431E-2 8.5925562482235401E-2 8.3354601383973062E-2 8.1623975588894185E-2 8.022490056774155E-2 7.8826814758246558E-2 7.7086626583810297E-2 7.5479487377329513E-2 7.3575749542159868E-2 7.217502583491113E-2 7.0836622462095059E-2 6.9501021855642359E-2 6.7975822348137871E-2 6.5916613414105762E-2 6.373869908115426E-2 6.1838753224005658E-2 5.9467118274762108E-2 5.801858267072639E-2 5.6572355227225216E-2 5.5020611873573474E-2 5.3232611802350234E-2 5.1300516566327628E-2 4.8604255324761488E-2 4.6863902281715608E-2 4.5853917179243177E-2 4.406822526855448E-2 4.2958494657267837E-2 4.1521664724514448E-2 4.0456283769212967E-2 3.8874864065831377E-2 3.7632579092418086E-2 3.6683430506891837E-2 3.5577821610845453E-2 3.4533378968964475E-2 3.3280707273145742E-2 3.2082607087108E-2 3.108877910837737E-2 3.0194366901241724E-2 2.957545014076482E-2 2.826804206655548E-2 2.7057906472016193E-2 2.6060616252483744E-2 2.5228359511171167E-2 2.4180289759878761E-2 2.3432280878077024E-2 2.2914098838071988E-2 2.1988031857891173E-2 2.122370098373802E-2 2.0378254753656622E-2 1.9508078232133682E-2 1.8612347076121156E-2 1.810867347376182E-2 1.7642589914393624E-2 1.6977674811835455E-2 1.6270718213826672E-2 1.5531117631115058E-2 1.4728207502313106E-2 1.4084230713175442E-2 1.3540494038680814E-2 1.317976152085357E-2 1.2528200775673834E-2 1.207662565395134E-2 1.1454081784063009E-2 1.0887098635613334E-2 1.0297198750427833E-2 9.9005248757055558E-3 9.6395378666925182E-3 9.0123776757351001E-3 8.6537884498327899E-3 8.0549856597283368E-3 7.4080412356176866E-3 6.8878807722973258E-3 5.5794834864303245E-3 5.3480079585375229E-3 4.9139089094337157E-3 4.3531907681492316E-3 3.9197511934838652E-3 3.4192100947071243E-3 3.0011033007355118E-3 2.5155652454333043E-3 2.1769251212938362E-3 1.9156083750615784E-3 1.699960233691362E-3 1.1929892591398211E-3 8.9292838964915298E-4 3.4210236494128368E-4 1.5744952217779562E-4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Date

% tested

% positive 44262 44261 44260 44259 44258 44257 44256 44255 44254 44253 44252 44251 44250 44249 44248 44247 44246 44245 44244 44243 44242 44241 44240 44239 44238 44237 44236 44235 44234 44233 44232 44231 44230 44229 44228 44227 44226 44225 44224 44223 44222 44221 44220 44219 44218 44217 44216 44215 44214 44213 44212 44211 44210 44209 44208 44207 44206 44205 44204 44203 44202 44201 44200 44199 44198 44197 44196 44195 44194 44193 44192 44191 44190 44189 44188 44187 44186 44185 44184 44183 44182 44181 44180 44179 44178 44177 44176 44175 44174 44173 44172 44171 44170 44169 44168 44167 44166 44165 44164 44163 44162 44161 44160 44159 44158 44157 44156 44155 44154 44153 44152 44151 44150 44149 44148 44147 44146 44145 44144 44143 44142 44141 44140 44139 44138 44137 44136 44135 44134 44133 44132 44131 44130 44129 44128 44127 44126 44125 44124 44123 44122 44121 44120 44119 44118 44117 44116 44115 44114 44113 44112 44111 44110 44109 44108 44107 44106 44105 44104 44103 44102 44101 44100 44099 44098 44097 44096 44095 44094 44093 44092 44091 44090 44089 44088 44087 44086 44085 44084 44083 44082 44081 44080 44079 44078 44077 44076 44075 44074 44073 44072 44071 44070 44069 44068 44067 44066 44065 44064 44063 44062 44061 44060 44059 44058 44057 44056 44055 44054 44053 44052 44051 44050 44049 44048 44047 44046 44045 44044 44043 44042 44041 44040 44039 44038 44037 44036 44035 44034 44033 44032 44031 44030 44029 44028 44027 44026 44025 44024 44023 44022 44021 44020 44019 44018 44017 44016 44015 44014 44013 44012 44011 44010 44009 44008 44007 44006 44005 44004 44003 44002 44001 44000 43999 43998 43997 43996 43995 43994 43993 43992 43991 43990 43989 43988 43987 43986 43985 43984 43983 43982 43981 43980 43979 43978 43977 43976 43975 43974 43973 43972 43971 43970 43969 43968 43967 43966 43965 43964 43963 43962 43961 43960 43959 43958 43957 43956 43955 43954 43953 43952 43951 43950 43949 43948 43947 43946 43945 43944 43943 43942 43941 43940 43939 43938 43937 43936 43935 43934 43933 43932 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Date

% positive

Deaths 44262 44261 44260 44259 44258 44257 44256 44255 44254 44253 44252 44251 44250 44249 44248 44247 44246 44245 44244 44243 44242 44241 44240 44239 44238 44237 44236 44235 44234 44233 44232 44231 44230 44229 44228 44227 44226 44225 44224 44223 44222 44221 44220 44219 44218 44217 44216 44215 44214 44213 44212 44211 44210 44209 44208 44207 44206 44205 44204 44203 44202 44201 44200 44199 44198 44197 44196 44195 44194 44193 44192 44191 44190 44189 44188 44187 44186 44185 44184 44183 44182 44181 44180 44179 44178 44177 44176 44175 44174 44173 44172 44171 44170 44169 44168 44167 44166 44165 44164 44163 44162 44161 44160 44159 44158 44157 44156 44155 44154 44153 44152 44151 44150 44149 44148 44147 44146 44145 44144 44143 44142 44141 44140 44139 44138 44137 44136 44135 44134 44133 44132 44131 44130 44129 44128 44127 44126 44125 44124 44123 44122 44121 44120 44119 44118 44117 44116 44115 44114 44113 44112 44111 44110 44109 44108 44107 44106 44105 44104 44103 44102 44101 44100 44099 44098 44097 44096 44095 44094 44093 44092 44091 44090 44089 44088 44087 44086 44085 44084 44083 44082 44081 44080 44079 44078 44077 44076 44075 44074 44073 44072 44071 44070 44069 44068 44067 44066 44065 44064 44063 44062 44061 44060 44059 44058 44057 44056 44055 44054 44053 44052 44051 44050 44049 44048 44047 44046 44045 44044 44043 44042 44041 44040 44039 44038 44037 44036 44035 44034 44033 44032 44031 44030 44029 44028 44027 44026 44025 44024 44023 44022 44021 44020 44019 44018 44017 44016 44015 44014 44013 44012 44011 44010 44009 44008 44007 44006 44005 44004 44003 44002 44001 44000 43999 43998 43997 43996 43995 43994 43993 43992 43991 43990 43989 43988 43987 43986 43985 43984 43983 43982 43981 43980 43979 43978 43977 43976 43975 43974 43973 43972 43971 43970 43969 43968 43967 43966 43965 43964 43963 43962 43961 43960 43959 43958 43957 43956 43955 43954 43953 43952 43951 43950 43949 43948 43947 43946 43945 43944 43943 43942 43941 43940 43939 43938 43937 43936 43935 43934 43933 43932 43931 43930 43929 43928 43927 43926 43925 43924 43923 43922 43921 43920 43919 43918 43917 43916 43915 43914 43913 43912 43911 43910 43909 43908 43907 43906 43905 43904 43903 43902 43901 43900 43899 43898 43897 43896 43895 43894 43893 43892 43891 7955 7941 7930 7922 7919 7905 7879 7869 7856 7838 7805 7789 7762 7732 7715 7697 7677 7661 7630 7612 7580 7554 7536 7503 7467 7446 7413 7372 7349 7329 7287 7251 7220 7188 7154 7127 7107 7076 7037 6996 6963 6900 6865 6837 6792 6735 6689 6650 6596 6567 6541 6494 6449 6404 6367 6301 6272 6246 6216 6173 6132 6082 6027 5994 5967 5942 5895 5848 5803 5740 5712 5680 5659 5627 5568 5519 5471 5448 5412 5358 5319 5270 5206 5142 5117 5100 5064 5012 4962 4916 4865 4846 4820 4790 4764 4715 4673 4641 4625 4602 4569 4547 4518 4481 4448 4434 4415 4398 4372 4351 4335 4309 4302 4293 4273 4261 4249 4233 4221 4212 4201 4194 4182 4172 4162 4155 4152 4147 4137 4127 4115 4108 4099 4096 4091 4078 4070 4058 4050 4041 4037 4036 4032 4028 4022 4012 4003 3999 3995 3990 3979 3973 3967 3961 3958 3957 3950 3949 3949 3946 3938 3935 3925 3917 3909 3902 3895 3883 3879 3876 3869 3861 3855 3849 3839 3838 3836 3828 3824 3816 3807 3804 3799 3796 3789 3778 3766 3761 3755 3752 3746 3736 3722 3717 3707 3694 3691 3685 3674 3669 3661 3650 3641 3639 3636 3631 3620 3612 3604 3591 3585 3577 3565 3551 3536 3530 3523 3515 3506 3493 3488 3478 3458 3447 3440 3433 3422 3409 3405 3402 3382 3377 3368 3359 3347 3341 3334 3325 3319 3310 3303 3288 3275 3266 3246 3243 3236 3223 3212 3205 3190 3175 3168 3157 3142 3129 3108 3092 3074 3066 3052 3030 3016 2996 2982 2947 2939 2926 2900 2875 2844 2811 2776 2749 2740 2702 2668 2666 2646 2625 2596 2562 2538 2501 2458 2417 2379 2341 2306 2274 2221 2198 2164 2122 2090 2040 1991 1935 1881 1834 1788 1739 1692 1643 1580 1520 1476 1424 1372 1323 1276 1226 1168 1099 1045 1001 948 899 831 779 732 683 626 582 541 499 469 421 384 335 298 255 221 191 160 138 122 101 84 64 52 43 35 24 17 10 7 5 5 4 4 3 3 2 2 2 2

Date

Deaths