Problem 1 For this exercise, we will be computing the correlation between two data series

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1 ECON 256 Correlatios Worksheet Professor Jack Rossbach Problem For this exercise, we will be computig the correlatio betwee two data series Data Series Observatio Observatio 2 Observatio 3 X Y Q.) Compute the mea (average) value of both series. There are three observatios so 3 i the fuctios below. x i is the ith observatio of X. x x i ; y y i x 3 ( ) (60) 20 3 y 3 ( ) (39) 3 3 Q.2) Compute the Covariace betwee X ad Y. The covariace is defied as (x i x )(y i y ) ((5 20)(0 3) + (5 20)(25 3) + (30 20)(4 3)) 3 (( 5)( 3) + ( 5)(2) + (0)( 9)) 2 2 ( ) ( 35) Q.3) The variace of X is defied as Var[X] (x i x ) 2. If we compute it, we fid Var[X] 75. Similarly, Var[Y] 7. Usig the formula below, compute the correlatio, r, betwee X ad Y. Var[X] Var[Y] See Last Page for Additioal Discusso o the Formulas for Correlatio ad Covariace Problem 2

Look at the followig graphs. Classify each graph as havig a Positive correlatio or Negative correlatio, ad whether the correlatio is strog (close to or mius -), medium stregth (closer to 0.5 or -0.

2 Look at the followig graphs. Classify each graph as havig a Positive correlatio or Negative correlatio, ad whether the correlatio is strog (close to or mius -), medium stregth (closer to 0.5 or -0.5), or weak (close to 0). If the correlatio is weak, you may ot be able to tell whether it is positive or egative, but you ca still guess. Q2.) Is Correlatio Positive or Negative? Correlatio is Positive (Barely, so it s fie if could t tell) Is Correlatio Strog, Medium, or Weak? Weak. Correlatio is 0.05 which is close to zero Q2.2) Is Correlatio Positive or Negative? Correlatio is Positive Is Correlatio Strog, Medium, or Weak? Strog. Correlatio is 0.98 which is close to oe Q2.3) Is Correlatio Positive or Negative? Correlatio is Negative Is Correlatio Strog, Medium, or Weak? Either Medium Stregth or Strog is acceptable. Correlatio is 0.75 which is right o the boudary betwee beig Strog or Medium stregth.

3 Problem 3 Q3.) Below is the Uemploymet Rate vs GDP per Capita (PPP) i 200 for all coutries with data. Does the correlatio look Positive or Negative? Negative (but it s hard to tell) Does it look like a strog or weak correlatio? Weak (medium stregth would t be marked icorrectly. It s a tough graph to tell just by lookig) Q3.2) Usig the followig summary statistics, compute the correlatio betwee GDP per Capita ad Uemploymet for the World. I ll refer to the first series as GDP ad the secod as Uemp. Cov[GDP, Uemp] SD[GDP] Var[GDP] SD[Uemp] Var[Uemp] 6.05 What is the correlatio, r, betwee GDP per Capita ad the Uemploymet rate. Is the correlatio expected or uexpected? Cov(GDP, Uemp) Var[GDP] Var[Uemp] This meas there is a egative correlatio betwee GDP per capita ad Uemploymet. This is as most people expect, richer coutries have less uemployed people because the labor market fuctios better so people ca get jobs. Also, coutries with less uemploymet are richer, sice more people are workig ad producig goods ad services. Note also that the correlatio is pretty weak. Whe we compute the correlatio separately for rich ad poor coutries, we fid a positive correlatio for oe group ad a egative correlatio for the other group. Whe we combie the rich ad poor coutries, it weakes the overall correlatio.

4 Q3.3) Let s ow compute the correlatio betwee GDP per Capita ad the Uemploymet Rate separately for coutries that have GDP per capita s less tha $0,000/perso ad coutries that have GDP per capita s over that. We ll refer to the first set as Poor coutries ad the secod as Rich coutries. For Poor Coutires, we have the followig statistics. Cov[GDP, Uemp] 453 SD[GDP] Var[GDP] SD[Uemp] Var[Uemp] 6.46 What is the correlatio, r, betwee GDP per Capita ad the Uemploymet rate for the poor coutries? How might you explai this? Cov(GDP, Uemp) Var[GDP] Var[Uemp] This meas there is a positive correlatio betwee GDP per capita ad Uemploymet for poor coutries. This is surprisig, it meas that poor coutries with higher GDP per Capita have more uemployed people (as a fractio of the labor force) o average. The causality is ot clear, ad the correlatio is ot particularly strog, but oe possible cotributig factor is that if a coutry is very poor, it may lack ay sort of safety et for uemployed people. I this case, if you are uemployed you may starve to death or otherwise ot be able to survive. Sice there is o safety et, most people will either work or die, which meas less people will be uemployed compared to coutries that are slightly richer ad able to provide social safety ets such as uemploymet ad welfare beefits. Q3.4) For Rich Coutires, we have the followig statistics. Cov[GDP, Uemp] SD[GDP] Var[GDP] SD[Uemp] Var[Uemp] 5.63 What is the correlatio, r, betwee GDP per Capita ad the Uemploymet rate for the rich coutries? Cov(GDP, Uemp) Var[GDP] Var[Uemp] For Rich coutries, we are back to havig egative correlatio betwee uemploymet ad GDP as expected. Although there are differeces across coutries i how geerous the safety ets are, ulike poor coutries, every rich coutry has a social safety et of some sort so that uemployed people ad people that lose their jobs geerally do ot die.

5 Additioal Discussio o Formulas for Correlatio, Covariace, ad Variace I the week 8 slides, we give the formula for the correlatio betwee X ad Y as either: Or Var[X] Var[Y] (x i x )(y i y ) (x i x ) 2 (y i y ) 2 The reaso these are the same formulas are because the defiitios for the sample covariace ad variace (we will always use the sample statistics i this class, ad ot the populatio statistics) Therefore (x i x )(y i y ) Var(X) (x i x ) 2 Var(Y) (y i y ) 2 Var[X] Var[Y] (x i x )(y i y ) (x i x ) 2 (y i y ) 2 (x i x )(y i y ) ( 2 ) (x i x ) 2 (y i y ) 2 (x i x )(y i y ) ( ) (x i x ) 2 (y i y ) 2 So the formulas are the same. (x i x )(y i y ) (x i x ) 2 (y i y ) 2

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the