Correlation Causauation Linear Regression

NOTES:Scatter Plots and Correlation scatter plot is often used to present bivariate quantitative data. Each variable is represented on an axis and .ne axes are labeled accordingly. A scatter plot displays data as points on a grid using the associated numbers as coordinates or ordered pairs (x, y). The way the points are arranged by themselvesin a scatter plot may or may not suggest a relationship between the two variables. For instance, by reading the below, do you think there is a relationship between the hours spent Class1 100 95 90 studying and exam grades? graph C so 75 70 65 60 If y tends to increase as x increases, then the data have positive correlation. If y tends to decrease as x increases, then the data have negative correlation. 0 (h51.o Is 2025 3.0354.0 Mean Study Time (hours) Correlation Coefficient Positive Correlation Direction Negative Correlation Strength Weak Correlation No Correlation Neg*ve ST o Correlation In order to determine the direction and strength of the model, we use something called a correlation coefficient. It is represented by the letter r. Direction and correlation coefficients r is always between -1 and 1 • A positive r is a positive correlation, a negative r is a negative correlation Strength and correlation coefficients A r of 1 is a perfect positive correlation (an r of-I is a perfect negative correlation) The closer the value to 1 (or -l) the stronger the correlation Generally 0.8 —1 represent strong correlations 0.7-0.79 represent weak correlation ything below 0.7 generally has no correlation r = 0.93 r = -0.95 r = -0.75 wQqb r = 0. 61 no practice Problems: For each scatter plot, tell whether the data have a a) positive correlation, a negative correlation, or no correlation. b) If the correlation is positive or negative, determine if it is a strong or weak correlation 2. 3 4 2 n one 2 5. Positive, negative, or no correlation? a. Amount of exercise and percent of body fat b. A person's age and the number of medical conditions they have sqe c. Temperature and number of ice cream cones sold number d. The of students at Sprayberry and the umber of dogs in Atlanta e. Age of a tadpole and the length of its tail • 00 Correlation vs. Causation When a scatter lot shows a correlation between two variables, even if it'sa strong one, there i not necessarily a re a Ions 1 0th variables could be related to some third variable that actua y s th apparent correlation. Also, an apparent correlation simplycould be the resultof chance. Example 1: During the month of June the number of new babies born at the Utah Valley Hospital was recorded for a week. Over the same time period, the number of cakes sold at Carlo's Bakery in Hoboken, New Jersey was also recorded. What can be said about the correlation? Is there causation? Why or why not? Babies vs. Cakes Numberof Number of babiesborn cakessold 50 9 11 11 12 • 60 25 30 10 The* are •h 30 42 48 ImS.•es born does c.mp€5. ) •vs 20 z 52 56 10 CAVS o 2 4 6 s 10 12 14 Number ofbabies born at Utah Valley Hospital Example 2: An American medical researcher wants to see if there is a link between a person'ssocio-economic status (how much money they have) and certain types of cancer. Hisresearch seems •toindicate that there js a link (rich people seem to sufferfrom more cancers than poor people do). HisCausation Statement: Being rich willmake you more likely to get cancer. What can be said about the correlation? Is there causati , why or Doc-tore EEC,ut..R2t.- SO QporcD -O W.sov...) A O.xa. why not? u.mo A 40 xo 40 •to TH Da.Z.fZ • or NOTES: Scatter plots and line of best fit Kim works as waitress. Below is a table of how many tables she worksand what she makes in ti Tables Amount in ti s 2 13 6 10 12 13 15 42 65 78 92 102 1) Do you think the relationship is a correlation or a causation? Why? 9 OF t He SRE GEO 6 2) Do you think there is a positive or negative correlation coefficient? Why? 5 3 3) Graph the function,what kind this? 2 graph is CSCd4e.t 10 12 14 16 4) Can you predict how much money Kim willmake in tips if she served 22 tables? {S Sorre Line of best fit: SQt•e.r uS We . Calculator Steps: l) Go to [data]. If needed clear out the previousdata by pressing[data] and second time and pressing enter on whichever listyou want to clear 2) Enter all the x values into Ll and all the y values into 12 3) Go to [stat-reg/distr].You can do thisby pressing [2nd][data] 4) Select 2-Var stats (because we now have 2 variables) 5) Scroll down to find the variables ou need a b M (StJ.æ) e;ö.a..5) Wha ISthe line of best fit for tm's waitressing variables? C.80x 6) What is the correlation coefficient? r: .qQb 7) Based on your correlation coefficient is thisa strong or a weak model? Would it make a good predictor? Why or why not? 8) Using theYne of%est fit model, how much money would you predict that Kim would make if she served 22 tables? Jlqq.t€ An old myth says 1 dog year equals 7 human years, but that comparison is not accurate. An approximate comparison of ages is shown in the table: Age of Cat or 3 months 6 Dog (inyears) (.25 months years) (.5 ears 5 10 Approximate 15 2 4 6 8 24 32 40 48 56 14 18 20 21 72 90 94 101 Equivalent Humanage in ears 1. Use your calculator to determine the regression formula 2. What is the correlation coefficient? 3. Using e4i51 a. o •12years old t?etqifi nt age of the animal who is: b. Cat• 16year old .3Q( 12) 411.2-0 4. Usingyour equa I humanwhois: kvmun ould a cat or dog ave o e o have the maturityof a 4- a. 65 years pLua b. 78 years old lt.5 '4,us Y.3t 5. The documented world's 01 was named Bluey from Australia who died when he was 29 years and 6 months old. Find Bluey's approximate equivalent human age. q.32(aq.s) 4 S.ecrs ,