12.2 The Correlation Coefficient

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Ralph Clarence Ball
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1 12.2 The Correlation Coefficient In thi ection, we will be tuding the linear correlation coefficient which i the meaure of how cloel the two variable are related or are not related Linear Correlation Coefficient, r: For a et of n data point, the linear correlation coefficient, r, i defined b: 1 ( i ) ( i ) r n 1 Where and denote the ample tandard deviation of the -value and -value, repectivel. Eample 1 Do heavier people burn more calorie? The lean bod ma in kilogram and reting metabolic rate in kilocalorie for 12 ubject who are in a tud on dieting wa recorded. Lean bod ma i the weight without the fat. The reting metabolic rate (RMR) wa meaured in kilocalorie (Kcal) over a 24 hour period.

2 Lean Ma Reting Metabolic Rate i in kg in kcal Standardized Standardized Reting Lean Metabolic Ma Rate Z Z Product Z Z Average Z Z Standard n deviation 1 r ( Z ) Z

3 Thi procedure wa done in Microoft Ecel program, but ou can do thi in our calculator too. Procedure to put in data and find,,,and : 1. Pre STAT 2. Pre ENTER to put data into Edit 3. If there i old data in the calculator to clear it out croll uing the arrow ke to the column heading uch a L1 and puh CLEAR 4. Tpe in the data point into the column labeled L1 5. When finihed with the data point croll to column labeled L2 and tpe in the -data point. 6. When completed puh 2 nd MODE to activate quit 7. Pre STAT 8. Scroll to CALC 9. Scroll to 2 for 2-Variable Statitic 10. Pre 2 nd 1 to activate L1 comma, and 2 nd 2 to activate L2 and then puh ENTER Note that ou will alo get and. We will ue thi later. Now we need to find: Z, Z, and Z Z

4 Procedure to find: Z, Z, and Z Z Ea wa to calculate on TI-83 or TI Pre STAT 2. Pre ENTER to put data into Edit 3. Scroll uing the arrow ke to the column heading L3 thi column will be for z o L 3 ( L1 ) / and pre ENTER for our the lean ma data L 3 ( L )/ Scroll uing the arrow ke to the column heading L4 thi column will be for z o L 4 ( L2 ) / and pre ENTER for our the lean ma data L 4 ( L ) / Scroll uing the arrow ke to the column heading L5 = L3 L4 thi column will be forz Z and pre ENTER 6. The um from L5 which i Z need to be Z calculated. 7. Pre STAT 8. Scroll to CALC 9. Scroll to 1 for 1-Variable Statitic Pre ENTER 10. Pre 2 nd 5 to activate L5 and then puh ENTER 11. The calculator will dipla, but thi value i reall Z Z You can now calculate r, the correlation coefficient

5 r 1 n 1 Z Z For our problem 1 r r Do we have a hpothei tet procedure for our correlation coefficient? Of coure we do If r i the ample correlation coefficient, then ρ (rho) i the correponding population correlation coefficient. Hpothei Tet for the correlation coefficient: Aumption: 1. Simple random ample 2. Normal ditribution Step 1 H 0 : ρ=0 H 0 : the -variable and the -variable are uncorrelated in the population OR H 0 : there i not a linear relationhip between the and variable

6 H a : ρ 0 H 0 : the -variable and the -variable are correlated in the population OR H 0 : there i a linear relationhip between the and variable Step 2 The ignificance α= and the degree of freedom df=n-2. The t-ditribution i ued. Step 3 Calculate r uing the 1 ( i equationr n 1 ) ( i ) 1 n 1 ( Z Z ) Step 4 t r n r Step 5 Find the p-value for df=n-2 Step 6 Compare the p-value to α, If the p-value α, then reject the null hpothei. Otherwie do not reject H 0

7 Step 7 Interpret the reult in term of the p-value, whether or not H 0 wa rejected. Include the contet of the problem in the dicuion. Couple of thing to note: 1. A n get large, it can affect the value of the tet tatitic. Alwa look at our data in a catter plot to verif our reult. 2. There i alo a concern that the correlation between the two variable i reall cauation. Another variable ma be influencing the repone variable. An eample will be dicued in cla. Important note about the correlation coefficient: r reflect the lope of the catterplot. If the lope i poitive, then r i poitive. The magnitude of r indicate the trength of the linear relationhip. Value of r cloe to -1 or +1 indicate a trong linear relationhip between the two variable. A value of r cloe to 0 indicate a weak linear relationhip between the two variable. The ign of r ugget the tpe of linear relationhip. A poitive value of r ugget that the variable are poitivel linearl correlated. A negative value of r ugget that the variable are negativel linearl correlated.

8 The ign of r and the ign of the lope of the regreion line are the ame. If r i poitive, then o i the lope.

Unit 11 Using Linear Regression to Describe Relationships

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