Introduction

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Introduction In previous lessons, we have plotted and analyzed data that appears to have a linear relationship. The data points in some data sets were very close to a linear model, while other data sets had points that were farther from the linear model. The strength of the relationship between data that has a linear trend can be analyzed using the correlation coefficient. 1 4.3.2: Calculating and Interpreting the Correlation Coefficient

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Page 1: Introduction

IntroductionIn previous lessons, we have plotted and analyzed data that appears to have a linear relationship. The data points in some data sets were very close to a linear model, while other data sets had points that were farther from the linear model. The strength of the relationship between data that has a linear trend can be analyzed using the correlation coefficient.

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Introduction, continuedA correlation is a relationship between two events, such as x and y, where a change in one event implies a change in another event. The correlation coefficient, r, is a quantity that allows us to determine how strong this relationship is between two events. It is a value that ranges from –1 to 1; a correlation coefficient of –1 indicates a strong negative correlation, a correlation coefficient of 1 indicates a strong positive correlation, and a correlation coefficient of 0 indicates a very weak or no linear correlation.

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Introduction, continuedYou will use a calculator to calculate the correlation coefficient. Note that a correlation between two events does not imply that changing one event causes a change in the other event—only that a change might have taken place in the other event. This will be explored more later.

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Key Concepts• A correlation is a relationship between two events,

where a change in one event implies a change in another event.

• Correlation doesn’t mean that a change in the first event caused a change in the other event.

• The strength of a linear correlation can be measured using a correlation coefficient.

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Key Concepts, continued• Before determining the correlation coefficient, look at

the scatter plot of the data and make an initial assessment of the strength of a linear relationship between the two events.

• To calculate the correlation coefficient on a graphing calculator, follow the steps on the following slides.

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Key Concepts, continuedOn a TI-83/84:

Step 1: Set up the calculator to find correlations. Press [2nd], then [CATALOG] (the “0” key). Scroll down and select DiagnosticOn, then press [ENTER]. (This step only needs to be completed once; the calculator will stay in this mode until changed in this menu.)

Step 2: To calculate the correlation coefficient, first enter the data into a list. Press [2nd], then L1 (the “1” key). Scroll to enter data sets. Press [2nd], then L2 (the “2” key). Enter the second event in L2. 6

4.3.2: Calculating and Interpreting the Correlation Coefficient

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Key Concepts, continuedStep 3: Calculate the correlation coefficient. Press

[STAT], then select CALC at the top of the screen. Scroll down to 8:LinReg(a+bx), and press [ENTER].

The r value (the correlation coefficient) is displayed along with the equation.

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Key Concepts, continuedOn a TI-Nspire:

Step 1: Go to the lists and spreadsheet page. The icon looks like a table.

Step 2: Enter the data into the first column underneath the shaded row, pressing [enter] after each data value.

Step 3: Use the nav pad to arrow up to the first row below the shaded row and then arrow over to the right so that you are in the second column. Enter the data values, pressing [enter] after each data value.

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Key Concepts, continuedStep 4: Press the [menu] key.Step 5: Arrow down to 4: Statistics and press the

center click key.Step 6: Press the center click key again to select 1:

Stat Calculations.Step 7: Choose 3: Linear Regression (mx+b).Step 8: At the X List field, press [clear] and then type

in “a[]”. To type “[]”, press the [ctrl] key and then the [(] key.

Step 9: Press [tab] to go the Y List field and type in “b[]”.

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4.3.2: Calculating and Interpreting the Correlation Coefficient

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Key Concepts, continuedStep 10: Press [tab] to go the Results field and check

that results are listed in “c[]”. If not, change them.

Step 11: Press [tab] to “OK” and press the center click key.

Step 12: Arrow down until you see the “r” and look to the right. The number to the right is the correlation coefficient, r.

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Key Concepts, continued• A correlation coefficient of –1 indicates a strong

negative correlation.• A correlation coefficient of 1 indicates a strong

positive correlation.• A correlation coefficient of 0 indicates a very weak or

no linear correlation.• The correlation coefficient only assesses the strength

of a linear relationship between two variables. • The correlation coefficient does not assess causation

—that one event causes the other.

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Common Errors/Misconceptions• using the correlation coefficient to analyze data that is

not linear

• incorrectly using the correlation coefficient to assess the strength of a relationship

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Guided PracticeExample 1An education research team is interested in determining if there is a relationship between a student’s vocabulary and how frequently the student reads books. The team gives 20 students a 100-question vocabulary test, and asks students to record how many books they read in the past year. The results are in the table on the next slide. Is there a linear relationship between the number of books read and test scores? Use the correlation coefficient, r, to explain your answer.

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Guided Practice: Example 1, continued

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Books read Test score Books read Test score12 23 5 128 3 15 30

19 14 8 369 8 5 19

14 56 1 019 19 13 6315 25 4 96 30 16 782 6 16 16

14 42 7 9

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Guided Practice: Example 1, continued1. Create a scatter plot of the data.

Let the x-axis represent books read and the y-axis represent test score.

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Guided Practice: Example 1, continued

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Test

sco

re

Books read

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Guided Practice: Example 1, continued2. Describe the relationship between the

data using the graphical representation.It appears that the higher scores were from students who read more books, but the data does not appear to lie on a line. There is not a strong linear relationship between the two events.

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Guided Practice: Example 1, continued3. Calculate the correlation coefficient on

your graphing calculator. Refer to the steps in the Key Concepts section.The correlation coefficient, r, is approximately 0.48.

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Guided Practice: Example 1, continued4. Use the correlation coefficient to

describe the strength of the relationship between the data.A correlation coefficient of 1 indicates a strong positive correlation, and a correlation of 0 indicates no correlation. A correlation coefficient of 0.48 is about halfway between 1 and 0, and indicates that there is a weak positive linear relationship between the number of books a student read in the past year and his or her score on the vocabulary test.

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Guided Practice: Example 1, continued

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http://walch.com/ei/CAU4L3S2CorrCoef
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Guided PracticeExample 3Caitlyn thinks that there may be a relationship between class size and student performance on standardized tests. She tracks the average test performance of students from 20 different classes, and notes the number of students in each class in the table on the next slide. Is there a linear relationship between class size and average test score? Use the correlation coefficient, r, to explain your answer.

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Guided Practice: Example 3, continued

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4.3.2: Calculating and Interpreting the Correlation Coefficient

Class size Avg. test score Class size Avg. test score26 28 32 3336 25 27 3029 27 21 3326 32 28 2719 38 23 4134 32 29 2817 43 37 2314 42 14 3923 37 25 3117 41 33 30

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Guided Practice: Example 3, continued1. Create a scatter plot of the data.

Let the x-axis represent the number of students in each class and the y-axis represent the average test score.

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Guided Practice: Example 3, continued

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Aver

age

test

sco

re

Number of students

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Guided Practice: Example 3, continued2. Describe the relationship between the

data using the graphical representation.As the class size increases, the average test score decreases. It appears that there is a linear relationship with a negative slope between the two variables.

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Guided Practice: Example 3, continued3. Calculate the correlation coefficient on

your graphing calculator. Refer to the steps in the Key Concepts section.The correlation coefficient, r, is approximately –0.84.

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Guided Practice: Example 3, continued4. Use the correlation coefficient to

describe the strength of the relationship between the data.A correlation coefficient of –1 indicates a strong negative correlation, and a correlation of 0 indicates no correlation. A correlation coefficient of –0.84 is close to –1, and indicates that there is a strong negative linear relationship between class size and average test score.

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Guided Practice: Example 3, continued

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http://walch.com/ei/CAU4L3S2CorrCoef2

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