4.3.2: Warm-up, P.107 A new social networking company launched a TV commercial. The company tracked the number of users in thousands who joined the network after each week the commercial aired. Use the table of data on the next slide to answer the questions that follow : Calculating and Interpreting the Correlation Coefficient

2 Number of weeksNew users in thousands

1.Create a scatter plot of the data : Calculating and Interpreting the Correlation Coefficient New users in thousandsNumber of weeks

2.Does the data appear to have a linear or exponential relationship? Explain. The data appears to have a linear relationship. The shape of the data in the graph is linear because a line could be drawn on the graph with an approximately equal number of points above and below the graph : Calculating and Interpreting the Correlation Coefficient

4.3.2: 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 : Calculating and Interpreting the Correlation Coefficient

4.3.2: Key Concepts Correlation: 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 : Calculating and Interpreting the Correlation Coefficient

Key Concepts, continued Correlation coefficient of –1: indicates a perfect negative correlation. Correlation coefficient of 1: indicates a perfect positive correlation. Correlation coefficient of 0: indicates 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. Example: coefficients -.87 &.87 have equal strength!! : Calculating and Interpreting the Correlation Coefficient

Correlation Coefficient chart: copy!!! : Calculating and Interpreting the Correlation Coefficient Perfect Negative Correlation Strong Neg. Corr. Moderate Neg. Corr. Weak Neg. Corr. NO Correlation Weak Pos. Corr. Moderate Pos. Corr. Strong Pos. Corr. Perfect Positive Correlation negative correlation positive correlation

No relation: : Calculating and Interpreting the Correlation Coefficient Annual income # of children Weak, Negative Corr. Strong, Positive Corr. High School GPA College GPA Price ($) Quantity Sold Some form of technology is needed to calculate the Correlation Coefficient. Wolfram web site: “Pearson’s Correlation Coefficient Calculator” Or Or

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 : Calculating and Interpreting the Correlation Coefficient

Guided Practice Example #1: An 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 : Calculating and Interpreting the Correlation Coefficient

Guided Practice: Example #1, continued : Calculating and Interpreting the Correlation Coefficient Books readTest scoreBooks readTest score

Guided Practice: Example #1, continued 1.Create a scatter plot of the data. Let the x-axis represent books read and the y-axis represent test score : Calculating and Interpreting the Correlation Coefficient

Guided Practice: Example #1, continued : Calculating and Interpreting the Correlation Coefficient Test score Books read

Guided Practice: Example #1, continued 2.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 : Calculating and Interpreting the Correlation Coefficient

Guided Practice: Example #1, continued 3.Calculate the correlation coefficient on your graphing calculator. Refer to the website in the Key Concepts section. The correlation coefficient, r, is approximately : Calculating and Interpreting the Correlation Coefficient

Guided Practice: Example #1, continued 4.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 0 and 1, and indicates that there is a moderate positive linear relationship between the number of books a student read in the past year and his or her score on the vocabulary test : Calculating and Interpreting the Correlation Coefficient ✔

Guided Practice Example #3: Caitlyn 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 : Calculating and Interpreting the Correlation Coefficient

Guided Practice: Example #3, continued : Calculating and Interpreting the Correlation Coefficient Class sizeAvg. test scoreClass sizeAvg. test score

Guided Practice: Example #3, continued 1.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 : Calculating and Interpreting the Correlation Coefficient

Guided Practice: Example #3, continued : Calculating and Interpreting the Correlation Coefficient Average test score Number of students

Guided Practice: Example #3, continued 2.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 : Calculating and Interpreting the Correlation Coefficient

Guided Practice: Example #3, continued 3.Calculate the correlation coefficient on your graphing calculator. Refer to the website in the Key Concepts section. The correlation coefficient, r, is approximately – : Calculating and Interpreting the Correlation Coefficient

Guided Practice: Example #3, continued 4.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 : Calculating and Interpreting the Correlation Coefficient ✔

Homework Workbook(4.3.2):, P.115 #