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15.5 Linear Correlation Objectives: • Be able to construct a scatterplot to show the relationship between two variables. • Understand the properties of the linear correlation coefficient. • Be able to use your calculator to compute the linear correlation coefficient. Scatterplots • To determine whether there is a correlation between two variables, we obtain pairs of data, called data points, relating the first variable to the second. • To understand such data, we plot data points in a graph called a scatterplot. Scatterplots cont. (2) Example: An instructor wants to know whether there is a correlation between the number of times students attended tutoring sessions during the semester and their grades on a 50-point examination. The table shows the data for 10 students. Represent these data points by a scatterplot and interpret the graph. Solution: We plot the points (18, 42), (6, 31), (16, 46), and so on. As the number of tutoring sessions increases, the grades also generally increase. So the correlation coefficient is positive. Linear Correlation • Linear correlation exists between two variables if, when graphed, the points in the graph tend to lie in a straight line. • The linear correlation coefficient allows us to compute to what degree the points of a scatterplot lie along a straight line. • r = 1 means that the points all lie on a straight line and y increases as x increases. • r = -1 means that the points all lie on a straight line and y decreases as x increases. Linear Correlation cont. (2) • There is a positive correlation between the variables x and y if whenever x increases or decreases, then y changes in the same way. • We will say that there is a negative correlation between the variables x and y if whenever x increases or decreases, then y changes in the opposite way. Linear correlation coefficient, r Using your Calculator to Compute r 1) Enter the x-values in List1 and the y-values in List2. You must have the same number of entries in List1 and List2! 2) Go to STAT and then 3) ENTER 4: CALC LinReg(ax+b) Confidence of Correlation 1. Compute r for n pairs of data. 2. If the absolute value of r exceeds the number in the column labeled α = .05 on line n, then we are 95% confident that there is a significant linear correlation. (There is less than a 0.05 (5%) chance that the variables do not have significant linear correlation. ) 3. If the absolute value of r exceeds the number in the column labeled α = .01 on line n, then we are 99% confident that there is a significant linear correlation. (There is less than a 0.01 (1%) chance that the variables do not have significant linear correlation. Correlation Applications • Example A: You wish to determine if there is a relationship between the weight of the car and gas mileage. The table shows data regarding gas mileage of cars with various weights. Find the correlation coefficient for these data and determine whether there is linear correlation at the 95% or 99% level. • Example B: Find the correlation coefficient for the number of times students attended tutoring sessions during the semester and their grades on the 50-point examination. The table shows the data for 10 students. Determine whether there is linear correlation at the 95% or 99% level. Correlation Applications cont. (2) • Solution A: We will represent the weight by x and the gas mileage by y. So enter the weight in List1 and the gas mileage in List2. We now compute There are 10 pairs of data, so we look at line 10 of the table and find the value 0.765 under the α = 0.01 column. The absolute value of r is 0.85, so we are 99% confident that there is significant negative linear correlation between car weight and gas mileage. • Solution B: Enter the number of times students attended tutoring sessions during the semester in List1 and their grades on the 50-point examination in List2. • Compute the correlation coefficient, r . • There are again 10 pairs of data, so we look at line 10 of the table to find the confidence of correlation level.