Mod 2 Lesson 19 Interpreting Correlation
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1 Date: Mod 2 Lesson 19 Interpreting Correlation Example 1: Positive and Negative Relationships Linear relationships can be described as either positive or negative. Below are two scatter plots that display a linear relationship between two numerical variables x and y.
2 1. The relationship displayed in Scatter Plot 1 is a positive linear relationship. Does the value of the y variable tend to increase or decrease as the value of x increases? If you were to describe this relationship using a line, would the line have a positive or negative slope? 2. The relationship displayed in Scatter Plot 2 is a negative linear relationship. As the value of one of the variables increases, what happens to the value of the other variable? If you were to describe this relationship using a line, would the line have a positive or negative slope?
3 3. What does it mean to say that there is a positive linear relationship between two variables? 4. What does it mean to say that there is a negative linear relationship between two variables? Example 2: Some Linear Relationships are Stronger than Others Below are two scatter plots that show a linear relationship between two numerical variables x and y. 5. Is the linear relationship in Scatter Plot 3 positive or negative? 6. Is the linear relationship in Scatter Plot 4 positive or negative?
4 Example 2: Some Linear Relationships are Stronger than Others Below are two scatter plots that show a linear relationship between two numerical variables x and y. It is also common to describe the strength of a linear relationship. We would say that the linear relationship in Scatter Plot 3 is weaker than the linear relationship in Scatter Plot Why do you think the linear relationship in Scatter Plot 3 is considered weaker than the linear relationship in Scatter Plot 4? 8. What do you think a scatter plot that shows the strongest possible positive linear relationship would look like? Draw a scatter plot with 5 points that illustrates this.
5 9. How would a scatter plot that shows the strongest possible negative linear relationship look different from the scatter plot that you drew in the previous question? Exercises 10 12: Strength of Linear Relationships Consider the three scatter plots below. Place them in order from the one that shows the strongest linear relationship to the one that shows the weakest linear relationship. 11. Explain your reasoning for choosing the order in Exercise 10.
6 12. Which of the following two scatter plots shows the stronger linear relationship? (Think carefully about this one!) Example 3: The Correlation Coefficient The correlation coefficient is a number between 1 and +1 (including 1 and +1) that measures the strength and direction of a linear relationship. The correlation coefficient is denoted by the letter r. Several scatter plots are shown below. The value of the correlation coefficient for the data displayed in each plot is also given.
7 Looking at the scatter plots in Example 3, you should have discovered the following properties of the correlation coefficient: Property 1: The sign of r (positive or negative) corresponds to the direction of the linear relationship r = +1 Property 2: A value of indicates a perfect positive linear relationship, with all points in the scatter plot falling exactly on a straight line. r = 1 Property 3: A value of indicates a perfect negative linear relationship, with all points in the scatter plot falling exactly on a straight line. r Property 4: The closer the value of is to or, the stronger the linear relationship. Here is the data from a previous lesson on shoe length in inches and height in inches for 10 men.
8 16. Enter the shoe length and height data in your calculator. Find the value of the correlation coefficient between shoe length and height. Round to the nearest tenth. 17. Interpret the value of the correlation coefficient between shoe length and height for the data given above.
9 18. Based on the scatter plot, do you think that the value of the correlation coefficient between fat content and calories per serving will be positive or negative? Explain why you made this choice. 19. Based on the scatter plot, estimate the value of the correlation coefficient between fat content and calories.
10 20. Calculate the value of the correlation coefficient between fat content and calories per serving. Round to the nearest hundredth. Interpret this value. 21. Based on the scatter plot, do you think that the value of the correlation coefficient between sodium content and calories per serving will be positive or negative? Explain why you made this choice.
11 22. Based on the scatter plot, estimate the value of the correlation coefficient between sodium content and calories per serving. 23. Calculate the value of the correlation coefficient between sodium content and calories per serving. Round to the nearest hundredth. Interpret this value.
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