Grade Eight 3-5 Scatter Plots and Lines of Fit Lesson - 1st Term 2024 PDF

# Grade Eight ## 3-5 Scatter Plots and Lines of Fit ### Lesson (3-5) - Stu-book Page (112-119) - comp-book Page (63-66) ### Objectives - **Student able to** - Fit a function to linear data shown in a scatter plot and use fitted functions to solve problems in the context of the data. - Inte...

# Grade Eight ## 3-5 Scatter Plots and Lines of Fit ### Lesson (3-5) - Stu-book Page (112-119) - comp-book Page (63-66) ### Objectives - **Student able to** - Fit a function to linear data shown in a scatter plot and use fitted functions to solve problems in the context of the data. - Interpret the slope of a trend line within the context of data. ### Let's Start <br> Nicholas plotted data points to represent the relationship between screen size and cost of television sets. Everything about the televisions is the same, except for the screen size. | Screen (inches) | Cost ($) | |---|---| | 40 | 300 | | 42 | 350 | | 43 | 400 | | 48 | 480 | | 50 | 500 | - A. Describe any patterns you see. - B. What does this set of points tell you about the relationship of screen size and cost of the television? - C. Where do you think the point for a 46-inch television would be on the graph? How about for a 60-inch TV? Explain. <br> ### Example 1 Understand Association - A. What is the relationship between the hours after sunrise, *x*, and the temperature, *y*, shown in the scatter plot? - The temperature is generally increasing over time. - B. What is the relationship between the hours after sunset, *x*, and the temperature, *y*, shown in the scatter plot? - The temperature is generally decreasing over time. - C. What is the relationship between the hours after sunset, *x*, and the amount of rain, *y*, shown in the scatter plot? - There is no general trend in the data. When *y*-values tend to increase as *x*-values increase, the two data sets have a **positive association**. When *y*-values tend to decrease as *x*-values increase, the two data sets have a **negative association**. When there is no general relationship between *x*-values and *y*-values, the two data sets have **no association**. <br> ### Try It! 1. Describe the type of association each scatter plot shows. <br> ### Practice Describe the type of association each scatter plot shows. See Example 1 <br> ### Example 2 Understand Correlation How can the relationship between the hours after sunrise, *x*, and the temperature, *y*, be modeled? The data points on the scatter plot approximate a line. Sketch a trend line that best fits the data to determine whether a linear function can model the relationship. <br> - The scatter plot suggests a linear relationship. There is a **positive correlation** between hours after sunrise and the temperature. - When data with a negative association are modeled with a line, there is a **negative correlation**. - If the data do not have an association, they cannot be modeled with a linear function. <br> ### Try It! 2. How can the relationship between the hours after sunset, *x*, and the temperature, *y*, be modeled? If the relationship is modeled with a linear function, describe the correlation between the two data sets. <br> ### For each table, make a scatter plot of the data. Describe the type of association that the scatter plot shows. See Example 2 17. | *x* | *y* | |---|---| | 2 | 4 | | 3 | 4 | | 3 | 6 | | 5 | 8 | | 6 | 10 | 18. | *x* | *y* | |---|---| | 1 | 9 | | 2 | 7 | | 5 | 3 | | 6 | 2 | | 6 | 1 | 19. | *x* | *y* | |---|---| | 3 | 1 | | 4 | 9 | | 7 | 2 | | 8 | 8 | | 10 | 3 | <br> ### Example 3 Write the Equation of a Trend Line What trend line models the data in the scatterplot? A trend line models the data in a scatter plot by showing the general direction of the data. A trend line fits the data as closely as possible. **Step 1** Sketch a trend line for the data. A trend line approximates a balance of points above and below the line. **Step 2** Write the equation of this trend line. Select two points on the trend line to find the slope. - *m* = (500-50) / (16-2) ≈ 32.1 Use the slope and one of the points to write the equation in slope-intercept form. - *y*-50 = 32.1(*x*-2) - *y* = 32.1*x* - 14.2 <br> ### Try It! 3. a. What trend line, in slope-intercept form, models the data from the Example 2 *Try It*? b. Explain why there could be no data points on a trend line, yet the line models the data. <br> ### Example 4 Interpret Trend Lines The table shows the amount of time required to download a 100-megabyte file for various Internet speeds. Assuming the trend continues, how long would it take to download the 100-megabyte file if the Internet speed is 75 kilobytes per second? | Internet Speed (kilobytes/s (KB/s)) | Time to Download 100 Megabytes (min) | |---|---| |35| 6.65| |40| 5.82 | |45| 5.17 | |50| 4.65 | |55| 4.23 | |60| 3.88 | **Step 1** Make a scatter plot of the data and sketch a trend line. **Step 2** Find the equation of the trend line. Select two points on the trend line to find the slope: - *m* = (6-4) / (40-55) ≈ -0.13 Use the slope and one of the points to write the equation for the trend line. - *y* - 6 = -0.13(*x* - 40) - *y* = -0.13*x* + 11.2 <br> **Step 3** Use the equation of the linear model to find the *y*-value that corresponds to *x* = 75. - *y* = -0.13*x* + 11.2 - = -0.13 (75) + 11.2 - = 1.45 The download time would be 1.45 s. <br> ### Try It! 4. What is the *x*-intercept of the trend line? Is that possible in a real-world situation? Explain. <br> ### For each table, make a scatter plot of the data. Draw a trend line and write its equation. See Examples 3 and 4 20. | *x* | *y* | |---|---| |2| 3| |4| 6| |5| 5| |7| 7| |8| 9| |8| 8| 21. | *x* | *y* | |---|---| |3| 9| |5| 8| |5| 6| |6| 5| |6| 6| |8| 3| 22. | *x* | *y* | |---|---| |1| 1| |2| 3| |3| 5| |3| 6| |5| 8| |6| 9| <br> ### Concept Summary Scatter Plots and Trend Lines #### Table **Positive Association** | *x* | *y* | |---|---| |1| 2| |2| 3| |3| 3| |4| 4| |5| 6| |6| 7| |7| 7| **Negative Association** | *x* | *y* | |---|---| |1| 7| |2| 7| |2| 5| |4| 4| |5| 3| |6| 3| |6| 1| #### Graphs **Positive Correlation** *y = *x* + 1 **Negative Correlation** *y = -x + 8*