Can you use scatter plots and prediction equations? Can you model data using lines of regression? Then/Now

correlation coefficient scatter plot dot plot positive correlation bivariate data regression line correlation coefficient scatter plot dot plot positive correlation negative correlation line of fit prediction equation Vocabulary

Concept

Use a Scatter Plot and Prediction Equation A. EDUCATION The table below shows the approximate percent of students who sent applications to two colleges in various years since 1985. Make a scatter plot of the data and draw a line of fit. Describe the correlation. Example 1A

Answer: The data show a strong negative correlation. Use a Scatter Plot and Prediction Equation Graph the data as ordered pairs, with the number of years since 1985 on the horizontal axis and the percentage on the vertical axis. The points (3, 18) and (15, 13) appear to represent the data well. Draw a line through these two points. Answer: The data show a strong negative correlation. Example 1A

B. Use two ordered pairs to write a prediction equation. Use a Scatter Plot and Prediction Equation B. Use two ordered pairs to write a prediction equation. Find an equation of the line through (3, 18) and (15, 13). Begin by finding the slope. Slope formula Substitute. Simplify. Example 1B

Use a Scatter Plot and Prediction Equation C. Predict the percent of students who will send applications to two colleges in 2010. The year 2010 is 25 years after 1985, so use the prediction equation to find the value of y when x = 25. Example 1C

D. How accurate is this prediction? Use a Scatter Plot and Prediction Equation D. How accurate is this prediction? Answer: Except for the point at (6, 15), the line fits the data well, so the prediction value should be fairly accurate. Example 1D

A. SAFETY The table shows the approximate percent of drivers who wear seat belts in various years since 1994. Which shows the best line of fit for the data? Example 1A

A. B. C. D. Example 1A

B. The scatter plot shows the approximate percent of drivers who wear seat belts in various years since 1994. What is a good prediction equation for this data? Use the points (6, 71) and (12, 81). A. B. C. D. Example 1B

C. The equation represents the approximate percent of drivers y who wear seat belts in various years x since 1994. Predict the percent of drivers who will be wearing seat belts in 2010. A. 83% B. 87% C. 90% D. 95% Example 1C

A. There are no outliers so it fits very well. D. How accurate is the prediction about the percent of drivers who will wear seat belts in 2010? A. There are no outliers so it fits very well. B. Except for the one outlier the line fits the data very well. C. There are so many outliers that the equation does not fit very well. D. There is no way to tell. Example 1D

Regression Line INCOME The table shows the median income of U.S. families for the period 1970–2002. Use a graphing calculator to make a scatter plot of the data. Find an equation for and graph a line of regression. Then use the equation to predict the median income in 2015. Example 2

Step 1 Make a scatter plot. Regression Line Step 1 Make a scatter plot. Enter the years in L1 and the income in L2. Set the viewing window to fit the data. Use STAT PLOT to graph the scatter plot. Step 2 Find the equation of the line of regression. Find the regression equation by selecting LinReg(ax + b) on the STAT CALC menu. The regression equation is about y = 1349.87x – 2,650,768.34. The slope indicates that the income increases at a rate of about 1350 people per year. The correlation coefficient r is 0.997, which is very close to 1. So, the data fit the regression line very well. Example 2

Step 3 Graph the regression equation. Regression Line Step 3 Graph the regression equation. Copy the equation to the Y= list and graph. Notice that the regression line comes close to most of the data points. As the correlation coefficient indicated, the line fits the data well. Example 2

Step 4 Predict using the function. Regression Line Step 4 Predict using the function. Find y when x = 2015. Use VALUE on the CALC menu. Reset the window size to accommodate the x-value of 2015. Answer: According to the function, the median income in 2015 will be about $69,220. Example 2

The table shows the winning times for an annual dirt bike race for the period 2000–2008. Use a graphing calculator to make a scatter plot of the data. Find and graph a line of regression. Then use the function to predict the winning time in 2015. A. y = –15.75x + 31,890.25; about 154 seconds B. y = –14.75x + 29,825.67; about 104 seconds C. y = –14.6x + 29,604.72; about 186 seconds D. y = –14.95x + 30,233.25; about 99 seconds Example 2