Five-Minute Check (over Lesson 2–4)

CCSS

Then/Now

New Vocabulary

Key Concept: Scatter Plots

Example 1: Real-World Example: Use a Scatter Plot and Prediction Equation

Example 2: Real-World Example: Regression Line

Content Standards

F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Mathematical Practices

4 Model with mathematics.

5 Use appropriate tools strategically.

You wrote linear equations.

• Use scatter plots and prediction equations.

• Model data using lines of regression.

• bivariate data • regression line

• correlation coefficient• scatter plot

• dot plot

• positive correlation

• negative correlation

• line of fit

• prediction equation

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.

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.

Answer:The data show a strong negative correlation.

The points (3, 18) and (15, 13) appear to represent the data well. Draw a line through these two points.

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.

Use a Scatter Plot and Prediction Equation

Point-slope form

Substitute.

Distributive Property

Simplify.

Answer: One prediction equation is

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.

Answer: The model predicts that the percent in 2010 should be about 8.83%.

x = 25

Prediction equation

Simplify.

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.

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?

A. B.

C. D.

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.

A. 83%

B. 87%

C. 90%

D. 95%

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.

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.

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.

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 selectingLinReg(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.

Regression Line

Step 3 Graph the regression equation.

Copy the equation to the Y= list and graph.

Notice that the regression line comes closeto most of the data points. As the

correlationcoefficient indicated, the line fits the data

well.

Regression Line

Step 4 Predict using the function.

Find y when x = 2015. Use VALUE on theCALC menu. Reset the window size toaccommodate the x-value of 2015.

Answer: According to the function, the medianincome in 2015 will be about $69,220.

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; about186 seconds

D. y = –14.95x + 30,233.25; about

99 seconds

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.