1.5 Scatter Plots 1.5 Scatter Plots and Least-and Least-

Squares LinesSquares LinesObjectivesObjectives:: Create a scatter plot and draw an Create a scatter plot and draw an informal informal

inference about any correlation inference about any correlation between the between the

variables.variables.

Use a graphing calculator to find an Use a graphing calculator to find an equation for the equation for the

least-squares line and use it to least-squares line and use it to make predictions or make predictions or

estimates.estimates.

Standard:Standard: 2.2.11.C Construct and apply 2.2.11.C Construct and apply mathematical models, mathematical models,

including lines and curves of best fit, including lines and curves of best fit, to estimate to estimate

values of related quantities.values of related quantities.

In many real-world problems, In many real-world problems, you will find data that relate you will find data that relate 2 variables (and many times 2 variables (and many times more than two variables) more than two variables) such as time and distance or such as time and distance or age and height. You can view age and height. You can view the relationship between 2 the relationship between 2 variables with a variables with a scatter plotscatter plot..

There is a correlation between 2 There is a correlation between 2 variables when there appears to be a variables when there appears to be a line around which the data points line around which the data points cluster. The diagrams below show the cluster. The diagrams below show the 3 possible correlations.3 possible correlations.

Finding the Least-Squares Finding the Least-Squares LineLine

A scatter plot can help you see A scatter plot can help you see patterns in data involving 2 variables. patterns in data involving 2 variables. If you think there maybe a linear If you think there maybe a linear correlation between the variables, you correlation between the variables, you can use a calculator to find a can use a calculator to find a linear-linear-regression lineregression line, also called a , also called a least-least-squares linesquares line, that best fits the data., that best fits the data.

STAT (L1, L2)STAT (L1, L2) STAT CALC LIN REGRESSIONSTAT CALC LIN REGRESSION

The graph below shows the vertical The graph below shows the vertical distance from each point in a scatter distance from each point in a scatter plot to a fitted line. The fit of a plot to a fitted line. The fit of a least-least-squares linesquares line is based on is based on minimizing minimizing these vertical distances for a data these vertical distances for a data set.set.

Ex. 1 Create a scatter plot for the data Ex. 1 Create a scatter plot for the data shown below. Describe the correlation. shown below. Describe the correlation. Then find and graph an equation for the Then find and graph an equation for the least-squares line.least-squares line.

1.Create the

scatter plot.

2. Describe the correlation.

Correlation and Correlation and PredictionPrediction

The The correlation coefficientcorrelation coefficient, denoted , denoted by by rr, indicates how closely the data , indicates how closely the data points cluster around the least-points cluster around the least-squares line.squares line.

The correlation coefficient can vary The correlation coefficient can vary from -1, which is a perfect fit for a from -1, which is a perfect fit for a negative correlation, to +1, which is negative correlation, to +1, which is a perfect fit for a positive a perfect fit for a positive correlation.correlation.

The closer the correlation coefficient is to The closer the correlation coefficient is to -1 or +1, the better the least-squares line -1 or +1, the better the least-squares line

fits the data.fits the data.

Ex. 2 The winning times for the Ex. 2 The winning times for the men’s Olympic 1500-meter men’s Olympic 1500-meter freestyle swimming event are freestyle swimming event are given in the table. Notice that given in the table. Notice that there is not a winning time there is not a winning time recorded for the year 1940 (the recorded for the year 1940 (the Olympic games were not held Olympic games were not held during World War II). Estimate during World War II). Estimate what the winning time for this what the winning time for this event could have been in 1940.event could have been in 1940.

Ex. 2 Olympic Freestyle Swimming Ex. 2 Olympic Freestyle Swimming Event DataEvent Data

PSSA WARM-UP QUESTION PSSA WARM-UP QUESTION Standard 2.6.11 B Use Technology to Standard 2.6.11 B Use Technology to

analyze data.analyze data.

HINT: Use your calculator. STAT EDIT L1, HINT: Use your calculator. STAT EDIT L1, L2 type in given informationSTAT CALC to L2 type in given informationSTAT CALC to compute line of best fit & correlation (r).compute line of best fit & correlation (r).

Each day last week, the manager of a movie Each day last week, the manager of a movie theater recorded how many people attended theater recorded how many people attended a movie. He also recorded how many bags a movie. He also recorded how many bags of popcorn were sold. How can he graph of popcorn were sold. How can he graph the information and determine if there is a the information and determine if there is a correlation between these two sets of data?correlation between these two sets of data?

Number of people attending a movie

Number of bags of popcorn sold

175 76

100 43

213 101

249 133

362 197

331 185

250 148

y = .62x – 23.46

r = .99

Standard 2.8.11 A Identify and Standard 2.8.11 A Identify and represent patterns in data setsrepresent patterns in data sets

Look at the table below. Do you see Look at the table below. Do you see any patterns in the data? Can the any patterns in the data? Can the

data be represented by an equation? data be represented by an equation? Can the data be shown on a graph?Can the data be shown on a graph?

x y

1 -2

2 -1

3 0

4 1

HomeworkHomework

Integrated Algebra II- Section Integrated Algebra II- Section 1.5 Level A1.5 Level A

Honors Algebra II- Section 1.5 Honors Algebra II- Section 1.5 Level BLevel B