# 1.5 Scatter Plots and Least-Squares Lines

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1.5 Scatter Plots and Least-Squares Lines
Objectives: Create a scatter plot and draw an informal inference about any correlation between the variables. Use a graphing calculator to find an equation for the least-squares line and use it to make predictions or estimates. Standard: C Construct and apply mathematical models, including lines and curves of best fit, to estimate values of related quantities.

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

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

Finding the Least-Squares Line
A scatter plot can help you see patterns in data involving 2 variables. If you think there maybe a linear correlation between the variables, you can use a calculator to find a linear-regression line, also called a least-squares line, that best fits the data. STAT (L1, L2) STAT CALC LIN REGRESSION

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

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

Correlation and Prediction
The correlation coefficient, denoted by r, indicates how closely the data points cluster around the least-squares line. The correlation coefficient can vary from -1, which is a perfect fit for a negative correlation, to +1, which is a perfect fit for a positive correlation.

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

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

Ex. 2 Olympic Freestyle Swimming Event Data

PSSA WARM-UP QUESTION 
Standard B Use Technology to analyze data. HINT: Use your calculator. STAT EDIT L1, L2 type in given informationSTAT CALC to compute line of best fit & correlation (r). Each day last week, the manager of a movie theater recorded how many people attended a movie. He also recorded how many bags of popcorn were sold. How can he graph the information and determine if there is a 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 represent patterns in data sets
Look at the table below. Do you see any patterns in the data? Can the data be represented by an equation? Can the data be shown on a graph? x y 1 -2 2 -1 3 4

Homework Integrated Algebra II- Section 1.5 Level A Honors Algebra II- Section 1.5 Level B