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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.

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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.

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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.

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**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

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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.

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**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.

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**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.

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

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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.

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

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**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?

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**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

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**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

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

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