Study Resources For Algebra I. Unit 1C Analyzing Data Sets for Two Quantitative Variables
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1 Study Resources For Algebra I Unit 1C Analyzing Data Sets for Two Quantitative Variables This unit explores linear functions as they apply to data analysis of scatter plots. Information compiled and written by Ellen Mangels, Cockeysville Middle School July 2014 Algebra I Page 1 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
2 Topics: Each of the topics listed below link directly to their page. Scatterplots o Outliers o Clusters o Positive and Negative Correlation Line of Best Fit o Determine the Equation for the Line of Best Fit using 2 points o Interpolate o Extrapolate o Least Squares Regression Line o To see the Line of Best Fit on the Calculator o Correlation Coefficient (r) o Determination Coefficient (r 2 ) Residuals o Residual Plot o Constructing a Residual Plot o Residual Plot for Ice Cream Data Correlation vs. Causation Algebra I Page 2 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
3 Scatterplots A graph of plotted points that show the relationship between two sets of data. In this example, each dot represents one person's weight versus their height. Example: The local ice cream shop keeps track of how much ice cream they sell versus the noon temperature on that day. Here are their figures for the last 12 days: Remember that body temperature in Farhenheite is 98.6 F which is 37 C (Source: Video Scatter Plots (CC.9-12.N.Q.1) Video Scatter Plots (CC.9-12.S.ID.6a) Algebra I Page 3 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
4 Outliers "Outliers" are values that "lie outside" the other values. (Source: outlier (Source: Algebra I Page 4 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
5 Clusters (Source: (Source: Algebra I Page 5 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
6 Positive and Negative Correlation Scatter plots show how much one variable is affected by another. The relationship between two variables is called their correlation. (Source: Algebra I Page 6 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
7 Line of Best Fit Using the ice cream data again, draw a line that best fits the data. Try to have the line as close as possible to all points, and as many points above the line as below. (Source: Algebra I Page 7 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
8 Determine the Equation for the Line of Best Fit using 2 points Let's estimate two points on the line near actual values: (12, $180) and (25, $610) Notice these are not actual data values, but estimates of points on the line that are close to the actual values. Calculate the slope using the two points. Put that slope and the point (12, $180) into the "point-slope" formula: (Source: Algebra I Page 8 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
9 Interpolate Interpolation is where we find a value inside our set of data points. Here we use linear interpolation to estimate the sales at 21 C. Select an x value, 21 C, and estimate the y value on the graph, $480. Use the equation for the line of best fit to interpolate the data. y = 33x 216 y = 33 ( 21) 216 y = = 477 substitute the x value into the equation $477 is the answer from the equation. (Source: Algebra I Page 9 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
10 Extrapolate Extrapolation is where we find a value outside our set of data points. Here we use linear extrapolation to estimate the sales at 29 C (which is higher than any value we have). Be careful: Extrapolation can give misleading results because we are in "uncharted territory". Select an x value, 29 C, and estimate the y value on the graph, $750. Use the equation for the line of best fit to interpolate the data. y = 33x 216 y = 33 ( 29) 216 y = = 741 substitute the x value into the equation $741 is the answer from the equation. Don't use extrapolation too far! What sales would you expect at 0? y = 33(0) 216 = $216 Hmmm... Negative $216? We have extrapolated too far! (Source: Algebra I Page 10 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
11 Least Squares Regression Line The Least Squares Regression Line is the graphing calculator s version of the Line of Best Fit. 1. Type the data into the graphing calculator. Press STAT 2. Type the first column of data into L1 and the second column into L2. Select Edit: You may notice the last pair of data values seems to be missing. The calculator screen can only show a limited number of values at one time. 3. Calculate the equation. 4. The equation is: Press STAT Arrow over to CALC, then select 4: LinReg Which can be rounded and written as: y = 30x You many notice the regression equation values from the calculator are a bit different than the line of best fit equation we calculated using two points. The calculator uses the least squares regression formula. You do not need to know this formula for this course. Just know that the equation from the calculator is much more accurate for the line of best fit than an equation using only two points. For more details on using this formula, see page 528 in chapter 9 of your Carnegie Algebra book. y = ax + b Algebra I Page 11 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
12 To see the line of best fit on the calculator Enter the data in L1 and L2. (see steps) Set up the scatter plot o Press 2 nd Y= to access STAT PLOT o Select Plot1 Make sure On is hightlighted Make sure the first Type is highlighted o Press ZOOM 9 o To see the line on the scatter plot, type the equation in the Y= screen and press GRAPH. You can either use the rounded off version of the equation or use the following steps to insert the complete equation into Y=. Y= VARS 5: Statistics EQ ENTER GRAPH Algebra I Page 12 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
13 Correlation Coefficient (r) We already know that the correlation between the two variables can be postive, negative, or have no correlation at all. The correlation between the two variables can also be measured using the correlation coefficient. Before you calculate the Least Squares Regression equation on your graphing calculator, turn the diagnostics on. 2 nd CATALOG (the zero key) ALPHA D will skip down to the commands that begin with D Select DiagnosticOn Now when you calculate the equation for the line of best fit, you will see additional information. Look at the r value. If the r value is close to +1, the data has a strong postive correlation. If the r value is close to 0, the data has no correlation. If the r value is close to -1, the data has a strong negative correlation. With the ice cream data, the correlation coefficient, r = 0.96, is close to 1. There is a strong positive correlation between the temperature and ice cream sales. As the temperature goes up, more people buy ice cream. Algebra I Page 13 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
14 When the slope of the line in the plot is negative, the correlation is negative; and vice versa. The strongest correlations (r = 1.0 and r = -1.0) occur when data points fall exactly on a straight line. The correlation becomes weaker as the data points become more scattered. If the data points fall in a random pattern, the correlation is equal to zero. Correlation is affected by outliers. Compare the first scatterplot with the last scatterplot. The single outlier in the last plot greatly reduces the correlation (from 1.00 to 0.71). (Source: Algebra I Page 14 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
15 Determination Coefficient (r 2 ) The coefficient of determination is a measure of how well the regression line represents the data. (Source: With a value of 0 to 1, the coefficient of determination is calculated as the square of the correlation coefficient (r) between the sample and predicted data. The coefficient of determination shows how well a regression line fits the data. A value of 1 means every point on the regression line fits the data; a value of 0.5 means only half of the variation is explained by the regression line. The coefficient of determination is also commonly used to show how accurately a regression line can predict future outcomes. (Source: With the ice cream data, the coefficient of determination, r 2 = 0.92, is close to 1. This value shows that the equation, y = 30x 159, does a good job representing the data. Algebra I Page 15 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
16 Residuals The difference between the observed value of the dependent variable (y) and the predicted value (ŷ) is called the residual (e). Each data point has one residual. Residual = Observed value - Predicted value e = y ŷ Note: the symbol ŷ is read as y hat. Both the sum and the mean of the residuals are equal to zero. That is, Σ e = 0 and e = 0. Note: the symbol Σ represents the sum of a set of numbers. Note: when a variable is written with a line above it, such as e, it represents the mean of the set of numbers. (Source: Residual Plot A residual plot is a graph that shows the residuals on the vertical axis and the independent variable on the horizontal axis. If the points in a residual plot are randomly dispersed around the horizontal axis, a linear regression model is appropriate for the data; otherwise, a non-linear model is more appropriate. Below, the residual plots show three typical patterns. The first plot shows a random pattern, indicating a good fit for a linear model. The other plot patterns are non-random (U-shaped and inverted U), suggesting a better fit for a non-linear model. (Source: Algebra I Page 16 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
17 Constructing a Residual Plot Video Least Square Regression Line and Residual Plots Video Residual Plots Algebra I Page 17 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
18 Residual Plot for Ice Cream Data 1. Determine the Least Squares Regression Equation. 2. Go to the home screen 3. Press 2 ND STAT (to see the list), select 7:RESID, ENTER You will see a list of numbers beginning with a bracket: { The list of residual values will be easier to read in a list format. Store the list in L3. STO 2 ND L3 ENTER 5. Look at the list of Residual Values: STAT EDIT This list of values shows how far above or below the line of best fit that the original data points lie. STO is the key above ON L3 is found by pressing 2 ND and the number Set up a scatter plot using L1 and L3 Residual Plot Algebra I Page 18 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
19 7. You should notice that the residual graph looks like our scatterplot with the line of best fit turned horizontally. The fact that the residual graph is very scattered tells us that a linear regression was a good choice to represent this data. Residual Plot Linear Regression Line with Original Data Later in this course we will look at other regression equations such as Quadratic Regression and Exponential Regression. To see some of the other options for regression equations, look at STAT CALC. Options 5, 6, 7, 9, and 0 are other types of regression equations. Option #8 is the same as the Linear Regression option #4 written in a different way. Algebra I Page 19 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
20 Correlation vs. Causation Correlation says that two ideas are related. Causation says that one event causes the other event. Example: Studies show that students who put more time into completing homework do better on tests. There is definitely a correlation here and it would seem to make sense to then say that extra time studying causes a student to score better on a test. But be careful, just because two ideas are related, it doesn t always mean that one event causes the other event. (Source: Algebra I Page 20 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
21 BEWARE! Just because two ideas are related, doesn t mean that one event causes the other. Here are some examples: (Source: Algebra I Page 21 Unit 1C. Analyzing Data Sets for Two Quantitative Variables
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