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Scatter-plot, Best-Fit Line, and Correlation Coefficient

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Definitions: Scatter Diagrams (Scatter Plots) – a graph that shows the relationship between two quantitative variables. Explanatory Variable – predictor variable; plotted to the horizontal axis (x-axis). Response Variable – a value explained by the explanatory variable; plotted on the vertical axis (y-axis).

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Why might we want to see a Scatter Plot? Statisticians and quality control technicians gather data to determine correlations (relationships) between two events (variables). Scatter plots will often show at a glance whether a relationship exists between two sets of data. It will be easy to predict a value based on a graph if there is a relationship present.

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Types of Correlations: Strong Positive Correlation – the values go up from left to right and are linear. Weak Positive Correlation - the values go up from left to right and appear to be linear. Strong Negative Correlation – the values go down from left to right and are linear. Weak Negative Correlation - the values go down from left to right and appear to be linear. No Correlation – no evidence of a line at all.

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Examples of each Plot:

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How to create a Scatter Plot: We will be relying on our TI – 83 Graphing Calculator for this unit! 1 st, get Diagnostics ON, 2 nd catalog. Enter the data in the calculator lists. Place the data in L 1 and L 2. [STAT, #1Edit, type values in] 2 nd Y= button; StatPlot – turn ON; 1 st type is scatterplot. Choose ZOOM #9 ZoomStat.

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Let’s try one: SANDWICH Total Fat (g)Total Calories Grilled Chicken 5300 Hamburger9260 Cheeseburger13320 Quarter Pounder21420 Quarter Pounder with Cheese30530 Big Mac31560 Arch Sandwich Special31550 Arch Special with Bacon34590 Crispy Chicken25500 Fish Fillet28560 Grilled Chicken with Cheese20440

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The Correlation Coefficient: The Correlation Coefficient (r) is measure of the strength of the linear relationship. The values are always between -1 and 1. If r = +/- 1 it is a perfect relationship. The closer r is to +/- 1, the stronger the evidence of a relationship.

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The Correlation Coefficient: If r is close to zero, there is little or no evidence of a relationship. If the correlation coef. is over.90, it is considered very strong. Thus all Correlation Coefficients will be: -1< x < 1

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Salary with a Bachelors and Age AgeSalary (in thousands) 22$ 31 25$ 35 28$ 29.5 28 $ 36 31$ 48 35$ 52 39$ 78 45$ 55.5 49$ 64 55$ 85

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Find the Equation and Correlation Coefficient Place data into L1 and L2 Hit STAT Over to CALC. 4:Linreg(ax+b) Is there a High or Low, Positive or Negative correlation?

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Movie Cost V.Gross (millions) TITLE$ COSTU.S. GROSS 1. Titanic (1997)$200$600.8 2. Waterworld (1995)$175$88.25 3. Armageddon (1998)$140$201.6 4. Lethal Weapon 4 (1998)$140$129.7 5. Godzilla (1998)$125$136 6. Dante's Peak (1997)$116$67.1 7. Star Wars I: Phantom Menace (1999) $110$431 8. Batman and Robin (1997)$110$107 9. Speed 2 (1997)$110$48 10. Tomorrow Never Dies (1997)$110$125.3

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Finding the Line of Best Fit: STAT → CALC #4 LinReg(ax+b) Include the parameters L 1, L 2, Y 1 directly after it. – (Y 1 comes from VARS → YVARS, #Function, Y 1 ) Hit ENTER; the equation of the Best Fit comes up. Simply hit GRAPH to see it with the scatter.

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Using the Best-Fit Line to Predict. Once your line of “Best fit” is drawn on the calculator, it can be used to predict other values. On the TI-83/84: 1)2 nd Calc 2)1:Value 3)x= place in value

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Hypothesis Testing: Is there evidence that there is a relationship between the variables? To test this we will do a TWO-TAILED t-test Using Table 5 for the level of Significance, and d.f. = n – 2; degrees of freedom. Compare the answer from the following formula to determine if you will REJECT a particular correlation.

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TI-83/84 HELP TI Regression Models Rules for a Model Diagnostics On Correlation Coefficient Correlation Not Causation Residuals and Least Squares Graphing Residuals Linear Regression Linear Regression w/ Bio Data Exponential Regression Logarithmic Regression Power Regression

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