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

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).

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.

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.

Examples of each Plot:

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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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.

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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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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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.

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

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.

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