Linear Correlation To accompany Hawkes lesson 12.1 Original content by D.R.S.

Linear Correlation

Visual Assessment of Correlation A Scatter Plot of the (x,y) ordered pairs in your sample data can give you a notion of what the relationship might be. Do the points line up in a straight line? – Or in sort-of a straight-ish line? – Or all over the place with no apparent relationship between x and y? – Or in a curvy curve pattern?

Types of Relationships: HAWKES LEARNING SYSTEMS math courseware specialists Regression, Inference, and Model Building 12.1 Scatter Plots and Correlation Strong Linear Relationship Non-Linear Relationship No Relationship Weak Linear Relationship

The Horses Example Some horses were measured – Height (in hands?), Girth (inches), Length (inches), Weight (pounds) – Put these data values into your TI-84 lists L 1, L 2, L 3, L 4. Original data source and idea for this problem is “Elementary Statistics” by Johnson & Kuby, 10 th Edition, © Brooks-Cole-Thomson, Page 702.

Question: “Is Girth related to Weight?” We wonder: is the girth of a horse related to its weight? Significantly so? ρ (Greek letter rho) is the population parameter for the Correlation Coefficient r (our alphabet’s letter r) is the sample statistic for the Correlation Coefficient We use our sample r to estimate the population’s parameter ρ

The Correlation Coefficient

Pearson Correlation Coefficient,  – the parameter that measures the strength of a linear relationship for the population. Correlation Coefficient, r – measures how strongly one variable is linearly dependent upon the other for a sample. Correlation coefficient: HAWKES LEARNING SYSTEMS math courseware specialists Regression, Inference, and Model Building 12.1 Scatter Plots and Correlation When calculating the correlation coefficient, round your answers to three decimal places.

HAWKES LEARNING SYSTEMS math courseware specialists Regression, Inference, and Model Building 12.1 Scatter Plots and Correlation –1 ≤ r ≤ 1 Close to –1 means a strong negative correlation. Close to 0 means no correlation. Close to 1 means a strong positive correlation.

“Is a horse’s Girth significantly correlated to its Weight?”

LinRegTTest inputs Here are the inputs: Xlist and Ylist – where you put the data – Shortcut: 2 ND 2 puts L 2 Freq: 1 (unless…)

LinRegTTest Outputs, first screen t= the t statistic value for this test (the formula is in the book)

LinRegTTest Outputs, second screen b later, for Regression s much later, for advanced Regression

Making the Decision

How did the calculator get r and r 2 ? Here is the awful formula:

How did the calculator compute t ? Here is the awful formula:

Another test: Girth and Length Is there a significant relationship between a horse’s girth and length? What do you expect? – Think about people: do you expect a significant relationship between waist size and height?

TI-84 Inputs and Outputs for the Girth and Length question Inputs (Data already in lists) Outputs First screen Second screen

Girth and Length conclusions ConclusionsOutputs First screen Second screen

An extra problem type in Hawkes

Determine the significance: HAWKES LEARNING SYSTEMS math courseware specialists Regression, Inference, and Model Building 12.1 Scatter Plots and Correlation a. r  0.52, n  19,   0.05 r   0.456, Yes b. r  0.52, n  19,   0.01 r   0.575, No c. r  –0.44, n  35,   0.01 r   0.430, Yes Determine whether the following values of r are statistically significant.