Chapter 10rLinear Regression

Revisited

Correlation

• A numerical measure of the direction and strength of a linear association.– Like standard deviation was a numerical

measure of spread.

Correlation Coefficient - Facts

• The correlation coefficient is denoted by the letter r.– Safe to assume r is always correlation in this

class.

• The sign of the correlation coefficient give the direction of the association.– Positive is positive and negative is negative.

Correlation Coefficient - Facts

• The correlation coefficient is always between -1 and +1.– A low correlation is closer to zero and strong

closer to either -1 or +1.• Ex. r = 0.21 or -0.21 (weak), r = -0.98 or

0.98(strong).

– If correlation is equal to exactly -1 or +1 then the data points all fall on an exact straight line.

Correlation Coefficient - Facts

• Correlation coefficient has no units.– The correlation is just that the correlation.

• Learn it on its own scale, not as a percentage.

• Correlation doesn’t change if center or scale of original data is changed.– Depends only on the z-score.

What is STRONG/WEAK?

• Again a judgment call.

• Rule of thumb:– 0 to +/- 0.5 Weak– +/- 0.5 to +/- 0.80 Moderate– +/- 0.8 to +/- 1.0 Strong

Computing Correlation

• Use your technology to help you find this number.– Calculator

Hypothesis Testing for ρ (rho)

• Before we do a linear regression we can conclude whether or not there is a significant linear relationship between the variables or if r is due to chance.

• In order to do this we use a t Test for the correlation coefficient– Ho: ρ = 0

• No correlation between x and y variables

– Ha: ρ ≠ 0• Significant correlation between the variables

Example - Correlation HT

• Data was obtained in a study on the number of hours that nine people exercise each week and the amount of milk (in ounces) each person consumes each week. Test the significance of the correlation coefficient at α = 0.01.

Example - Correlation HT

Weekly Exercise Hours (X) Amount of Milk Consumed (Y)

3 48

0 8

2 32

5 64

8 10

5 32

10 56

2 72

1 48

Example - Correlation HT

• Step 1– Ho: ρ = 0– Ha: ρ ≠ 0

• Step 2– α = 0.01

• Step 3 (note d.f.)– t(n – 2) = t(9 – 2 = 7) = t(7)

Example - Correlation HT

• Step 4– Enter the lists into your calculator.– STAT -> TESTS -> LinRegTTest

• Make sure the right lists are there for X and Y• Check appropriate Ha (should be not equal)• Calculate

– Report the r value = 0.067

– t(7) = 0.178

– P-value = 0.864

Example - Correlation HT

• Step 5– 0 .864 > 0.01– DO NOT REJECT Ho

• Step 6– There is not significant evidence to suggest a

correlation between the variables.

• This means that you would probably not do the linear regression analysis on these variables.