Chapter 10r

1
Chapter 10r

• Linear Regression Revisited

2
Correlation

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

3
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.

4
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.

5
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 doesnt change if center or scale of
  original data is changed.
• Depends only on the z-score.

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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

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Computing Correlation

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

8
Hypothesis Testing for ? (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 ? 0
• No correlation between x and y variables
• Ha ? ? 0
• Significant correlation between the variables

9
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 a 0.01.

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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
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Example - Correlation HT

• Step 1
• Ho ? 0
• Ha ? ? 0
• Step 2
• a 0.01
• Step 3 (note d.f.)
• t(n 2) t(9 2 7) t(7)

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Example - Correlation HT

• Step 4
• Enter the lists into your calculator.
• STAT -gt TESTS -gt 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

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Example - Correlation HT

• Step 5
• 0 .864 gt 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.