Linear Regression To accompany Hawkes lesson 12.2 Original content by D.R.S.
Linear Correlation To accompany Hawkes lesson 12.1 Original content by D.R.S.
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Transcript of Linear Correlation To accompany Hawkes lesson 12.1 Original content by D.R.S.
![Page 1: Linear Correlation To accompany Hawkes lesson 12.1 Original content by D.R.S.](https://reader030.fdocuments.in/reader030/viewer/2022032705/56649dbd5503460f94aafa8d/html5/thumbnails/1.jpg)
Linear Correlation
To accompany Hawkes lesson 12.1Original content by D.R.S.
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Linear Correlation
• Input: A bunch of data points• Take two measurements from each
member in your sample.• Example: Weight and Blood Pressure
• Output: “There is / is not a significant linear relationship between and .
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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?
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Types of Relationships:
HAWKES LEARNING SYSTEMS
math courseware specialists
Regression, Inference, and Model Building
12.1 Scatter Plots and Correlation
Strong Linear
Relationship
Non-LinearRelationship
NoRelationship
Weak LinearRelationship
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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 L1, L2, L3, L4.
• Original data source and idea for this problem is “Elementary Statistics” by Johnson & Kuby, 10th Edition, © Brooks-Cole-Thomson, Page 702.
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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
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The Correlation Coefficient
• If , it means there is absolutely no relationship between Girth () and Height ()
• If , it means there is perfect positive correlation between girth and height.
• If , it means there is perfect negative correlation between girth and height.
• There’s an awful formula to compute .• Remember: sample estimates population .
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• 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.
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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.
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Hypothesis Test for significant
• Null Hypothesis: “No relationship”• Alternative:
“But there IS a significant relationship!”• There’s some level of significance specified in
advance, like or • It involves calculating a value and finding “what
is the -value of this ?”• And if -value < , reject the null hypothesis
– If so, then we say “Yes, significant relationship!”
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Hypothesis Test for significant
• Usually we do this two-tailed test:– Null Hypothesis : “No relationship”– Alternative Hypothesis: , “There is a significant
linear relationship.”• Be aware of a couple one-tailed variations:
– Test for significant POSITIVE correlation only:using and
– Test for significant NEGATIVE correlation only:using and
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“Is a horse’s Girth significantly correlated to its Weight?”
• Here’s how we do the Hypothesis Test for
• Let’s suppose that level of significance , requiring strong evidence.
• STATS, TESTS, F:LinRegTTest– Shortcut instead of scrolling: ALPHA F directly.– But it might be option E on TI-83/Plus.
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LinRegTTest inputs
• Here are the inputs:
• Xlist and Ylist – where you put the data– Shortcut: 2ND 2 puts L2
• Freq: 1 (unless…)
• β & : ≠ 0– This is the Alternative
Hypothesis
• RegEq: VARS, right arrow to Y-VARS, 1, 1– Just put it in for later
• Highlight “Calculate”• Press ENTER
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LinRegTTest Outputs, first screen
•
• t= the t statistic value for this test (the formula is in the book)
• p = the p-value for this t test statistic
• in this kind of a test• later – for regression
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LinRegTTest Outputs, second screen
• b later, for Regression• s much later, for
advanced Regression
• r2 = how much of the output variable (weight) is explained by the input variable (girth)
• r = the correlation coefficient for the sample– Close to – strong
positive relationship– Or – strong negative
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Making the Decision
• We will use the p-value method.• Compare the -value (as calculated by the TI-84)
to the Level Of Significance value for this experiment.
• In this example, (it was chosen during the design of the experiment) and the calculator computed p=5.3448432E-5 ,
• Since , reject the null hypothesis. There IS a significant linear rel. between girth and height.
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How did the calculator get r and r2?
• Here is the awful formula:
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How did the calculator compute t ?
• Here is the awful formula:
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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?
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TI-84 Inputs and Outputsfor the Girth and Length question
Inputs• (Data already in lists)
OutputsFirst screen
Second screen
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Girth and Length conclusions
Conclusions• What does the tell you in
this particular case?
• At the level of significance, is there a significant relationship between a horse’s girth and his length?
OutputsFirst screen
Second screen
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An extra problem type in Hawkes
• They tell you three pieces of information:– The Level of Significance chosen, – The correlation coefficient calculated, – The sample size,
• They ask you “Is this significant?”• Use Table I on Page 777 to determine this.
Lookup in column and row.• If , then Yes, significant.
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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, a = 0.05
r = 0.456, Yes
b. r = 0.52, n = 19, a = 0.01
r = 0.575, No
c. r = –0.44, n = 35, a = 0.01
r = 0.430, Yes
Determine whether the following values of r are statistically significant.