© Buddy Freeman, 2015 Let X and Y be two normally distributed random variables satisfying the...
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© Buddy Freeman, 2015 et X and Y be two normally distributed random ariables satisfying the equality of variance ssumption both ways. or clarity let us examine this concept further. e assume that X is a normally distributed random ariable (think bell curve). Pick any two distinct alues of X, call them X 1 and X 2 . The variance of he population of Y values that correspond to X 1 ust be equal to the variance of the population of values that correspond to X 2 . Likewise, we ssume that Y is a normally distributed random ariable (think bell curve again). If you pick any wo distinct values of Y, call them Y 1 and Y 2 , then he variance of the population of X values that orrespond to Y 1 must be equal to the variance of the opulation of X values that correspond to Y .
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Transcript of © Buddy Freeman, 2015 Let X and Y be two normally distributed random variables satisfying the...
© Buddy Freeman, 2015
Let X and Y be two normally distributed random variables satisfying the equality of variance assumption both ways. For clarity let us examine this concept further. We assume that X is a normally distributed random variable (think bell curve). Pick any two distinct values of X, call them X1 and X2. The variance of
the population of Y values that correspond to X1
must be equal to the variance of the population of Y values that correspond to X2. Likewise, we
assume that Y is a normally distributed random variable (think bell curve again). If you pick any two distinct values of Y, call them Y1 and Y2, then
the variance of the population of X values that correspond to Y1 must be equal to the variance of the
population of X values that correspond to Y2.
© Buddy Freeman, 2015
1. Calculate the sample coefficient of determination (r2). This value (often expressed as a percentage) represents the proportion of the variation in Y that is attributable to the relationship expressed between X and Y in the regression model.
SSTSSR
r 2
© Buddy Freeman, 2015
One Way to Calculate r2
What is the interpretation of the coefficient of determination (r2)?
92.89% of the variation in water consumption may be attributedto the linear relationship between the number of commercialsand water consumption.
9289.050000000
5651000)&cov( 22
2
SSY
SSXYX
SST
SSRr
© Buddy Freeman, 2015
2. Calculate the sample correlation coefficient (r). with the sign of the cov(X&Y). REMEMBER: correlation coefficients arealways between minus 1 and plus 1. Minus 1is perfect negative correlation and plus 1 is perfect positive correlation.
2rr
© Buddy Freeman, 2015
3. To answer the QUESTION: "Does a linearrelationship exist between X and Y at acertain level of significance?" we can use the test statistic: NOTE: Algebraically, this equation is exactly equal to the t-test used in regression analysis.
)2()1( 22
nr
rtn
© Buddy Freeman, 2015
POINT: The sample correlation coefficient (the Pearson correlation coefficient) can be calculated directly using the formula:
SSYSSX
YXr
&cov
