© Buddy Freeman, 2015 Let X and Y be two normally distributed random variables satisfying the equality of variance assumption both ways. For clarity let.

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Presentation transcript:

© 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 X 1 and X 2. The variance of the population of Y values that correspond to X 1 must be equal to the variance of the population of Y values that correspond to X 2. 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 Y 1 and Y 2, then the variance of the population of X values that correspond to Y 1 must be equal to the variance of the population of X values that correspond to Y 2.

© Buddy Freeman, Calculate the sample coefficient of determination (r 2 ). 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.

© Buddy Freeman, 2015 One Way to Calculate r 2 What is the interpretation of the coefficient of determination (r 2 )? 92.89% of the variation in water consumption may be attributed to the linear relationship between the number of commercials and water consumption.

© Buddy Freeman, Calculate the sample correlation coefficient (r). with the sign of the cov(X&Y). REMEMBER: correlation coefficients are always between minus 1 and plus 1. Minus 1 is perfect negative correlation and plus 1 is perfect positive correlation.

© Buddy Freeman, To answer the QUESTION: "Does a linear relationship exist between X and Y at a certain level of significance?" we can use the test statistic: NOTE: Algebraically, this equation is exactly equal to the t-test used in regression analysis.

© Buddy Freeman, 2015 POINT: The sample correlation coefficient (the Pearson correlation coefficient) can be calculated directly using the formula: