Let X and Y be two normally distributed random variables satisfying the equality of variance

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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

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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.

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.

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 attributed

to the linear relationship between the number of commercials

and water consumption.

2. 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.

3. 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.

POINT: The sample correlation coefficient

(the Pearson correlation coefficient)

can be calculated directly using the

formula: