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The t Statistic:
used in place of z-Score when population standard…
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The value 
is called the coefficient of determination because it measures the proportion of variability in one variable that can be determined from the relationship with the other variable. A correlation of r= 80 (or −0.80), for example, means that = 0.64 (or 64%) of the variability in the Y scores can be predicted from the relationship with X.
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Hypothesis tests with correlations: (there is no population correlation); (There is a real correlation).
Directional Hypothesis tests with correlations: (the population correlation is not positive); (the population correlation is positive).
The statistical technique for finding the best-fitting straight line for a set of data is called regression, and the resulting straight line is called the regression line. 1) Compute distance between actual data point and the point on the line: . To determine the total error between the line and the data, we add the squared errors for all of the data points: .
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Regression linear Equation for Y: The best-fitting line is the one that has the smallest total squared error. Linear equation is: . To find b: or . To find a:
Standard Error of Estimate: gives a measure of the standard distance between the predicted Y values on the regression line and the actual Y values in the data. The standard error of estimate is directly related to the magnitude of the correlation between X and Y.
Wider confidence interval: more confidence and great t value. Narrower confidence interval: less confidence., more precise interval, smaller t value.
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As the sample size increases, the standard error decreases, and the interval gets smaller.
Strength or consistency of the relationship: the correlation measures the consistency of the relationship. A perfect correlation always is identified by a correlation of 1.00 and indicates a perfectly consistent relationship. At the other extreme, a correlation of 0 indicates no consistency at all.
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In graphic displays of linear correlation, data is enclosed in an envelope to help you to see the overall trend in the data. As a rule of thumb, when the envelope is shaped roughly like a football, the correlation is around 0.7. Envelopes that are fatter than a football indicate correlations closer to 0, and narrower shapes indicate correlations closer to 1.00.
The regression equation simply describes the best-fitting line and is used for making predictions. However, and the standard error of estimate indicates how accurate these predictions will be.
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Whenever the correlation between two variables is significant, you can conclude that the regression equation is also significant. Similarly, if a correlation is not significant, the regression equation is also not significant.
Criteria for interpreting value of proposed by Cohen.
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The correlation from the sample will help to determine which of two interpretations is more likely. A sample correlation near zero supports the conclusion that the population correlation is also zero. A sample correlation that is substantially different from zero supports the conclusion that there is a real, nonzero correlation in the population.
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Adding a constant to (or subtracting a constant from) each X and/or Y, mutiplying or dividing each X or Y by a positive constant does not change the pattern and does not change the value of the correlation. Multiplying by a negative constant produces a mirror image of the pattern and, therefore, changes the sign of the correlation.
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Hypothesis test with correlations: standard error for r: + t statistic: The t statistic has degrees of freedom defined by df = n-2.
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