Least-squares regression fits a straight line to data in order to predict a response variable from an explanatory variable . Inference about regression requires additional conditions.
The simple linear regression model says that there is a populationregression line that describes how the mean response in an entire population varies as changes. The observed response for any has a Normal distribution with mean given by the population regression line and with the same standard deviation for any value of .
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The parameters of the simple linear regression model are the intercept , the slope , and the model standard deviation . The slope and intercept of the least-squares line estimate the slope and intercept of the population regression line.
The parameter is estimated by the regression standard error
where the differences between the observed and predicted responses are the residuals
Prior to inference, always examine the residuals for Normality, constant variance, and any other remaining patterns in the data. Plots of the residuals are commonly used as part of this examination.
The regression standard error has degrees of freedom. Inference about and uses distributions with degrees of freedom.
Confidence intervals for the slope of the population regression line have the form . In practice, use software to find the slope of the least-squares line and its standard error .
To test the hypothesis that the population slope is zero, use the statistic , also given by software. This null hypothesis says that straight-line dependence on has no value for predicting .
The test for zero population slope also tests the null hypothesis that the population correlation is zero. This statistic can be expressed in terms of the sample correlation, .