Definition formulae The total sum of squares denoted

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Definition formulae The total sum of squares, denoted by SSTo, is defined as The

Definition formulae The total sum of squares, denoted by SSTo, is defined as The residual sum of squares, denoted by SSResid, is defined as 2

Calculation Formulae Recalled SSTo and SSResid are generally found as part of the standard

Calculation Formulae Recalled SSTo and SSResid are generally found as part of the standard output from most statistical packages or can be obtained using the following computational formulas: 3

Coefficient of Determination The coefficient of determination, denoted by r 2, gives the proportion

Coefficient of Determination The coefficient of determination, denoted by r 2, gives the proportion of variation in y that can be attributed to an approximate linear relationship between x and y. 4

Estimated Standard Deviation, se The statistic for estimating the variance s 2 is where

Estimated Standard Deviation, se The statistic for estimating the variance s 2 is where 5

Estimated Standard Deviation, se The estimate of s is the estimated standard deviation The

Estimated Standard Deviation, se The estimate of s is the estimated standard deviation The number of degrees of freedom associated with estimating s 2 or s in simple linear regression is n-2. 6

Example continued 7

Example continued 7

Example continued 8

Example continued 8

Example continued With r 2=0. 627 or 62. 7%, we can say that 62.

Example continued With r 2=0. 627 or 62. 7%, we can say that 62. 7% of the observed variation in %Fat can be attributed to the probabilistic linear relationship with human age. The magnitude of a typical sample deviation from the least squares line is about 5. 75(%) which is reasonably large compared to the y values themselves. This would suggest that the model is only useful in the sense of provide gross ballpark estimates for %Fat for humans based on age. 9

Properties of the Sampling Distribution of b When the four basic assumptions of the

Properties of the Sampling Distribution of b When the four basic assumptions of the simple linear regression model are satisfied, the following conditions are met: 1. The mean value of b is b. Specifically, mb=b and hence b is an unbiased statistic for estimating b 2. The standard deviation of the statistic b is 10 3. The statistic b has a normal distribution (a consequence of the error e being normally distributed)

Estimated Standard Deviation of b The estimated standard deviation of the statistic b is

Estimated Standard Deviation of b The estimated standard deviation of the statistic b is When then four basic assumptions of the simple linear regression model are satisfied, the probability distribution of the standardized variable Is the t distribution with df = n - 2 11

Confidence interval for b When then four basic assumptions of the simple linear regression

Confidence interval for b When then four basic assumptions of the simple linear regression model are satisfied, a confidence interval for b, the slope of the population regression line, has the form b (t critical value) sb Where the t critical value is based on df = n - 2. 12