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# Population parameters and sample estimates

Population parameter

Sample estimate

Multiplier (Test statistic)

Standard error

Description

This is the point where the regression line intercepts the y-axis. It is the predicted response when the predictor variable is equal to zero.

This is the slope of the regression line. For every one-unit increase in the predictor, we see that the response increases by this value. The critical t-value is =T.INV.2T(.05, n-2) for a 95% significance level.

SE fit from Minitab

This is the mean for the responses.

SE fit from Minitab

Another way to look at this is the 95% confidence band around the regression line. This is the band within which we are sure — to a 95% level of significance — that the confidence band actually exists. It deals with issues of as well as at the same time. For this reason, we use the same standard error as for but with a multiplier that gives us a wider band.

We calculate W as,

Where F is the critical value for our (usually 0.05, according to 95% significance level) and degrees of freedom 2 and n-2. We use the same standard error as the confidence interval for in the test above.

This is the predicted response for a particular value of the predictor variable x.

MSE

n.a.

n.a.

This is the variance for the regression model. It relates to the differences between observed responses and expected responses.

S

n.a.

n.a.

This is the standard deviation for the regression model. It is the square root of the variance. For the population, this is ; for the sample, this is .

(in MLR)

## SSR

Take each of the predicted responses, and square their differences from the mean of the observed responses, then sum the squares.

## SSE

Take each of the observed predictor values and response values, and square the differences between the observed response values and predicted responses we obtain from the regression equation, then sum the squares.

## MSE

Where n = sample size, p = number of predictor variables. For a simple linear regression, p = 1.

## SSTO

Take each of the observed responses, and square their differences from the mean of the observed responses, then sum the squares.

## R-squared

This is the percentage of the variation in the response that is explained by the variation in the predictor.

## Predicted vs observed

The observed response for a given response value at row in the data is as follows,

represents the error that is then squared to obtain the variance.

The predicted response is as follows,