Question

Exercise \(5.48\) described a regression situation in which \(y=\) hardness of molded plastic and \(x=\) amount of time elapsed since termination of the molding process. Summary quantities included \(n=15\), SSResid = \(1235.470\), and \(\mathrm{SSTo}=25,321.368\) a. Calculate a point estimate of \(\sigma .\) On how many degrees of freedom is the estimate based? b. What percentage of observed variation in hardness can be explained by the simple linear regression model relationship between hardness and elapsed time?

Short answer

The point estimate of standard deviation (sigma) and the degrees of freedom are derived from Steps 1 and 2 respectively, while Steps 3 and 4 provide the calculation and answer for the percentage of observed variation in hardness explained by the regression model.

Step-by-step solution

Compute Estimate of Sigma: Point estimate of standard deviation

The point estimate of standard deviation \( \sigma \) is calculated using the formula for standard error of the estimate in linear regression which is \( \sigma = \sqrt{\frac{SSResid}{n - 2}} \). Using the provided Residual Sum of Squares (SSResid) of \(1235.470\) and sample size (n) of \(15\), we substitute these values into the formula: \( \sigma = \sqrt{\frac{1235.470}{15 - 2}} \).

Compute Degrees of Freedom

The degrees of freedom for the estimate is computed as \( n - 2 \), which is \( 15 - 2 = 13 \). This is because we are working with a simple linear regression model with two parameters (intercept and slope), hence we subtract 2 from our sample number.

Computing R-squared as Percentage of Variation

R-squared, the coefficient of determination, is used to determine the percentage of variation in hardness that can be explained by the regression model. It is calculated using the formula \( R^2 = \frac{SSTo - SSResid}{SSTo} \). Here, Total Sum of Squares (SSTo) is \( 25321.368 \) and Residual Sum of Squares (SSResid) is \( 1235.470 \). Substituting these values into the formula will help us determine the required percentage.

Calculation and Answer

After performing the calculations in Steps 1 and 3, round off the results to a suitable number of decimal places for accuracy in the final answers.

Key concepts

Standard Error of the Estimate

The standard error of the estimate in simple linear regression reflects how much data points deviate from the regression line. It's essentially the average distance that the observed data points are from the regression line. Think of it as the measure of the spread of points around the fitted line.

To calculate it, one would use the formula: ewline ewline \[ \sigma = \sqrt{\frac{SSResid}{n - 2}} \] ewline The numerator, SSResid, stands for the residual sum of squares, indicating the variance unexplained by the regression model. The denominator, 'n - 2', represents the degrees of freedom, which accounts for the number of independent values while considering the parameters (slope and intercept) estimated in the model.

In our case, with an SSResid of 1235.470 and a sample size of 15, the calculation yields: ewlineewline \[ \sigma = \sqrt{\frac{1235.470}{15 - 2}} \] ewline ewline Using this figure, we can judge the accuracy of the regression model's predictions.

Degrees of Freedom

Degrees of freedom in the context of regression analysis are the number of values that can vary in calculating a statistic, given that certain restrictions are already in place. For instance, in simple linear regression with one explanatory variable, calculating the standard error of the estimate requires finding how many data points effectively contribute to the variation not explained by the regression model.

The formula used is: ewline ewline \[ df = n - k \] ewline ewline Here, 'df' represents the degrees of freedom, 'n' is the total number of observations, and 'k' is the number of parameters estimated (in simple linear regression, 'k' would typically be 2, accounting for the slope and intercept). So with our sample size of 15, we have: ewline ewline \[ df = 15 - 2 \] ewline ewline This tells us that we have 13 degrees of freedom for the estimation of sigma, implying there are 13 independent pieces of information used to estimate the variance that's not attributed to the regression line.

R-squared

R-squared, or the coefficient of determination, is a statistical measure in regression that represents the proportion of the variance for the dependent variable that's explained by the independent variables in the model. In simple terms, it tells us how well the data fit the regression model. The higher the R-squared value, the better the model explains the variability of the output.

Calculated as: ewline ewline \[ R^2 = \frac{SSTo - SSResid}{SSTo} \] ewline The Total Sum of Squares (SSTo) reflects the total variance in the dependent variable, while the Residual Sum of Squares (SSResid) represents the variance not explained by the model. In our exercise, by substituting the given values into the formula, we ascertain the percentage of variance in hardness that elapsed time can account for.

Residual Sum of Squares (SSResid)

The residual sum of squares (SSResid) is a measure used in statistical analysis to quantify the discrepancy between the data and an estimation model. It's the sum of the squares of the differences between the observed values and the values predicted by the model. The lower the SSResid, the closer your model's predictions are to the actual data points.

We typically calculate it with: ewline ewline \[ SSResid = \sum_{i=1}^{n} (y_i - \hat{y_i})^2 \] ewline ewline Where \(y_i\) is the observed value and \( \hat{y_i} \) is the predicted value from the regression model. In our exercise, a SSResid of 1235.470 indicates the total variation that the simple linear regression model does not explain. It's pivotal in calculating both the standard error of the estimate and the R-squared value.

Total Sum of Squares (SSTo)

The total sum of squares (SSTo) indicates the total variation in the observed data. It's calculated by summing the squares of the differences between each observed value and the overall mean of the data. It forms the denominator in the R-squared formula and is a critical component in assessing the model's goodness of fit.

You can find it using the equation: ewline ewline \[ SSTo = \sum_{i=1}^{n} (y_i - \bar{y})^2 \] ewline ewline Where \( \bar{y} \) is the mean of the observed data. It gives us a basis to compare how much of the total variation is explained by the model (R-squared) and how much remains as the residual variance (SSResid). Our exercise shows a SSTo of 25,321.368, setting the stage for evaluating the explanatory power of the regression model.