Question

in Fish In Exercise 9.21 , we see that the conditions are met for using the \(\mathrm{pH}\) of a lake in Florida to predict the mercury level of fish in the lake. The data are given in FloridaLakes. Computer output is shown for the linear model with several values missing: The regression equation is AvgMercury \(=1.53-0.152 \mathrm{pH}\) Predictor Constant \(\mathrm{pH}\) \(\begin{array}{rrrr}\text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ 1.5309 & 0.2035 & 7.52 & 0.000 \\ -0.15230 & \text { * (c) }^{* *} & -5.02 & 0.000\end{array}\) stant \(S=* *(b)^{* *} \quad R-S q=* *(a)^{* x}\) Analysis of Variance Source Regression Residual Erri Total \(\begin{array}{rrrrrr} & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { ion } & 1 & 2.0024 & 2.0024 & 25.24 & 0.000 \\ \text { Error } & 51 & 4.0455 & 0.0793 & & \\ & 52 & 6.0479 & & & \end{array}\) (a) Use the information in the ANOVA table to compute and interpret the value of \(R^{2}\). (b) Show how to estimate the standard deviation of the error term, \(s_{\epsilon}\). (c) Use the result from part (b) and the summary statistics below to compute the standard error of the slope, \(S E,\) for this model: $$ \begin{array}{lrrrrr} \text { Variable } & \text { N } & \text { Mean } & \text { StDev } & \text { Minimum } & \text { Maximum } \\ \text { pH } & 53 & 6.591 & 1.288 & 3.600 & 9.100 \\ \text { AvgMercury } & 53 & 0.5272 & 0.3410 & 0.0400 & 1.3300 \end{array} $$

Short answer

To get the necessary answers, you follow three main steps. Firstly, compute \(R^{2}\) which is approximately 0.3315. Secondly, calculate the standard deviation of the error term, \(s_{\epsilon}\) which is found to be approximately 0.2816. Lastly, compute the standard error of the slope, which is found to be approximately 0.0303

Step-by-step solution

Compute \(R^{2}\)

To compute the big R-squared (\(R^{2}\)), we use the formula \[R^{2} = \frac {SS_{Regression}} {SS_{Total}}\] Here, \(SS_{Regression}\) is 2.0024 and \(SS_{Total}\) is 6.0479. Substitute these values into the formula and calculate \(R^{2}\).

Estimate the standard deviation of the error term, \(s_{\epsilon}\)

\[s_{\epsilon}\] is calculated as the squareroot of the Mean Square Error (MSE). The MSE is obtained from ANOVA table and equals 0.0793. Calculate s by taking the squareroot of the Mean Square Error.

Compute the standard error of the slope (SE)

The formula to calculate the standard error of the slope is \[SE_{slope} = \frac{s_{\epsilon}}{\sqrt{SSX}}\] where \(s_{\epsilon}\) is the standard deviation of the error term found in step 2, and \(SSX\) is the sum of squares of the differences from the mean value of pH. The latter, \(SSX\), is calculated as follows: \[\sum{(x_i - \overline{x})^2}\]where \(x_i\) is each individual value of pH, and \(\overline{x}\) is the mean of pH. Since we are missing individual values for pH, we make use of the fact that \(SSX = (n-1) \times \text{{StDev}}^2\), where n is the number of observations of pH, and StDev is the standard deviation of pH. Substitutes these values into the formula for SE_{slope} and calculate it.

Key concepts

Understanding R-Squared (R^2) in Linear Regression Analysis

The metric R-Squared, symbolized as R², is a statistical measure in linear regression that determines the proportion of the variation in the dependent variable that can be explained by the independent variable(s). Imagine you are trying to predict the weight of a group of people based on their heights. An R² close to 1 indicates that the regression model explains a large portion of the variation in weight, while an R² near 0 suggests that height provides little information about weight in this scenario.

It acts as a goodness-of-fit measure, with higher values usually indicating a better model, but beware - it doesn't mean the model is the right one! It's possible for a model with a high R² to be misleading, especially if it's based on a nonsensical relationship or if we've just got too many predictors throwing in their two cents.

To calculate R², you take the sum of squares due to regression (SS_Regression) divided by the total sum of squares (SS_Total). It's like figuring out what percentage of your pizza has been eaten by comparing the eaten slices (SS_Regression) to the whole pie (SS_Total).

Deciphering the Standard Deviation of the Error Term (s_epsilon)

The standard deviation of error term, denoted as s_ε, is the estimate of the variability or dispersion of the observed values around the regression line. Think of throwing darts at a dartboard - s_ε tells you how spread out those darts are from the bullseye, which represents the predictions made by your regression model.

To get this value, we take the square root of the Mean Square Error (MSE) from the ANOVA table. MSE is like an average of the squares of the 'darts' distances from the bullseye. Actually calculating s_ε is like measuring the average distance itself, not just the squared distance, making it easier to understand and more tangible, a bit like comparing apples to apples instead of apples to apple squares. It's a crucial component for assessing model performance and for calculating other important statistics such as SE_slope.

Standard Error of the Slope (SE_slope): Nailing Down the Precision

The standard error of the slope, or SE_slope, measures the precision of the estimate of the slope coefficient in your regression model. In simpler terms, it tells you how accurate your estimation of the relationship between the independent and dependent variable is - are you just in the ballpark, or did you hit it out of the park?

An easy way to think about SE_slope is like this: If you recorded the heights and shoe sizes of everyone in your class to find out the general trend, SE_slope would tell you how confident you can be about the strength of the connection between those two measurements. Small values of SE_slope suggest a more precise estimate, meaning you can be pretty confident about the trend you've observed. To calculate SE_slope, you need the standard deviation of the error term (s_ε) and the total sum of squares for the independent variable, which is like the variability of the class heights in this metaphor.

ANOVA Table in Linear Regression: Breaking Down Variability

The ANOVA (Analysis of Variance) table in the context of linear regression analysis breaks down the variability in the data into components that can help us assess how well our regression model works. It's like checking which ingredients in a recipe are making your cake rise.

The ANOVA table typically includes figures like the degrees of freedom (DF), sum of squares (SS), mean squares (MS), and an F-statistic. Here's a quick bite of what those ingredients are serving up:

• DF: Like telling you how many independent pieces of information went into making your cake.
• SS: The sum of squares is like quantifying how much sugar and flour you've added to the mix - for regression, how much the model and the errors contribute to the total variability.
• MS: Mean Squares are like an average - if you divided all that sugar and flour evenly into every bite of your cake. For our analysis, MS is SS divided by DF.
• F-statistic: It compares the model to a scenario where there's no relationship (plain cake with no rising agent). A high F-statistic means your model adds significant lift, just like baking powder in the cake.

Understanding the ANOVA table arms you with the info to say how much faith to put in your regression model's ability to predict outcomes - like knowing if that cake will be a showstopper.

Sum of Squares (SS): Measuring Variability in Data

Sum of Squares (SS) is a measure used to describe how much the values in a data set vary around their mean. It's like measuring the roughness of a road by checking out how many potholes there are and how deep they go.

In linear regression, we deal with two main types of sum of squares:

• The SS_Regression represents how much variance in your dependent variable can be explained by the model. It's like showing how many of those bumps in the road are because of potholes versus something else.
• The SS_Residual (or Error) measures the variance that the model fails to explain, which is the remaining bumpiness after you've patched up the potholes.
• The SS_Total is the overarching measurement of variability, or in our road analogy, the total bumpiness including both potholes and other flaws in the road.

The SS is a foundational figure in many statistical tests and procedures, such as ANOVA, allowing you to judge whether the potholes (your model's predictions) are the main reason for the road's condition, or if other factors are also at play.