91Ó°ÊÓ

Exercise \(5.46\) presented data on \(x=\) squawfish length and \(y=\) maximum size of salmonid consumed, both in \(\mathrm{mm}\). Use the accompanying MINITAB output along with the values \(\bar{x}=343.27\) and \(S_{x x}=69,112.18\) to answer the following questions. The regression equation is Size \(=-89.1=0.729\) length \(\begin{array}{lrrrr}\text { Predictor } & \text { Coef } & \text { Stdev } & \text { t-ratio } & p \\ \text { Constant } & 89.09 & 16.83 & 5.29 & 0.000 \\\ \text { length } & 0.72907 & 0.04778 & 15.26 & 0.000 \\ s=12.56 & R-s q=96.3 \% & R-s q(a d j)=95.9 \% \text { Analysis of } V a\end{array}\) Variance \(\begin{array}{lrrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\ \text { Regression } & 1 & 36736 & 36736 & 232.87 & 0.000 \\ \text { Error } & 9 & 1420 & 158 & & \\\ \text { Total } & 10 & 38156 & & & \end{array}\) a. Does there appear to be a useful linear relationship between length and size? b. Does it appear that the average change in maximum size associated with a 1 -mm increase in length is less than \(.8 \mathrm{~mm}\) ? State and test the appropriate hypotheses. c. Estimate average maximum size when length is 325 \(\mathrm{mm}\) in a way that conveys information about the precision of estimation. d. How would the estimate when length is \(250 \mathrm{~mm}\) compare to the estimate of Part (c)? Answer without actually calculating the new estimate.

Step by step solution

01

Identify the Relationship

To understand if there is a useful linear relationship between length and size, the coefficient of determination, \( R^{2} \), is often evaluated, representing the proportion of variance in the dependent variable (in this case, maximum size of salmonid) that can be predicted from the independent variable (the squawfish length). Here, \( R^{2} = 96.3% \), suggesting a very strong linear relationship.

02

Hypotheses Test

The question asks whether it appears that the average change in maximum size associated with a 1 mm increase in length is less than 0.8 mm. This is a test on the slope of the regression line, represented by the coefficient of 'length' in the regression equation. The null hypothesis is that the slope is equal to 0.8, while the alternative hypothesis is that the slope is not equal to 0.8. Here, the estimated slope from the regression output is 0.72907, which is indeed less than 0.8. The t-ratio is 15.26 and the corresponding p-value is 0.000. Because the p-value is smaller than any reasonable significance level (generally 0.05), we can reject the null hypothesis and conclude that the average change in maximum size associated with a 1 mm increase in length is statistically less than 0.8 mm.

03

Estimate Average Maximum Size

To estimate the average maximum size when length is 325 mm, we substitute 325 for 'length' into the regression equation. We get Size = -89.1 + (0.729 * 325) = 147.25 mm. The standard error around this estimate can be found in the regression output, represented by 's = 12.56'. This means that if we took many samples of squawfish of 325mm in length, approximately 68% of the time the average maximum size of salmonid consumed would be between 134.69 and 159.81 mm.

04

Compare Estimates

To answer part d, without calculating a new estimate, consider the positive slope of the regression line of 0.729. This means that as the squawfish length increases, the maximum size of salmonid consumed also tends to increase. Thus, we can say that the prediction for maximum size with a length of 250 mm will be less than for a length of 325 mm (estimated in step 3).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficient of Determination

Understanding the coefficient of determination, denoted as \( R^2 \), is crucial in regression analysis. It provides insight into the goodness of fit of the model. In simple terms, \( R^2 \) indicates how well the independent variable—in this case, squawfish length—predicts the dependent variable, which is the maximum size of the salmonid consumed.
A value of \( R^2 \) close to 1, such as 96.3% found in this exercise, signals a strong predictive power, meaning most of the variance in salmonid size can be explained by the length of the squawfish.

• High \( R^2 \) values suggest a strong correlation.
• It does not imply causation, only correlation.
• Ideal for understanding the effectiveness of the regression model.

In practice, while \( R^2 \) provides a broad picture, it should be considered alongside other diagnostics to verify model validity.

Hypothesis Testing

Hypothesis testing is a powerful statistical tool used for making inferences about population parameters based on sample data. In the context of linear regression, it often involves testing if the slope of the regression line is significantly different from a hypothesized value.
Here, the hypothesis aims to determine if the change in salmonid size per mm of squawfish length is less than 0.8 mm.

• The null hypothesis (\( H_0 \)) assumes the slope is equal to 0.8.
• The alternative hypothesis (\( H_a \)) posits the slope is less than 0.8.
• A low p-value (0.000 in this case) indicates strong evidence to reject \( H_0 \).

This exercise shows that with 0.72907 as the estimated slope and a p-value of 0.000, there is substantial evidence to conclude that the slope is less than 0.8, indicating a lesser average change in size than hypothesized.

Estimation and Prediction

Estimation and prediction are key objectives in regression analysis, allowing us to infer future outcomes or unknown parameters. In this context, you estimate the average maximum size of salmonid for a squawfish of a specific length.
For example, substituting 325 mm into the regression equation provides an estimated average size of 147.25 mm, and the standard error (\( s = 12.56 \)) offers a measure of the precision of this estimate.

• Point estimation gives a specific value (like 147.25 mm).
• Standard error quantifies the estimate's accuracy.
• A prediction interval indicates the range within which future observations are likely to lie.

This method of estimation allows students to grasp how changes in independent variables could impact outcomes in real-world scenarios.

Regression Analysis

Regression analysis is a statistical approach to modeling the relationship between a dependent variable and one or more independent variables. In simple linear regression, such as this scenario, it involves a straight line that best fits the data points on a graph.
The equation identified here, Size = \(-89.1 + 0.729\times\text{length}\), showcases how each unit change in squawfish length results in an average change in salmonid size.

• The regression coefficient (0.729) indicates the rate of change.
• The constant term (89.1) is the intercept—where the line crosses the y-axis.
• Regression helps to evaluate trends and make forecasts.

A deep understanding of these components allows for effective data-driven decisions and is an invaluable tool for analysis of varied datasets in numerous fields.