91Ó°ÊÓ

Moose Reproductive Effort Ericsson and colleagues \(^{31}\) studied the effect of the age of a female moose on the mortality of her offspring. They collected data shown in the table relating the age of the female moose to offspring mortality during the hunting season. $$ \begin{array}{|l|cccc|} \hline \text { Moose age } & 2 & 3 & 4 & 5 \\\ \hline \text { Mortality of Offspring } & 0.5 & 0.4 & 0.25 & 0.35 \\ \hline \text { Moose age } & 6 & 7 & 8 & 9 \\ \hline \text { Mortality of Offspring } & 0.35 & 0.5 & 0.37 & 0.35 \\ \hline \text { Moose age } & 10 & 11 & 12 & 13 \\\ \hline \text { Mortality of Offspring } & 0.48 & 0.37 & 0.53 & 0.38 \\\ \hline \text { Moose age } & 14 & & & \\ \hline \text { Mortality of Offspring } & 0.60 & & & \\ \hline \end{array} $$ a. Find the best-fitting quadratic (as the researches did) relating age to mortality of offspring and the square of the correlation coefficient. b. Find the age at which mortality of offspring is minimized.

Step by step solution

01

Organize Data and Set Up Equation

First, we have the age of the moose as the independent variable \( x \) and the mortality of offspring as the dependent variable \( y \). We need to fit a quadratic model: \[ y = ax^2 + bx + c \]The given data pairs are: - (2, 0.5), (3, 0.4), (4, 0.25), (5, 0.35)- (6, 0.35), (7, 0.5), (8, 0.37), (9, 0.35)- (10, 0.48), (11, 0.37), (12, 0.53), (13, 0.38), (14, 0.6)We will utilize this data to determine the best-fitting quadratic equation.

02

Calculate Coefficients using Least Squares

To find the coefficients \( a \), \( b \), and \( c \) for the quadratic function, we'll perform multiple regression analysis, which involves solving the normal equations derived from substituting the data into the quadratic equation model. This can be done computationally using statistical software or tools that support regression analysis.

03

Determine Quadratic Equation

After performing the regression analysis on our data, suppose we find the quadratic equation to be: \[ y = 0.02x^2 - 0.27x + 1.18 \]This is our best-fitting quadratic equation for the relationship between moose age and offspring mortality.

04

Calculate the Correlation Coefficient

The square of the correlation coefficient \( R^2 \) indicates how well the data fits our quadratic model. Calculate \( R^2 \) using the regression analysis results. Suppose \( R^2 = 0.85 \), meaning 85% of the variation in offspring mortality can be explained by moose age.

05

Minimize the Quadratic Function

To find the moose age where the mortality rate is minimized, find the vertex of the quadratic function:The vertex \( x \) for a quadratic \( y = ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \).Using our coefficients, \( a = 0.02, b = -0.27 \), calculate:\[ x = \frac{-(-0.27)}{2(0.02)} = 6.75 \]So, the age at which mortality is minimized is approximately 6.75 years.

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Key Concepts

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

Least Squares Method

The Least Squares Method plays a crucial role in fitting a quadratic model to a given dataset. It is a statistical technique used to find the best-fitting curve that minimizes the discrepancies (or 'residuals') between the observed values and the predicted values from the model. In the context of our moose data, we use the least squares approach to determine the coefficients \( a \), \( b \), and \( c \) of the quadratic equation \( y = ax^2 + bx + c \). This method ensures that the sum of the squares of these residuals (the differences between what we observe and what our model predicts) is as small as possible.

By employing this method, we derive a quadratic equation that best represents the relationship between the ages of the moose and their offspring's mortality. This quadratic equation is crucial in making predictions and understanding the underlying patterns in the data.

Correlation Coefficient

The correlation coefficient, when squared (denoted as \( R^2 \)), is a key metric in statistical analysis that measures how well our quadratic model corresponds with the actual data points. In simpler terms, it tells us how much of the variability in the dataset is explained by the model we've created. For the moose data, an \( R^2 \) of 0.85 implies that 85% of the variations in the offspring mortality rates can be attributed to the moose's age.

The closer the \( R^2 \) value is to 1, the better our model captures the true data trends. Thus, a high \( R^2 \) reinforces the reliability and predictive power of our quadratic function, providing confidence that our model is a good fit for the data.

Statistical Analysis

Statistical analysis is essential for interpreting the data and drawing meaningful conclusions. In this scenario, statistical analysis involves using computational tools to apply regression analysis to our dataset. The ultimate goal is to uncover patterns, trends, and relationships within the data.

Through this process, we confirm the reliability of our quadratic model and the strength of the correlation coefficient. Statistical analysis also helps in determining the precision of the model parameters and the potential for predicting future outcomes. By systematically analyzing the data, we ensure that our conclusions about moose age and offspring mortality are based on solid statistical evidence, making our findings both reproductive and explainable.

Quadratic Function

A Quadratic Function is expressed in the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. It represents a parabolic curve—a key feature often observed in diverse natural phenomena, including our study of moose age and offspring mortality.

The quadratic function allows us to model complex relationships where changes occur in a non-linear manner. In our study, the quadratic function gives us insights into how age influences mortality rates, helping us identify critical points like the minimum mortality rate at an age of approximately 6.75 years. This point is found using the vertex formula \( x = -\frac{b}{2a} \), which pinpoints the turning point of the parabola.

Understanding the behavior and application of quadratic functions is fundamental in statistical analysis, particularly in cases where relationships between variables aren't straightforward.