91Ó°ÊÓ

In a study, 60 colts were measured every 14 days from birth. The ordered pairs \((d, l)\) represent the average length \(l\) (in centimeters) of the 60 colts \(d\) days after birth. (Spreadsheet at LarsonPrecalculus.com) $$\begin{aligned} &(14,81.2)\\\ &(56,98.3)\\\ &(98,110.0) \end{aligned}$$ $$\begin{aligned} &(28,87.1)\\\ &(70,102.4) \end{aligned}$$ $$\begin{aligned} &(42,93.7)\\\ &(84,106.2) \end{aligned}$$ (a) Use the regression feature of a graphing utility to find a linear model for the data and to identify the correlation coefficient. (b) According to the correlation coefficient, does the model represent the data well? Explain. (c) Use the graphing utility to plot the data and graph the model in the same viewing window. How closely does the model fit the data? (d) Use the model to predict the average length of a colt 112 days after birth.

Step by step solution

01

Apply Linear Regression to the Data

Given measured pairs \((day, length)\), use a regression feature of a graphing utility to find the best fit line or linear model for the data. The aim is to determine a model in the form \(l = md + b\), where \(l\) is the average length of the colts, \(d\) is number of days, \(m\) is the slope, and \(b\) is the y-intercept.

02

Calculate the Correlation Coefficient

After obtaining the linear model, calculate the correlation coefficient, denoted by \(r\). The coefficient should lie between -1 and 1, where 1 means perfect positive correlation, -1 means perfect negative correlation and 0 implies no correlation. This gives a quantified measure of how well the model fits the data.

03

Assessment of Model Representation

Evaluate the worth of the derived model based on the computed correlation coefficient. If \(r\) is close to 1 or -1, it implies the data points closely follow a straight line, suggesting the model represents the data well. If \(r\) is close to zero, it means there's no linear relationship between the variables so the model isn't a good fit.

04

Use the Model for Prediction

Finally, use the model to predict the length of a colt 112 days after birth. Substitute \(d = 112\) into the linear equation \(l = md + b\) to predict the average length of a colt at day 112.

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Key Concepts

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

Correlation coefficient

The correlation coefficient is a key metric in determining the strength and direction of a linear relationship between two variables. In this context, it measures how well the length of the colts correlates with the days since their birth. This coefficient, generally represented by the symbol \( r \), provides a value between -1 and 1.

• A value of 1 indicates a perfect positive linear relationship, meaning as days increase, the length also increases at a constant rate.
• A value of -1 would indicate a perfect negative linear relationship.
• An \( r \) value close to 0 suggests little to no linear relationship.

Once the linear model is determined using regression, the correlation coefficient tells us how closely the model fits the data. A higher \( r \) value means the data points plot closer to the line, validating the model's accuracy in representing the data.

Linear model

A linear model is an equation that attempts to capture the relationship between two variables with a straight line. In the exercise, the goal is to establish a model that relates the average length of colts (\( l \)) with the number of days since birth (\( d \)).
The general form of a linear model is \( l = md + b \), where:

• \( l \) is the dependent variable (average length).
• \( d \) is the independent variable (days).
• \( m \) is the slope, indicating how much \( l \) changes with each additional day.
• \( b \) is the y-intercept, representing the initial average length at day zero.

The slope \( m \) provides insight into the growth rate, and creating this model allows us to make interpretations about an average colt's growth over time.

Data fitting

Data fitting, in this context, involves using linear regression to find the best-fit line for the data. This process helps in establishing a linear model that minimizes the differences between the observed data points and the predictions made by the model.
Fitting a line involves using statistical methods to determine the values of \( m \) (slope) and \( b \) (y-intercept) that best reduce the sum of the squared differences (often called residuals) between the actual lengths and those predicted by the model.
A good fit means the line accurately represents the data, closely aligning with the actual points plotted on the graph. Using visual assessment along with numeric methods like the correlation coefficient provides a strong basis for evaluating the success of the data fitting.

Prediction with linear models

Once a linear model is derived, it can be used to make predictions about new, untested data points. In the exercise, the model is applied to predict the average length of colts at 112 days. This involves substituting \( d = 112 \) into the linear equation \( l = md + b \) to find the estimated length.
Prediction is a powerful tool because it allows us to extend beyond the initial dataset and infer information about similar events. Although predictions are based on past data, their accuracy hinges on the quality of the model and how closely the data follows a linear trend. High correlation coefficients lend credibility to the predictions, reinforcing that deviations from the expected length are minimal.