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Chapter 5: Problem 45

Example 5.15 described a study that involved substituting sunflower meal for a portion of the usual diet of farm-raised sea breams (Aquaculture [2007]: 528-534). This paper also gave data on \(y=\) feed intake (in grams per 100 grams of fish per day) and \(x=\) percentage sunflower meal in the diet (read from a graph in the paper). \(\begin{array}{rrrrrrrrr}x & 0 & 6 & 12 & 18 & 24 & 30 & 36 \\ y & 0.86 & 0.84 & 0.82 & 0.86 & 0.87 & 1.00 & 1.09\end{array}\) A scatterplot of these data is curved and the pattern in the plot resembles a quadratic curve. a. Using a statistical software package or a graphing calculator, find the equation of the least-squares quadratic curve that can be used to describe the relationship between percentage sunflower meal and feed intake. \(\quad \hat{y}=0.866-0.008 x+0.0004 x^{2}\) b. Use the least-squares equation from Part (a) to predict feed intake for fish fed a diet that included \(20 \%\) sunflower meal.

Short Answer

Expert verified
The predicted feed intake for fish fed a diet that includes 20% sunflower meal is approximately \(0.87\) grams per 100 grams of fish per day (calculation may vary slightly depending on rounding).

Step by step solution

01

Apply Quadratic Regression on Given Data

Quadratic regression can be applied using calculation software or a graphing calculator. In this case, Quadratic regression is applied on the given data \((x, y)\). The least-square quadratic curve equation is found to be \(\hat{y}=0.866-0.008x+0.0004x^{2}\)
02

Predict the Feed Intake

To predict the feed intake for a 20 percent sunflower meal diet, substitute \(x=20\) into the least-square equation. This yields \(\hat{y}=0.866-0.008*20+0.0004*20^{2}\) and calculate the equation to predict the feed intake value

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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 is a powerful mathematical approach used to find the best-fitting curve to a given set of data points by minimizing the squared differences between observed and predicted values. In the context of quadratic regression, the result is a quadratic equation of the form \( \hat{y}=ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are coefficients determined via the least squares calculations. This method helps in capturing the underlying pattern in a curvilinear data series, making it ideal for our aquaculture nutrition data analysis. By plotting and analyzing the given data \(x, y\), which denote percentages of sunflower meal and feed intake, respectively, we determine the best fit quadratic equation based on the least squares principle, \( \hat{y}=0.866-0.008x+0.0004x^{2} \). This equation is then used for predictive modeling of feed intake under various diet compositions.
Scatterplot Analysis
Scatterplot Analysis is a visualization technique for exploring relationships between two variables by plotting data points on a two-dimensional graph. Here, one axis represents the percentage of sunflower meal in the diet (\(x\)) and the other axis represents feed intake (\(y\)). By plotting the data from the study, we can identify patterns or trends in these variables.
  • The data appears to follow a curved pattern, suggesting a non-linear relationship, which is why a quadratic model rather than a linear one is more appropriate.
  • The scatterplot helps visualize the distribution and fit of the quadratic regression curve, \( \hat{y}=0.866-0.008x+0.0004x^{2} \).
Thus, Scatterplot Analysis is crucial in guiding the choice of the regression model and verifying the adequacy of the fit once the model is constructed.
Predictive Modeling
Predictive Modeling uses statistical techniques to develop models that forecast outcomes based on input variables. In our case, the quadratic regression model serves as our predictive model, allowing us to estimate the feed intake (\(\hat{y}\)) of sea bream when given a certain percentage of sunflower meal (\(x\)) in their diet.
To predict for a 20% sunflower meal diet, substitute 20 into the model equation: \(\hat{y}=0.866-0.008 \times 20 + 0.0004 \times (20)^2\). Calculate to find \(\hat{y}\), which predicts the feed intake metric.

This predictive modeling approach can be utilized to optimize diets and enhance feed efficiency based on various diet compositions within aquaculture nutrition studies, supporting the development of sustainable feeding strategies.
Aquaculture Nutrition
Aquaculture Nutrition focuses on the dietary requirements and feeding practices for farm-raised aquatic animals, aiming for optimal growth and health. In this context, understanding how different ingredients, like sunflower meal, impact feed intake is vital.
The study involving sea bream allows us to discern the effects of substituting sunflower meal on dietary consumption patterns. Using the quadratic regression model developed, we can assess how feed intake varies with changing percentages of sunflower meal, which plays a critical role in formulating cost-effective, nutritious diet plans.
  • Ensures feeds are nutritionally balanced to support growth.
  • Helps in reducing cost by identifying viable alternative ingredients.
  • Aims to improve sustainability in aquaculture practices.
Thus, insights from predictive modeling within aquaculture nutrition contribute to more sustainable and efficient farming practices.

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Most popular questions from this chapter

For each of the following pairs of variables, indicate whether you would expect a positive correlation, a negative correlation, or a correlation close to \(0 .\) Explain your choice. a. Maximum daily temperature and cooling costs b. Interest rate and number of loan applications c. Incomes of husbands and wives when both have fulltime jobs d. Height and IQ e. Height and shoe size f. Score on the math section of the SAT exam and score on the verbal section of the same test g. Time spent on homework and time spent watching television during the same day by elementary school children h. Amount of fertilizer used per acre and crop yield (Hint: As the amount of fertilizer is increased, yield tends to increase for a while but then tends to start decreasing.)

With a bit of algebra, we can show that $$ \text { SSResid }=\left(1-r^{2}\right) \sum(y-\bar{y})^{2} $$ from which it follows that $$ s_{e}=\sqrt{\frac{n-1}{n-2}} \sqrt{1-r^{2}} s_{y} $$ Unless \(n\) is quite small, \((n-1) /(n-2) \approx 1,\) so $$ s_{e} \approx \sqrt{1-r^{2}} s_{y} $$ a. For what value of \(r\) is \(s_{e}\) as large as \(s_{y}\) ? What is the least- squares line in this case? \(r=0, \hat{y}=\bar{y}\) b. For what values of \(r\) will \(s_{e}\) be much smaller than \(s_{y}^{?}\) c. A study by the Berkeley Institute of Human Development (see the book Statistics by Freedman et al., listed in the back of the book) reported the following summary data for a sample of \(n=66\) California boys: \(r \approx .80\) At age 6 , average height \(\approx 46\) inches, standard deviation \(\approx 1.7\) inches. At age 18 , average height \(\approx 70\) inches, standard deviation \(\approx 2.5\) inches. What would \(s_{e}\) be for the least-squares line used to predict 18 -year-old height from G-year-old height? d. Referring to Part (c), suppose that you wanted to predict the past value of G-year-old height from knowledge of 18 -year-old height. Find the equation for the appropriate least-squares line. What is the corresponding value of \(s_{e} ? \hat{y}=7.92+.544 x, s_{c}=1.02\)

The accompanying data resulted from an experiment in which weld diameter \(x\) and shear strength \(y\) (in pounds) were determined for five different spot welds on stccl. \(\Lambda\) scattcrplot shows a strong lincar pattcrn. With \(\sum(x-\bar{x})^{2}=1000\) and \(\sum(x-\bar{x})(y-\bar{y})=8577,\) the least- squares line is \(\hat{y}=-936.22+8.577 x\) \(\begin{array}{llllrr}x & 200.1 & 210.1 & 220.1 & 230.1 & 240.0 \\ y & 813.7 & 785.3 & 960.4 & 1118.0 & 1076.2\end{array}\) a. Because \(1 \mathrm{lb}=0.4536 \mathrm{~kg}\), strength observations can be re-expressed in kilograms through multiplication by this conversion factor: new \(y=0.4536(\) old \(y)\). What is the equation of the least-squares line when \(y\) is expressed in kilograms? \(\quad \hat{y}=-424.7+3.891 x\) b. More generally, suppose that each \(y\) value in a data set consisting of \(n(x, y)\) pairs is multiplied by a conversion factor \(c\) (which changes the units of measurement for \(y\) ). What effect does this have on the slope \(b\) (i.e., how does the new value of \(b\) compare to the value before conversion), on the intercept \(a\), and on the equation of the least-squares line? Verify your conjectures by using the given formulas for \(b\) and \(a\). (Hint: Replace \(y\) with \(c y\), and see what happensand remember, this conversion will affect \(\bar{y}\).)

An article on the cost of housing in California that appeared in the San Luis Obispo Tribune (March 30 . 2001 ) included the following statement: "In Northern California, people from the San Francisco Bay area pushed into the Central Valley, benefiting from home prices that dropped on average \(\$ 4000\) for every mile traveled east of the Bay area." If this statement is correct, what is the slope of the least-squares regression line, \(\hat{y}=a+b x,\) where \(y=\) house price (in dollars) and \(x=\) distance east of the Bay (in miles)? Explain.
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