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Chapter 8: Problem 8

Milk Storage the table shows the number of days that milk will keep as a function of the temperature. Use the method of least squares to find the best- fitting linear model for the data. Number of Days Milk Can Be Stored Safely $$ \begin{array}{|l|l|l|l|l|} \hline \text { Temperature ('F) } & 30 & 38 & 45 & 50 \\ \hline \text { Days } & 24 & 10 & 5 & 0.5 \\ \hline \end{array} $$

Short Answer

Expert verified
The best-fitting linear model is \( y = -1.568x + 73.739 \).

Step by step solution

01

Organize the Data

The data provided gives us pairs of temperature and the number of days milk can be stored. We have the following points: (30, 24), (38, 10), (45, 5), and (50, 0.5).
02

Calculate the Means

Calculate the mean of the temperatures (\( x \)) and the mean of the days (\( y \)). \[ \text{Mean of temperatures} \ (\bar{x}) = \frac{30 + 38 + 45 + 50}{4} = 40.75 \]\[ \text{Mean of days} \ (\bar{y}) = \frac{24 + 10 + 5 + 0.5}{4} = 9.875 \]
03

Calculate the Slope (m)

Use the formula for the slope \( m \) of the line: \[ m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \]Calculate the numerator and the denominator separately. For the points provided:\( (30, 24), (38, 10), (45, 5), (50, 0.5) \), first calculate \( (x_i - \bar{x})(y_i - \bar{y}) \) for each pair and total them up, then do the same for \( (x_i - \bar{x})^2 \).Numerator: \( (-10.75)(14.125) + (-2.75)(0.125) + (4.25)(-4.875) + (9.25)(-9.375) = -372.5 \)Denominator: \( (-10.75)^2 + (-2.75)^2 + (4.25)^2 + (9.25)^2 = 237.5 \)Thus, \( m = \frac{-372.5}{237.5} = -1.568 \).
04

Calculate the Y-intercept (b)

Use the formula \( b = \bar{y} - m\bar{x} \).Substitute the values of \( m \), \( \bar{y} \), and \( \bar{x} \):\[ b = 9.875 - (-1.568)(40.75) \]\[ b = 9.875 + 63.864 = 73.739 \]
05

Write the Equation

With the slope \( m = -1.568 \)and the y-intercept \( b = 73.739 \), the equation of the best-fitting line is:\[ y = -1.568x + 73.739 \]. This is the linear model predicting the number of days milk can be stored based on temperature.

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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 fundamental approach in linear regression used to find the best-fitting line for a set of data points. The main goal here is to minimize the sum of the squares of the differences ("errors") between the observed values and those predicted by the linear model. This line of best fit is the one that reduces these errors to their lowest possible value.

Here's how it works in simple terms:
  • First, you organize the data into pairs of variables. In our case, we're looking at temperature and the number of days milk can be stored.
  • The method calculates a straight line that goes through this data as "closely" as possible according to predefined mathematical rules.
  • Each "error" is calculated as the vertical distance from a data point to the line; squared errors avoid discrepancies caused by positive/negative deviations.
Overall, the least squares method is powerful because it provides a clear, mathematical way to relate dependent and independent variables in datasets. This makes it invaluable in fields like data analysis, where understanding patterns and trends is crucial.
Data Analysis
Data analysis is the process of inspecting, cleansing, and modeling data with the goal of discovering useful information. Understanding this step is essential for applying methods like least squares effectively. In the context of our exercise, the data involves temperature readings and their effect on milk storage longevity.

Key steps in data analysis:
  • Data Collection: Gather relevant data; in our scenario, temperature and milk preservation days were recorded.
  • Data Cleaning: This involves removing inaccuracies and verifying consistency. Our example is pre-cleaned, with each point providing paired data.
  • Modeling: Using mathematical techniques, such as linear regression, to establish relationships between variables. In this case, examining how temperature affects milk storage.
Using data analysis helps us draw conclusions and forecast future scenarios. For example, by determining the impact of temperature on milk storage, we can plan better refrigeration practices.
Slope-Intercept Form
The slope-intercept form of a linear equation is a way of describing a straight line using the general form: \[y = mx + b\]In this formula:
  • \( m \) is the slope of the line, representing how much y (the dependent variable) changes for a one-unit change in x (the independent variable).
  • \( b \) is the y-intercept, showing where the line crosses the y-axis.
For our exercise, the linear model calculated was \( y = -1.568x + 73.739 \), where the slope \( m = -1.568 \) tells us that for each degree Fahrenheit increase in temperature, the storage days decrease by approximately 1.568 days. The y-intercept \( b = 73.739 \) indicates mathematically, the number of days when "temperature" is zero (though contextually impractical, it helps in plotting).

Understanding slope-intercept form helps interpret real-world linear relationships through a simple algebraic structure.
Temperature and Milk Storage
The relationship between temperature and milk storage is a classic example in data analysis and linear regression. In this exercise, data shows how varying temperatures affect how long milk stays fresh.

Key Points:
  • Inverse Relationship: As temperature increases, the number of days milk can be stored decreases. This is evident from the negative slope in the best-fitting line.
  • Predictive Modeling: By understanding this relationship, predictions can be made about milk storage times for other temperatures not in the data set.
Practical applications include refining storage guidelines in the dairy industry, ensuring that milk remains fresh for as long as possible by maintaining optimal low temperatures. Thus, such analytically-driven models assist businesses and consumers in deciding on the ideal storing conditions.

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Most popular questions from this chapter

Parasite Propagation Boll weevils have long presented a threat to cotton crops in the southern United States. Research has been done to determine the optimal conditions for reproduction of Catolaccus grandis, a parasite that attaches to boll weevils and kills them. The number of eggs produced in 30 days by one female \(C .\) grandis under conditions of 16 hours of light each day, a constant temperature of \(y^{\circ} \mathrm{C},\) and a relative humidity of \(x \%\) can be modeled by $$ \begin{aligned} f(x, y)=&-4191.6877+299.7038 x+23.1412 y \\ &-5.2210 x^{2}-0.0937 y^{2}-0.4023 x y \text { eggs } \end{aligned} $$ (Source: J. A. Morales-Ramos, S. M. Greenberg, and E. G. King, "Selection of Optimal Physical Conditions for Mass Propagation of Catolaccus grandis," Environmental Entomology, vol. \(25,\) no. 1 \((1996),\) pp. \(165-173)\) a. Calculate the point where the partial derivatives of \(f\) are both equal to zero. b. Write a sentence interpreting the point in part \(a\) and characterizing it as a maximum, a minimum, or a saddle point.

Generic Production A Cobb-Douglas function for the production of a certain commodity is $$ q(x, y)=x^{0.8} y^{0.2} $$ where \(q\) is measured in units of output, \(x\) is measured in units of capital, and \(y\) is measured in units of labor. The price of labor is \(\$ 3\) per unit. The price for capital is \(\$ 10\) per unit. The producer has a budget of \(\$ 200\) to spend on labor and capital to produce this commodity. a. Write an equation for the budget constraint. b. How much of the budget should be spent on each of labor and capital to maximize production while satisfying the budget constraint.

Animal Experiments The number of animal experiments in England declined between 1970 and \(1980 .\) The numbers for selected years are shown in the table. Use the method of least squares to find the best-fitting linear model for the data. Animal Experiments Conducted in England $$ \begin{array}{|l|c|c|c|c|} \hline \text { Year } & 1970 & 1975 & 1978 & 1980 \\ \hline \text { Experiments (millions) } & 5.5 & 5 & 4.8 & 4.6 \\ \hline \end{array} $$

Single Young Adults The percentage of adults 20 to 24 years of age who had not yet been married is given in the table. Percentage of 20 - to 24 -Year-OId Adults Who Have Not Yet Been Married $$ \begin{array}{|c|c|c|c|c|c|} \hline \text { Year } & 1960 & 1970 & 1980 & 1990 & 2000 \\ \hline \text { Not Yet Married (\%) } & 41 & 46 & 60 & 72 & 78 \\ \hline \end{array} $$a. Use the method of least squares to find the multivariable function \(f\) with inputs \(a\) and \(b\) for the bestfitting line \(y=a x+b,\) where \(x\) is the number of years since \(1970 .\) b. Write the minimum value of \(f(a, b)\). c. Write the linear model that best fits these data. d. Use the model to estimate the percentage of 20 to 24 year old adults 20 - to 24 -year-old who had never been married in \(2010 .\)

Locate and classify any critical points. $$ R(k, m)=3 k^{2}-2 k m-20 k+3 m^{2}-4 m+60 $$
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