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Chapter 22: Problem 2

Find the equation of the indicated least squares curve. Sketch the curve and plot the data points on the same graph. For the points in the following table, find the least-squares curve \(y=m \sqrt{x}+b\). $$\begin{array}{l|c|c|c|c|c} x & 0 & 4 & 8 & 12 & 16 \\\\\hline y & 1 & 9 & 11 & 14 & 15\end{array}$$

Short Answer

Expert verified
The least-squares curve is formed by solving the normal equations and plotting it with the data points.

Step by step solution

01

Set Up the System of Equations

For the given points, we want to find the least squares curve in the form \(y = m\sqrt{x} + b\). Using the least squares method, we set up two linear equations by minimizing the sum of squared differences \(S = \sum (y_i - (m\sqrt{x_i} + b))^2\). This results in setting partial derivatives to zero: \(\frac{\partial S}{\partial m} = 0\) and \(\frac{\partial S}{\partial b} = 0\).
02

Calculate Necessary Sums

Calculate the following sums based on the given data: 1. \(\sum \sqrt{x_i}\)2. \(\sum (\sqrt{x_i})^2\)3. \(\sum y_i\)4. \(\sum y_i\sqrt{x_i}\) Using the table values: - \(\sum \sqrt{x_i} = \sqrt{0} + \sqrt{4} + \sqrt{8} + \sqrt{12} + \sqrt{16}\)- \(\sum (\sqrt{x_i})^2 \)- \(\sum y_i = 1 + 9 + 11 + 14 + 15\) - \(\sum y_i\sqrt{x_i}\) by calculating each \(y_i\sqrt{x_i}\) and summing.
03

Solve the Normal Equations

Using the sums calculated, set up the equations \(\begin{align*} bN + m\sum \sqrt{x} &= \sum y \ b\sum \sqrt{x} + m\sum (\sqrt{x})^2 &= \sum y\sqrt{x} \end{align*}\). Solve these linear equations for \(m\) and \(b\) by substituting the previously calculated sums.
04

Find the Values of m and b

Plug the appropriate values into the equations derived in Step 3 and solve them to find the values of \(m\) and \(b\). This involves solving a set of 2 simultaneous linear equations based on the calculated sums.
05

Form the Least Squares Curve Equation

With the found values of \(m\) and \(b\), substitute them back into the equation \(y = m\sqrt{x} + b\) to form the final least squares curve equation.
06

Sketch the Curve and Plot Data Points

Using graphing software or by hand, plot the equation of the curve \(y = m\sqrt{x} + b\). Also, plot the original data points \((0, 1), (4, 9), (8, 11), (12, 14), (16, 15)\) on the same graph to visualize the fit of the curve.

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Key Concepts

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

Curve Fitting
Curve fitting is a mathematical process used to find a curve that best fits a series of data points. In the context of the least squares method, the curve-fitting process aims to minimize the sum of the squared differences between the data points and the curve. This specific problem aims to fit the curve in the form of \(y = m\sqrt{x} + b\), which means we are trying to model our data with a particular type of equation.

This method is crucial when data points do not perfectly align with a simple curve or line, which is common in real-world data.
  • The least squares method ensures the model correctly represents the data by reducing the differences between the predicted and actual values.
  • It effectively minimizes the error, thus providing a best-fit line or curve among many potential options.
A correctly fitted curve allows for a clearer understanding of the underlying trends or relationships within the data.
System of Equations
In the process of least squares curve fitting, setting up a system of equations is an essential step. These equations help find the values of the parameters (in this case, \(m\) and \(b\)) that define the curve.

The system of equations derives from the minimization of the sum of squared differences \(S\), between the observed outcomes and the fitted values. This involves returning the partial derivatives of \(S\) with respect to each parameter to zero, creating a system of simultaneous equations.
  • For this problem, the equations are based on the sums that involve the square root of \(x\) and \(y\).
  • Solving these equations provides the values that make the least squares curve represent the data as accurately as possible.
Solving a system of equations is often done using techniques like substitution or elimination, ensuring coherent and logical results.
Partial Derivatives
Partial derivatives are used in the least squares method to minimize the difference between the predicted and observed data points. These derivatives help in determining the direction and rate of changes in the linear parameters that are being optimized.

When working with multiple variables as in this current problem, we take the derivative of the sum of squared differences \(S\) with respect to each parameter (\(m\) and \(b\)), separately.
  • Taking the derivative with respect to \(m\) and \(b\) allows us to form equations that result in finding the minimum value for \(S\).
  • Setting these derivatives to zero ensures that we are finding a "stationary point," often indicating a minimum in the context of least squares.
Partial derivatives simplify handling problems involving multiple variables by focusing on each variable individually, ensuring an accurate and efficient path to the solution.
Graphing Data
Graphing the data is the final step in completing a curve-fitting task. Visual representation helps to clearly understand how the least squares curve fits the observed data points. A graph provides a tangible visual interpretation of the relationship between variables indicated by the fitted curve.

In our exercise, the task is to sketch the calculated least-squares curve along with the original data points.
  • Plot the data points such as \((0, 1), (4, 9), (8, 11), (12, 14), (16, 15)\) to provide reference positions from the table.
  • Overlay the derived curve using the equation \(y = m\sqrt{x} + b\) with the specific values of \(m\) and \(b\) determined, to illustrate the fit.
Graphing highlights how well the theoretical model aligns with empirical data, showcasing the strengths and possible discrepancies, if any, in the underlying model.

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Most popular questions from this chapter

Find the indicated measure of central tendency. A test of air pollution in a city gave the following readings of the concentration of sulfur dioxide (in parts per million) for 18 consecutive days: $$\begin{aligned} &0.14,0.18,0.27,0.19,0.15,0.22,0.20,0.18,0.15\\\ &0.17,0.24,0.23,0.22,0.18,0.32,0.26,0.17,0.23 \end{aligned}$$ Find the median and the mode of these readings.

Find the indicated measure of central tendency. The weekly salaries (in dollars) for the workers in a small factory are as follows: $$\begin{aligned} &600,750,625,575,525,700,550\\\ &750,625,800,700,575,600,700 \end{aligned}$$ Find the median and the mode of the salaries.

Find the equation of the least-squares line for the given data. Graph the line and data points on the same graph. If gas is cooled under conditions of constant volume, it is noted that the pressure falls nearly proportionally as the temperature. If this were to happen until there was no pressure, the theoretical temperature for this case is referred to as absolute zero. In an elementary experiment, the following data were found for pressure and temperature under constant volume. $$\begin{array}{l|c|r|r|r|r|r}T\left(^{\circ} C\right) & 0.0 & 20 & 40 & 60 & 80 & 100 \\\\\hline P(\mathrm{kPa}) & 133 & 143 & 153 & 162 & 172 & 183\end{array}$$ Find the least-squares line for \(P\) as a function of \(T,\) and from the graph determine the value of absolute zero found in this experiment. Check the values and curve with a calculator.

Find the equation of the least-squares line for the given data. Graph the line and data points on the same graph. The speed \(\nu\) (in \(\mathrm{m} / \mathrm{s}\) ) of sound was measured as a function of the temperature \(T\) (in \(^{\circ} \mathrm{C}\) ) with the following results. Find \(\nu\) as a function of \(T\). $$\begin{array}{l|r|r|r|r|r|r|r}T\left(^{\circ} \mathrm{C}\right) & 0 & 10 & 20 & 30 & 40 & 50 & 60 \\\\\hline v(\mathrm{m} / \mathrm{s}) & 331 & 337 & 344 & 350 & 356 & 363 & 369\end{array}$$

Find the indicated quantities. In testing a computer system, the number of instructions it could perform in 1 ns was measured at different points in a program. The numbers of instructions were recorded as follows: 19,21,22,25,22,20,18,21,20,19,22,21,19,23,21 Form a frequency distribution table for these values.
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