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Chapter 22: Problem 6

Find the equation of the indicated least squares curve. Sketch the curve and plot the data points on the same graph. The increase in length \(y\) of a certain metallic rod was measured in relation to particular increases \(x\) in temperature. Find the leastsquares curve \(y=m x^{2}+b .\) Compare the equation with that using the quadratic regression feature of a calculator. $$\begin{array}{l|r|r|r|r|r}x\left(^{\circ} \mathrm{C}\right) & 50.0 & 100 & 150 & 200 & 250 \\\\\hline y(\mathrm{cm}) & 1.00 & 4.40 & 9.40 & 16.4 & 24.0\end{array}$$

Short Answer

Expert verified
The least squares curve is \(y = 0.0035x^2 - 2.12\).

Step by step solution

01

Define the Variables

We are given a set of data points with temperature increase values \(x\) and corresponding rod length increases \(y\). Our goal is to fit a curve of the form \(y = mx^2 + b\) using the least squares method. The data points provided are: \((50, 1.00), (100, 4.40), (150, 9.40), (200, 16.4), (250, 24.0)\).
02

Calculate the Sums Needed for the Equation

To find the least squares curve, we need the following sums: \(\sum x^2\), \(\sum x^4\), \(\sum y\), \(\sum x^2y\), and \(\sum x^2\). Calculate each for the given data set:- \(\sum x^2 = 50^2 + 100^2 + 150^2 + 200^2 + 250^2 = 87500\)- \(\sum x^4 = 50^4 + 100^4 + 150^4 + 200^4 + 250^4 = 1525000000\)- \(\sum y = 1.00 + 4.40 + 9.40 + 16.4 + 24.0 = 55.2\)- \(\sum x^2y = 50^2 \times 1.00 + 100^2 \times 4.40 + 150^2 \times 9.40 + 200^2 \times 16.4 + 250^2 \times 24.0 = 1105000\)
03

Set Up the Normal Equations

The normal equations for this problem are derived from the least squares method, specifically for a quadratic form:\[\begin{align*}N\cdot b + \sum x^2\cdot m &= \sum y \\sum x^2\cdot b + \sum x^4\cdot m &= \sum x^2y\end{align*}\]Substitute the sums calculated in the previous step:\[\begin{align*}5b + 87500m &= 55.2 \87500b + 1525000000m &= 1105000\end{align*}\]
04

Solve the Normal Equations

Solve the system of linear equations from Step 3 using substitution or matrix methods (e.g., Gaussian elimination or Cramer's Rule). Simplifying, we find that:\(b \approx -2.12\) and \(m \approx 0.0035\).
05

Finalize the Equation

Substitute the solution values back into the quadratic equation \(y = mx^2 + b\). The least squares quadratic curve equation becomes:\(y = 0.0035x^2 - 2.12\).
06

Verification Using Calculator

Use the quadratic regression feature of a calculator for verification. Enter the data points \((x, y)\) and obtain the quadratic regression parameters. Confirm that the regression matches the derived quadratic equation to ensure accuracy.
07

Plot the Data and Curve

Using graphing software or graph paper, plot the original data points: \((50, 1.00), (100, 4.40), (150, 9.40), (200, 16.4), (250, 24.0)\). Then, plot the least squares quadratic curve \(y = 0.0035x^2 - 2.12\). Compare the plotted curve against individual data points to visually verify the fit.

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Key Concepts

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

Quadratic Equation
A quadratic equation is a type of polynomial equation of the degree two, typically written in the form \(ax^2 + bx + c = 0\). Here, each of \(a\), \(b\), and \(c\) is a constant, and \(x\) represents the variable. Quadratic equations are characteristic because their graph forms a distinctive U-shape known as a parabola.
  • Parabolas can either open upwards or downwards depending on the sign of \(a\).
  • The vertex of a parabola is the highest or lowest point, while the axis of symmetry is a vertical line that passes through the vertex.
In our exercise, the quadratic equation takes the form of \(y = mx^2 + b\), focusing on modeling data with variations influenced more significantly by the square of the independent variable. This application is crucial in fitting curves to specific datasets.
Data Analysis
Data analysis involves systematically applying statistical and logical techniques to describe and evaluate data. It helps to extract meaningful insights that inform future decisions.
  • Cleaning: Ensures data accuracy by identifying and correcting errors.
  • Exploration: Visual inspections and computations provide an overview.
  • Modeling: Employ statistical methods to predict or describe outcomes.
In the provided exercise, data analysis allows us to determine how the increase in temperature relates to the elongation of a metal rod. By calculating necessary sums like \(\sum x^2\) and \(\sum y\), we can prepare for generating a model that reflects the data trend.
Understanding these metrics paves the way for more accurate predictions, minimizing errors in curve fitting through disciplined preparation.
Quadratic Regression
Quadratic regression expands upon linear regression by modeling data as a parabolic curve rather than a straight line. It is more useful when the data shows a curvilinear relationship.
  • Quadratic regression finds the best-fit curve using a method called "least squares," minimizing the sum of the squares of the differences between the observed and predicted values.
  • This process yields coefficients that define the curve.
The quadratic regression in this exercise arrives at an equation \(y = 0.0035x^2 - 2.12\).
This equation not only fits the observed data points but also anticipates values for unmeasured temperatures, highlighting the method鈥檚 strength in predicting nonlinear trends.
Statistical Methods
Statistical methods are mathematical techniques for analyzing and interpreting data to discover underlying patterns or trends. Among many processes, the least squares method forms the backbone for curve fitting in data analysis.
  • Least squares minimize variance between predicted and actual values, effectively balancing influential data points.
  • Normal equations formulate from these, providing solutions that characterize the predictive model.
In solving the presented problem, normal equations are employed to dissect and understand relationships within data, yielding parameters \(m\) and \(b\) for the equation. Techniques like matrix operations, Gaussian elimination, or Cramer's Rule further assist in solving these equations.
Verifying this method through calculator regression ensures accuracy and offers practical assurance of analyses conducted in more theoretical contexts.

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Most popular questions from this chapter

Find the equation of the indicated least squares curve. Sketch the curve and plot the data points on the same graph. The average daily temperatures \(T\) (in \(^{\circ} \mathrm{F}\) ) for each month in Minneapolis (National Weather Service records) are given in the following table. $$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c}t & \mathrm{J} & \mathrm{F} & \mathrm{M} & \mathrm{A} & \mathrm{M} & \mathrm{J} & \mathrm{J} & \mathrm{A} & \mathrm{S} & \mathrm{O} & \mathrm{N} & \mathrm{D} \\\\\hline T\left(^{\circ} \mathrm{F}\right) & 11 & 18 & 29 & 46 & 57 & 68 & 73 & 71 & 61 & 50 & 33 & 19\end{array}$$ Find the least-squares curve \(T=m \cos \left[\frac{\pi}{6}(t-0.5)\right]+b .\) Assume the average temperature is for the 15 th of each month. Then the values of \(t\) (in months) are \(0.5,1.5, \ldots, 11.5 .\) (The fit is fairly good.) Compare the equation using the sinusoidal regression feature of a calculator. What are the main reasons for the differences in the equations?

Solve the given problems. With \(59 \%\) of the area under the normal curve between \(z_{1}\) and \(z_{2}\) to the left of \(z_{2}=1.1,\) find \(z_{1}\)

Use the following sets of numbers. They are the same as those used in Exercise 22.2. A: 3,6,4,2,5,4,7,6,3,4,6,4,5,7,3 B: 25,26,23,24,25,28,26,27,23,28,25 C: 0.48,0.53,0.49,0.45,0.55,0.49,0.47,0.55,0.48,0.57,0.51,0.46,0.53,0.50,0.49,0.53 D: 105, 108, 103, 108, 106, 104, 109, 104, 110, 108, 108, 104, 113,106,107,106,107,109,105,111,109,108 to find the standard deviation s for the indicated sets of numbers. Set \(A\)

Solve the given problems. The residents of a city suburb live at a mean distance of \(16.0 \mathrm{km}\) from the center of the city, with a standard deviation of \(4.0 \mathrm{km}\) What percent of the residents live between \(12.0 \mathrm{km}\) and \(18.0 \mathrm{km}\) of the center of the city?

Find the equation of the indicated least squares curve. Sketch the curve and plot the data points on the same graph. A fraction \(f\) of annual hot-water loads at a certain facility are heated by solar energy. The fractions \(f\) for certain values of the collector area \(A\) are given in the following table. Find the leastsquares curve \(f=m \sqrt{A}+b .\) Compare the equation with that using the power regression feature of a calculator. $$\begin{array}{l|l|l|l|l|l} A\left(\mathrm{m}^{2}\right) & 0 & 12 & 27 & 56 & 90 \\\\\hline f & 0.0 & 0.2 & 0.4 & 0.6 & 0.8\end{array}$$
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