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Chapter 12: Problem 6

You are given five points with these coordinates: $$ \begin{array}{c|rrrrrrr} x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 1 & 1 & 3 & 5 & 5 \end{array} $$ a. Use the data entry method on your scientific or graphing calculator to enter the \(n=5\) observations. Find the sums of squares and cross-products, \(S_{x x} S_{x y},\) and \(S_{y y}\) b. Find the least-squares line for the data. c. Plot the five points and graph the line in part b. Does the line appear to provide a good fit to the data points? d. Construct the ANOVA table for the linear regression.

Short Answer

Expert verified
Answer: No, the least-squares line is not a good fit for the data points, as indicated by the negative error sum of squares (SSE) in the ANOVA table.

Step by step solution

01

Calculate sums and products of x and y values

First, we need to calculate the summations of \(x, y, x^2, y^2,\) and \(xy\) as follows: $$ \sum x = -2-1+0+1+2 = 0 $$ $$ \sum y = 1+1+3+5+5 = 15 $$ $$ \sum x^2 = (-2)^2+(-1)^2+0^2+1^2+2^2 = 10 $$ $$ \sum y^2 = 1^2+1^2+3^2+5^2+5^2 = 61 $$ $$ \sum xy = (-2)(1)+(-1)(1)+0(3)+1(5)+2(5)=14 $$
02

Calculate the sums of squares and cross-products

Now we will find the sums of squares and cross-products, using the calculated summations: $$ S_{x x} = \sum x^2 - \frac{(\sum x)^2}{n} = 10 - \frac{(0)^2}{5} = 10 $$ $$ S_{x y} = \sum xy - \frac{(\sum x)(\sum y)}{n} = 14 - \frac{(0)(15)}{5}= 14 $$ $$ S_{y y} = \sum y^2 - \frac{(\sum y)^2}{n} = 61 - \frac{(15)^2}{5} = 16 $$
03

Find the least-squares line

To find the least-squares line, we need to calculate the slope \(b_1\) and the intercept \(b_0\) using the formulas: $$ b_1 = \frac{S_{xy}}{S_{xx}} = \frac{14}{10} = 1.4 $$ $$ b_0 = \bar{y} - b_1 \bar{x} = \frac{\sum y}{n} - 1.4 \frac{\sum x}{n} = \frac{15}{5} - 1.4 \cdot \frac{0}{5} = 3 $$ Thus, the least-squares line is given by the equation: $$ y = 3 + 1.4x $$
04

Plot the points and the line

Using graph paper or a graphing calculator, plot the given points and draw the least-squares line \(y = 3 + 1.4x\). Examine the plot and decide if the line appears to provide a good fit to the data points. Note that a detailed assessment would require additional steps, such as calculating and analyzing residuals.
05

Construct the ANOVA table

To construct the ANOVA table, we first need to calculate the total sum of squares (SST), the regression sum of squares (SSR), and the error sum of squares (SSE). We already have \(S_{yy}\), which is SST. $$ \text{SST} = S_{yy} = 16 $$ $$ \text{SSR} = b_1^2 S_{xx} = (1.4)^2 (10) = 19.6 $$ $$ \text{SSE} = \text{SST} - \text{SSR} = 16 - 19.6 = -3.6 $$ Since the SSE is negative, the line does not provide a good fit to the data points. Now, we can create the ANOVA table: $$ \begin{array}{|c|c|c|c|c|} \hline \text{Source } & \text{d.f.} & \text{Sum of Squares } & \text{Mean Square} & \text{F} \\ \hline \text{Regression} & 1 & 19.6 & 19.6 & {S}_{\mathrm{b}_{1}} \mathrm{/S}_{e} \mathrm{/}{S}_{b}_{1}\\ \hline \text{Error} & 3 & -3.6 & -1.2 & \\ \hline \text{Total} & 4 & 16 & & \\ \hline \end{array} $$

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Key Concepts

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

Sums of Squares
In statistics, sums of squares are a fundamental concept used to understand variability. For linear regression, these sums help gauge how well your model fits your data. You will encounter three main sums of squares:
  • Sum of Squares Total (SST): This quantifies the total variation in the dependent variable, considering it includes both the variation explained by the regression (SSR) and the residual variation (SSE). It is represented as the total deviation of the observed values from the mean.
  • Sum of Squares for Regression (SSR): This measures how much of SST is explained by the regression line. It's essentially how much of the variation your model accounts for. A higher SSR means a better fit of the model.
  • Sum of Squares for Error (SSE): This denotes the variation caused by variables not explained by your regression model. A lower SSE indicates a closer fit of the line to the data points.
To calculate these, we start with individual sums of squares for each variable and cross-products between variables, such as \(S_{xx}\), \(S_{yy}\), and \(S_{xy}\). In a nutshell, understanding sums of squares is key to dissecting variance in data and assessing the quality of a model's prediction power.
Least-Squares Line
The least-squares line in linear regression aims to minimize the distance between the data points and the regression line itself. Effectively, this approach attempts to find the best-fitting line by reducing the total of the squared differences (errors) between the observed values and the values predicted by the line. The equation takes the form:\[ y = b_0 + b_1x \]where:
  • \(b_0\) is the y-intercept, showing where the line crosses the y-axis when \(x = 0\).
  • \(b_1\) is the slope, indicating the rate of change in \(y\) with respect to a unit change in \(x\).
To determine these coefficients, you use the sums of square calculations of \(S_{xx}\) and \(S_{xy}\). For example, \(b_1\) is derived from \(b_1 = \frac{S_{xy}}{S_{xx}}\) and \(b_0\) is found via \(b_0 = \bar{y} - b_1\bar{x}\). Here, \(\bar{x}\) and \(\bar{y}\) are the mean values of \(x\) and \(y\) respectively. The least-squares method ensures that the line has the smallest possible variation from the actual data points.
ANOVA Table
An ANOVA (Analysis of Variance) table in the context of linear regression is used to summarize the partitioning of the sums of squares into components associated with the regression model and the error term. It's a convenient way to compile the results of a linear regression analysis.An ANOVA table typically consists of:
  • Source: Categories of variance source, such as Regression and Error.
  • Degrees of Freedom (d.f.): Represents the number of values that are free to vary.
    • Regression has 1 degree of freedom because we estimate a single slope.
    • Error (usually termed residual) has \(n - 2\) degrees of freedom in simple linear regression, where \(n\) is the number of data points.
  • Sum of Squares: Discusses how the total variation is divided between the different components, as calculated earlier.
  • Mean Square: It's the Sum of Squares divided by their respective degrees of freedom.
  • F-Value: This statistic (F-test) evaluates the overall significance of the regression model. It checks if the explained variance by the model is significant compared to the unexplained variance.
This table is essential for determining whether the predictive variable significantly accounts for the outcome variable's variation.

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Most popular questions from this chapter

An experiment was conducted to investigate the effect of a training program on the length of time for a typical male college student to complete the 100 -yard dash. Nine students were placed in the program. The reduction \(y\) in time to complete the 100 -yard dash was measured for three students at the end of 2 weeks, for three at the end of 4 weeks, and for three at the end of 6 weeks of training. The data are given in the table. $$ \begin{array}{l|l|l|l} \text { Reduction in Time, } y(\mathrm{sec}) & 1.6, .8,1.0 & 2.1,1.6,2.5 & 3.8,2.7,3.1 \\ \hline \text { Length of Training, } x(\mathrm{wk}) & 2 & 4 & 6 \end{array} $$ Use an appropriate computer software package to analyze these data. State any conclusions you can draw.

Graph the line corresponding to the equation \(y=2 x+1\) by graphing the points corresponding to \(x=0,1,\) and \(2 .\) Give the \(y\) -intercept and slope for the line.

A horticulturalist devised a scale to measure the freshness of roses that were packaged and stored for varying periods of time before transplanting. The freshness measurement \(y\) and the length of time in days that the rose is pack-aged and stored before transplanting \(x\) are given below. $$ \begin{array}{l|lllll} x & 5 & 10 & 15 & 20 & 25 \\ \hline y & 15.3 & 13.6 & 9.8 & 5.5 & 1.8 \\ & 16.8 & 13.8 & 8.7 & 4.7 & 1.0 \end{array} $$ a. Fit a least-squares line to the data. b. Construct the ANOVA table. c. Is there sufficient evidence to indicate that freshness is linearly related to storage time? Use \(\alpha=.05 .\) d. Estimate the mean rate of change in freshness for a 1 -day increase in storage time usig a \(98 \%\) confidence interval. e. Estimate the expected freshness measurement for a storage time of 14 days with a \(95 \%\) confidence interval. f. Of what value is the linear model in reference to \(\bar{y}\) in predicting freshness?

Professor Isaac Asimov was one of the most prolific writers of all time. Prior to his death, he wrote nearly 500 books during a 40-year career. In fact, as his career progressed, he became even more productive in terms of the number of books written within a given period of time. \({ }^{1}\) The data give the time in months required to write his books in increments of 100 : $$ \begin{array}{l|ccccc} \text { Number of Books, } x & 100 & 200 & 300 & 400 & 490 \\ \hline \text { Time in Months, } y & 237 & 350 & 419 & 465 & 507 \end{array} $$ a. Assume that the number of books \(x\) and the time in months \(y\) are linearly related. Find the least-squares line relating \(y\) to \(x\). b. Plot the time as a function of the number of books written using a scatterplot, and graph the leastsquares line on the same paper. Does it seem to provide a good fit to the data points? c. Construct the ANOVA table for the linear regression.

You can refresh your memory about regression lines and the correlation coefficient by doing the MyApplet Exercises at the end of Chapter \(3 .\) a. Graph the line corresponding to the equation \(y=0.5 x+3\) by graphing the points corresponding to \(x=0,1,\) and 2 . Give the \(y\) -intercept and slope for the line. b. Check your graph using the How a Line Works applet.
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