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Chapter 12: Problem 76

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.

Short Answer

Expert verified
2. What is the y-intercept of the line? 3. What is the slope of the line?

Step by step solution

01

Graph the line

To graph the line, we can use the given equation, \(y=0.5x+3\), to find the y-value corresponding to each x-value. Then, we can plot these points on the coordinate plane: - For \(x=0\), we have \(y=0.5(0)+3=3\) - For \(x=1\), we have \(y=0.5(1)+3=3.5\) - For \(x=2\), we have \(y=0.5(2)+3=4\) Plot these points on the coordinate plane and draw a straight line passing through the points.
02

Identify the y-intercept and slope

The y-intercept is the point where the line intersects the y-axis, and this occurs when \(x=0\). Using our results from Step 1, the y-intercept is at the point \((0,3)\). The slope of the line can be found using the equation \(y=0.5x+3\). Since the line is in slope-intercept form (i.e., \(y=mx+b\)), the slope is the coefficient of the x term, which in this case is \(0.5\).
03

Part a Summary

The line corresponding to the equation \(y=0.5x+3\) passes through the points \((0,3)\), \((1,3.5)\), and \((2,4)\). The y-intercept is at the point \((0,3)\), and the slope is \(0.5\).
04

Check your work using an applet

Open the How a Line Works applet and enter the equation of the line, \(y=0.5x+3\). The applet should show a visualization of the line, its slope, and its y-intercept. Confirm that the points \((0,3)\), \((1,3.5)\), and \((2,4)\) are on the line, and that the y-intercept and slope match the values found in Steps 1 and 2.

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Key Concepts

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

Graphing
Graphing is a fundamental concept when dealing with linear regression, as it allows us to visualize mathematical relationships. When you graph a line, you're essentially plotting points on a coordinate plane. For a specific linear equation like \( y = 0.5x + 3 \), the task begins with determining points that lie on this line.
To do this, you can select a set of \( x \)-values, substitute them into the equation, and solve for the corresponding \( y \).
For example:
  • When \( x = 0 \), \( y = 0.5(0) + 3 = 3 \).
  • When \( x = 1 \), \( y = 0.5(1) + 3 = 3.5 \).
  • When \( x = 2 \), \( y = 0.5(2) + 3 = 4 \).
Once these points \((0,3)\), \((1,3.5)\), and \((2,4)\) are plotted, you then connect them with a straight line. This line represents all possible solutions to the equation, forming a visual representation of the relationship between \( x \) and \( y \).
Graphing helps in understanding the nature of the equation and predicting values.
Y-Intercept
In linear equations, the y-intercept represents the point where the line crosses the y-axis. It is crucial because it provides the starting value of \( y \) when \( x \) is zero. With the equation \( y = 0.5x + 3 \), when you set \( x = 0 \), you find that \( y = 3 \).
This means the y-intercept is the point \((0,3)\).
Knowing the y-intercept helps in graphing the equation accurately, as it gives a fixed point on the graph. The intercept will remain the same irrespective of changes to the slope. Therefore, it's a reliable reference point whenever you're plotting or analyzing the graph of a line.
Slope
The concept of slope is essential in understanding how a line behaves on a graph. The slope tells us how much \( y \) changes for a unit change in \( x \). In the equation \( y = 0.5x + 3 \), the slope is represented by the coefficient of \( x \), which is \( 0.5 \).
This implies that for every one unit increase in \( x \), the value of \( y \) increases by 0.5 units.
A positive slope, like 0.5, indicates that the line is rising as you move from left to right. Conversely, a negative slope would mean the line is falling. Understanding the slope is vital for predicting trends and relationships in data sets. It shows how one variable affects another and can provide insights into cause and effect within your graphed data.

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Most popular questions from this chapter

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.

How does the coefficient of correlation measure the strength of the linear relationship between two variables \(y\) and \(x ?\)

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.

If you play tennis, you know that tennis racquets vary in their physical characteristics. The data in the accompanying table give measures of bending stiffness and twisting stiffness as measured by engineering tests for 12 tennis racquets: $$ \begin{array}{ccc} & \text { Bending } & \text { Twisting } \\ \text { Racquet } & \text { Stiffness, } x & \text { Stiffness, } y \\ \hline 1 & 419 & 227 \\ 2 & 407 & 231 \\ 3 & 363 & 200 \\ 4 & 360 & 211 \\ 5 & 257 & 182 \\ 6 & 622 & 304 \\ 7 & 424 & 384 \\ 8 & 359 & 194 \\ 9 & 346 & 158 \\ 10 & 556 & 225 \\ 11 & 474 & 305 \\ 12 & 441 & 235 \end{array} $$ a. If a racquet has bending stiffness, is it also likely to have twisting stiffness? Do the data provide evidence that \(x\) and \(y\) are positively correlated? b. Calculate the coefficient of determination \(r^{2}\) and interpret its value.

The makers of the Lexus EX1274 automobile have steadily increased their sales since their U.S. launch in \(1989 .\) However, the rate of increase changed in 1996 when Lexus introduced a line of trucks. The sales of Lexus from 1996 to 2005 are shown in the table: \({ }^{18}\) $$ \begin{aligned} &\begin{array}{l|rrrrrrrrrrr} \text { Year } & 1996 & 1997 & 1998 & 1999 & 2000 & 2001 & 2002 & 2003 & 2004 & 2005 \\ \hline \text { Sales of thousands } & 80 & 100 & 155 & 180 & 210 & 224 & 234 & 260 & 288 & 303 \end{array}\\\ &\text { vehicles } \end{aligned} $$ a. Plot the data using a scatterplot. How would you describe the relationship between year and sales of Lexus? b. Find the least-squares regression line relating the sales of Lexus to the year being measured? c. Is there sufficient evidence to indicate that sales are linearly related to year? Use \(\alpha=.05\) d. Predict the sales of Lexus for the year 2006 using a \(95 \%\) prediction interval. e. If they are available, examine the diagnostic plots to check the validity of the regression assumptions. f. If you were to predict the sales of Lexus in the year \(2015,\) what problems might arise with your prediction?
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