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Chapter 3: Problem 24

Identify the values of the \(y\) -intercept \(a\) and the slope \(b\), and sketch the following regression lines, for values of \(x\) between 0 and 10 a. \(\hat{y}=7+0.5 x\) b. \(\hat{y}=7+x\) c. \(\hat{y}=7-x\) d. \(\hat{y}=7\)

Short Answer

Expert verified
a: \( y \)-intercept 7, slope 0.5 b: \( y \)-intercept 7, slope 1 c: \( y \)-intercept 7, slope -1 d: \( y \)-intercept 7, slope 0

Step by step solution

01

Identify Y-intercept and Slope for Line (a)

For the line given by equation (a), \( \hat{y} = 7 + 0.5x \), the \( y \)-intercept \( a \) is 7, and the slope \( b \) is 0.5.
02

Identify Y-intercept and Slope for Line (b)

For the line given by equation (b), \( \hat{y} = 7 + x \), the \( y \)-intercept \( a \) is 7, and the slope \( b \) is 1.
03

Identify Y-intercept and Slope for Line (c)

For the line in equation (c), \( \hat{y} = 7 - x \), the \( y \)-intercept \( a \) is 7, and the slope \( b \) is -1.
04

Identify Y-intercept and Slope for Horizontal Line (d)

For the line given by equation (d), \( \hat{y} = 7 \), the \( y \)-intercept \( a \) is 7, and it has no slope (or a slope of 0 since the line is horizontal).
05

Sketch the Lines

To sketch the lines, start at the \( y \)-intercept (7 on the \( y \)-axis) for each. - For (a), rise 0.5 for every 1 unit in \( x \).- For (b), rise 1 for every 1 unit in \( x \).- For (c), fall 1 for every 1 unit in \( x \).- For (d), draw a horizontal line at \( y = 7 \) for all \( x \).

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Key Concepts

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

Understanding the Y-intercept
The concept of the y-intercept is crucial in linear regression and graphing lines. It represents the point where the line crosses the y-axis. This point is denoted by the letter \( a \) in the equation of a line, which is often expressed in the form \( \hat{y} = a + bx \). For each line in the exercise, this is where \( x = 0 \). The value of \( y \) at this point will be \( a \).

Here's how to identify it for each given line:
  • For equation (a), \( \hat{y} = 7 + 0.5x \), the y-intercept is 7.
  • In equation (b), \( \hat{y} = 7 + x \), the y-intercept again is 7.
  • Similarly, in equation (c), \( \hat{y} = 7 - x \), the y-intercept is also 7.
  • Lastly, for equation (d), \( \hat{y} = 7 \), the y-intercept is, unsurprisingly, 7.
No matter the slope, the y-intercept in this exercise remains constant at 7. This means that all lines cross the y-axis at the point where y equals 7.
Defining the Slope
Slope is another foundational concept associated with linear regression and graphing lines. It indicates how steep the line is, as well as its direction - which tells you how much \( y \) will rise or fall with each increase in \( x \).

The slope \( b \) in the equation of a line is the coefficient of \( x \). Here's how it works with each of the given lines:
  • In equation (a), \( \hat{y} = 7 + 0.5x \), the slope is 0.5. This means for every unit increase in \( x \), \( y \) increases by 0.5.
  • For equation (b), \( \hat{y} = 7 + x \), the slope is 1. Here, for each increase in \( x \), \( y \) rises by 1 unit.
  • With equation (c), \( \hat{y} = 7 - x \), the slope is -1, indicating \( y \) decreases by 1 for every unit increase in \( x \).
  • Finally, for equation (d), \( \hat{y} = 7 \), the slope is 0 because the line is horizontal, meaning \( y \) does not change as \( x \) changes.
Recognizing the slope helps you understand how different lines behave relative to the x-axis.
Graphing Lines with Linear Equations
Graphing lines using linear equations involves both the y-intercept and the slope. The y-intercept gives you a starting point, and the slope directs the tilt of the line.

When graphing the lines from the exercise, remember:
  • All lines intersect the y-axis at (0, 7), as this is consistent across all equations.
  • For equation (a), start at the y-intercept 7 and move up 0.5 for every step to the right by 1 on the x-axis. This gives a gentle rise.
  • In equation (b), start at the same y-intercept. Move up 1 for each move to the right on the x-axis for a steeper line.
  • With equation (c), also start at 7. For every unit step right on the x-axis, move down 1 on the y-axis, creating a descending line.
  • For equation (d), the line is horizontal at y = 7, so it runs parallel to the x-axis with no slope change.
Graphing these lines helps visualize the relationship defined by their linear equations and better grasp the concepts of slope and y-intercept.

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Most popular questions from this chapter

Each month, the owner of Fay's Tanning Salon records in a data file the monthly total sales receipts and the amount spent that month on advertising. a. Identify the two variables. b. For each variable, indicate whether it is quantitative or categorical. c. Identify the response variable and the explanatory variable.

The weight (in carats) and the price (in millions of dollars) of the 9 most expensive diamonds in the world was collected from www.elitetraveler.com. Let the explanatory variable \(x=\) weight and the response variable \(y=\) price. The regression equation is \(\hat{y}=109.618+0.043 x\). a. Princie is a diamond whose weight is 34.65 carats. Use the regression equation to predict its price. b. The selling price of Princie is \(\$ 39.3\) million. Calculate the residual associated with the diamond and comment on its value in the context of the problem. c. The correlation coefficient is \(0.053 .\) Does it mean that a diamond's weight is a reliable predictor of its price?

An article in the September \(16,2006,\) issue of The Economist showed a scatterplot for many nations relating the response variable \(y=\) annual oil consumption per person (in barrels) and the explanatory variable \(x=\) gross domestic product (GDP, per person, in thousands of dollars). The values shown on the plot were approximately as shown in the table. a. Create a data file and use it to construct a scatterplot. Interpret. b. Find and interpret the prediction equation. c. Find and interpret the correlation. d. Find and interpret the residual for Canada.

According to data selected from GSS in \(2014,\) the correlation between \(y=\) email hours per week and \(x=\) ideal number of children is -0.0008 a. Would you call this association strong or weak? Explain. b. The correlation between email hours per week and Internet hours per week is \(0.33 .\) For this sample, which explanatory variable, ideal number of children or Internet hours per week, seems to have a stronger association with \(y ?\) Explain.

Data used in this exercise was published by www.bloomberg.com for the most government debt per person for 58 countries and their respective population sizes in \(2014 .\) When using population size (in millions) as the explanatory variable \(x,\) and government debt per person (in dollars) as the response variable \(y,\) the regression equation is predicted as government debt per person \(=19560.405-13.495\) population. a. Interpret the slope of the regression equation. Is the association positive or negative? Explain what this means. b. Predict government debt per person at the (i) minimum population size \(x\) value of 4 million, (ii) at the maximum population size \(x\) value of 1367.5 million. c. For India, government debt per person \(=\$ 946,\) and population \(=1259.7\) million. Find the predicted government debt per person and the residual for India. Interpret the value of this residual.
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