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Chapter 26: Problem 16

The equation of the least-squares regression line for predicting selling price from appraised value is (a) price \(=100.2+1.2186 \times\) appraised value. (b) price \(=1.2186+100.2 \times\) appraised value. (c) price \(=114.0+0.1446 \times\) appraised value.

Short Answer

Expert verified
Option (a) is correct: price = 100.2 + 1.2186 脳 appraised value.

Step by step solution

01

Analyze the Equation Format

Recall that the equation of a least-squares regression line is typically written in the form of \( y = a + bx \), where \( a \) is the y-intercept, and \( b \) is the slope of the line.
02

Evaluate Each Option

Option (a) is written as \( \text{price} = 100.2 + 1.2186 \times \text{appraised value} \), which correctly follows the format \( y = a + bx \) with \( a = 100.2 \) and \( b = 1.2186 \).Option (b) is \( \text{price} = 1.2186 + 100.2 \times \text{appraised value} \), which incorrectly places \( a \) and \( b \) in reverse positions.Option (c) is \( \text{price} = 114.0 + 0.1446 \times \text{appraised value} \), which follows the correct structure but with different parameters for \( a \) and \( b \).
03

Select the Correct Equation

Of the given options, option (a) is formatted correctly and uses common numerical parameter values that align with the notation of \( y = a + bx \).
04

Verification

Ensure all transformations and mathematical symbols align with the regression line format, which option (a) does, signifying it as most likely the correct representation.

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Key Concepts

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

Regression Line
In the world of statistics, a regression line is a tool used to predict the relationship between two variables. Imagine plotting a series of data points on a graph - these data points might show some degree of variation but could trend in a certain direction. The regression line, also known as the line of best fit, is drawn through these points to showcase this trend.
By predicting outcomes, this line helps in understanding the correlation and causation between variables. When dealing with least-squares regression lines, the line is calculated to minimize the distance of the data points from the line itself. This provides the closest average approximation of all points. Think of it as the best representation of your data in a linear form.
Slope
The slope of a line is a measure that indicates how steep the line is. In the context of a regression line, it shows the relationship between the changes in the dependent variable and the independent variable. When you look at the equation of a line, the slope is typically represented by the variable "b" in the formula \( y = a + bx \).
  • A positive slope means as the independent variable increases, the dependent variable also increases.
  • A negative slope suggests the opposite 鈥� as one variable goes up, the other goes down.
Understanding the slope is critical to interpreting the nature of the relationship between the variables. It tells us how one variable is expected to change as the other variable changes. When analyzing the slope, pay close attention as it highlights the rate of change occurring between your variables.
Y-Intercept
The y-intercept is an essential part of the regression equation that indicates where the line crosses the y-axis. In the equation \( y = a + bx \), the "a" represents the y-intercept. This point reveals the value of the dependent variable when the independent variable is zero. It sets the baseline from which the effects of changes in the independent variable are measured.
Whenever you analyze a regression line, knowing the y-intercept helps you understand where your prediction begins. Imagine any scenario where no variable contributes to change - this is where the y-intercept plays its pivotal role. It is a constant added to account for the position of the line relative to the y-axis.
Equation Format
Understanding the correct equation format is crucial when working with regression lines. The standard form is \( y = a + bx \), where:
  • \( y \) represents the dependent variable (what you want to predict).
  • \( a \) is the y-intercept (baseline value).
  • \( b \) is the slope (rate of change).
  • \( x \) represents the independent variable (predictor).
By ensuring that you use this format correctly, you can accurately represent the behavior of your data. Each variable in this formula has a specific role, and understanding their functions allows for proper interpretation of the relationship between variables. The format enables predictions and insights, ensuring clarity and consistency in data analysis.

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Most popular questions from this chapter

The slope \(\beta\) of the population regression line describes (a) the average selling price in a population of units when a unit's appraised value is 0 . (b) the average increase in selling price in a population of units when appraised value increases by \(\$ 1000\). (c) the exact increase in the selling price of an individual unit when its appraised value increases by \(\$ 1000\).

We know that there is a strong linear relationship. Let's check the other conditions for inference. Figure \(26.14\) includes a table of the two variables, the predicted values for each \(x\) in the data, the residuals, and related quantities. (a) Round the residuals to the nearest whole number and make a stemplot. The distribution is single peaked and symmetric and appears close to Normal. (b) Make a residual plot of residuals against boats registered. Use a vertical scale from \(-20\) to 20 to show the pattern more clearly. Add the "residual \(=0\) " line. There is no clearly nonlinear pattern. The spread about the line may be a bit greater for larger values of the explanatory variable, but the effect is not large. (c) It is reasonable to regard the number of manatees killed by boats in successive years as independent. The number of boats grew over time. Someone says that pollution also grew over time and may explain more manatee deaths. How would you respond to this idea?

A strain of genetically engineered cotton, known as Bt cotton, is resistant to certain insects, which results in larger yields of cotton. Farmers in northern China have increased the number of acres planted in Bt cotton. Because Bt cotton is resistant to certain pests, farmers have also reduced their use of insecticide. Scientists in China were interested in the longterm effects of Bt cotton cultivation and decreased insecticide use on insect populations that are not affected by Bt cotton. One such insect is the mirid bug. Scientists measured the number of mirid bugs per 100 plants and the proportion of Bt cotton planted at 38 locations in northern China for the 12 -year period from 1997-2008. The scientists reported a regression analysis as follows: \({ }^{8}\) number of mirid bugs per 100 plants $$ \begin{aligned} =0.54+6.81 & \times \mathrm{B} \text { t cotton planting proportionr } 2=0.90, \mathrm{P}<0.0001 \\ &=0.54+6.81 \times \text { Bt cotton planting proportion } \\ r^{2} &=0.90, \quad P<0.0001 \end{aligned} $$ (a) What does the slope \(b=6.81\) say about the relation between Bt cotton planting proportion and number of mirid bugs per 100 plants? (b) What does \(r^{2}=0.90\) add to the information given by the equation of the least-squares line? (c) What null and alternative hypotheses do you think the P-value refers to? What does this P-value tell you? (d) Does the large value of \(r^{2}\) and the small \(P\)-value indicate that increasing the proportion of acres planted in Bt cotton causes an increase in mirid bugs?

Exercise \(26.1\) gives data on wine consumption and the risk of breast cancer. Software tells us that the leastsquares slope is \(b=0.009012\) with standard error \(\mathrm{SE}_{b}=0.001112\). Because there are only four observations, the observed slope \(b\) may not be an accurate estimate of the population slope \(\beta\). Give a \(90 \%\) confidence interval for \(\beta\).

A 95\% confidence interval for the population slope \(\beta\) is (a) \(1.2186 \pm 424.4836\). (b) \(1.2186 \pm 0.2905\). (c) \(1.2186 \pm 0.1446\).
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