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Chapter 9: Problem 1

In the least-squares line \(\hat{y}=5-2 x,\) what is the value of the slope? When \(x\) changes by 1 unit, by how much does \(\hat{y}\) change?

Short Answer

Expert verified
The slope is -2, and \(\hat{y}\) changes by -2 for each unit increase in \(x\).

Step by step solution

01

Identify the Slope in the Equation

The given least-squares line is \( \hat{y} = 5 - 2x \). In the equation of a line \( \hat{y} = mx + b \), \(m\) represents the slope. Here, it is \(-2\).
02

Understand the Meaning of the Slope

The slope \(-2\) indicates that for every 1 unit increase in \(x\), the value of \(\hat{y}\) decreases by 2 units. This is because the slope is negative.
03

Relate Slope to Change in \(\hat{y}\)

For a 1 unit change in \(x\), \(\hat{y}\) changes by exactly the value of the slope. Since the slope is \(-2\), \(\hat{y}\) changes by \(-2\) units.

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Key Concepts

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

Slope
The concept of slope is fundamental when discussing the equation of a line, particularly in the context of least squares regression. The slope of a line, symbolized as \( m \), indicates the rate at which \( y \) changes with respect to changes in \( x \). It's essentially a measure of steepness. In the given least-squares line equation \( \hat{y} = 5 - 2x \), the slope \( m \) is \(-2\).
  • A negative slope, like \(-2\), tells us that for every unit increase in \( x \), the output, \( \hat{y} \), decreases by 2 units.
  • If the slope were positive, \( \hat{y} \) would increase as \( x \) increases.
  • A slope of zero would mean that \( \hat{y} \) stays constant regardless of any changes in \( x \).
Understanding the slope is crucial, as it provides insight into the relationship between variables. In practical terms, if you're looking at a graph, a steep slope indicates rapid change, while a flatter slope shows slower change.
Equation of a Line
In mathematics, the equation of a line is a useful tool to describe a straight line. It typically takes the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form is straightforward and allows for easy interpretation of linear relationships.
  • Slope \( (m) \): As discussed, it reflects how steep the line is and the direction of the line's tilt.
  • Y-Intercept \( (b) \): This is the value of \( y \) when \( x = 0 \). It tells you where the line crosses the y-axis.
    In the equation \( \hat{y} = 5 - 2x \), the y-intercept is 5, meaning the line crosses the y-axis at the point (0, 5).
Recognizing the components of the equation helps us understand not only the relationship between \( x \) and \( y \) but also predict \( y \) for any given value of \( x \). These predictions are the essence of the least squares line in statistics, where the goal is to best fit the line to data points.
Change in Variables
The change in variables, particularly in the least squares line context, ties back to the slope. As the independent variable \( x \) changes, the dependent variable \( \hat{y} \) responds. This relationship is quantitatively defined by the slope. In our example, each unit of change in \( x \) leads to a corresponding change in \( \hat{y} \) by the amount of the slope, \(-2\).
  • If \( x \) increases by 1, \( \hat{y} \) decreases by 2 (because of the negative slope).
  • If \( x \) decreases by 1, \( \hat{y} \) increases by 2, again due to the negative slope.
It's important to note this change facilitates prediction: you can tell how \( \hat{y} \) will behave with any change in \( x \). Modeling this relationship accurately is essential in applications like forecasting or analyzing trends in data.

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Most popular questions from this chapter

Over the past 50 years, there has been a strong negative correlation between average annual income and the record time to run I mile. In other words, average annual incomes have been rising while the record time to run 1 mile has been decreasing. (a) Do you think increasing incomes cause decreasing times to run the mile? Explain. (b) What lurking variables might be causing the change in one or both of the variables? Explain.

Please do the following. (a) Draw a scatter diagram displaying the data. (b) Verify the given sums \(\Sigma x, \Sigma y, \Sigma x^{2}, \Sigma y^{2},\) and \(\Sigma x y\) and the value of the sample correlation coefficient \(r\) (c) Find \(\bar{x}, \bar{y}, a,\) and \(b .\) Then find the equation of the least- squares line \(\hat{y}=a+b x\) (d) Graph the least-squares line on your scatter diagram. Be sure to use the point \((\bar{x}, \bar{y})\) as one of the points on the line. (e) Interpretation Find the value of the coefficient of determination \(r^{2} .\) What percentage of the variation in \(y\) can be explained by the corresponding variation in \(x\) and the least-squares line? What percentage is unexplained? Answers may vary slightly due to rounding. Cricket Chirps: Temperature Anyone who has been outdoors on a summer evening has probably heard crickets. Did you know that it is possible to use the cricket as a thermometer? Crickets tend to chirp more frequently as temperatures increase. This phenomenon was studied in detail by George W. Pierce, a physics professor at Harvard. In the following data, \(x\) is a random variable representing chirps per second and \(y\) is a random variable representing temperature ('F). These data are also available for download at the Online Study Center.Complete parts (a) through (e), given \(\Sigma x=249.8, \Sigma y=1200.6\) \(\Sigma x^{2}=4200.56, \Sigma y^{2}=96,725.86, \Sigma x y=20,127.47,\) and \(r \approx 0.835\) (f) What is the predicted temperature when \(x=19\) chirps per second?

Describe the relationship between two variables when the correlation coefficient \(r\) is (a) near -1 (b) near 0 (c) near 1

When drawing a scatter diagram, along which axis is the explanatory variable placed? Along which axis is the response variable placed?

In baseball, is there a linear correlation between batting average and home run percentage? Let \(x\) represent the batting average of a professional baseball player, and let \(y\) represent the player's home run percentage (number of home runs per 100 times at bat). A random sample of \(n=7\) professional baseball players gave the following information (Reference: The Baseball Encyclopedia, Macmillan Publishing Company). (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or high? positive or negative? (c) Use a calculator to verify that \(\Sigma x=1.957, \Sigma x^{2} \approx 0.553, \Sigma y=30.1\) \(\Sigma y^{2}=150.15,\) and \(\Sigma x y \approx 8.753 .\) Compute \(r .\) As \(x\) increases, does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.
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