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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, indicating that y decreases by 2 when x increases by 1.

Step by step solution

01

Identify the Slope in the Equation

The equation of a line in the context of a least-squares line or linear regression is usually written in the form \( \hat{y} = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In the given equation, \( \hat{y} = 5 - 2x \), the term with \( x \) is \( -2x \). Hence, the coefficient of \( x \), which is \(-2\), is the slope.
02

Understand the Meaning of Slope Change

The slope of a line \( m \) indicates how much the dependent variable \( \hat{y} \) changes for a one-unit change in the independent variable \( x \). A slope of \(-2\) means that for every increase of 1 unit in \( x \), \( \hat{y} \) decreases by 2 units.

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Key Concepts

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

Understanding Least-Squares in Linear Regression
Least-squares is a fundamental method in statistics, widely used to find the best-fitting line through a set of data points. This method minimizes the sum of the squares of the differences between the observed values and the values predicted by the line, hence the name "least-squares." The goal is to find the line that has the smallest possible vertical distance from all the data points. This approach ensures that the discrepancies between the observed data and the fitted line are minimized.

When you see a least-squares line equation, such as \(\hat{y} = 5 - 2x\), it is derived from this optimization process. In this equation, \(\hat{y}\) represents the predicted or estimated dependent variable for given values of the independent variable \(x\).
  • The least-squares method is crucial for predicting the outcome based on input data.
  • It assumes a linear relationship between the dependent and independent variable.
  • It helps in understanding trends and making data-driven decisions.
Exploring Slope-Intercept Form
The slope-intercept form is a popular way of expressing the equation of a straight line. It is written as \(\hat{y} = mx + b\), where:
  • \(m\) is the slope, which indicates the steepness and direction of the line.
  • \(b\) is the y-intercept, which shows where the line intersects the y-axis when \(x = 0\).
In the equation \(\hat{y} = 5 - 2x\), it follows the slope-intercept format:
  • The slope \(m\) is \(-2\), meaning the line slopes downward as you move from left to right. This tells us that for each increase in \(x\), \(\hat{y}\) decreases by 2 units.
  • The y-intercept \(b\) is \(5\), showing that the line crosses the y-axis at the point (0, 5).
Having the equation in slope-intercept form makes it easier to quickly understand how changes in \(x\) influence \(\hat{y}\) and visualize the line's characteristics on a graph.
Independent vs. Dependent Variables
In the context of linear regression and equations such as \(\hat{y} = 5 - 2x\), it's important to distinguish between dependent and independent variables. These two types of variables have specific roles in a mathematical model:
  • The **independent variable** represents the input or cause. It is plotted along the x-axis. In our equation, \(x\) is the independent variable. You can think of it as the variable we manipulate to observe the effect on \(\hat{y}\).
  • The **dependent variable** indicates the output or effect. It is plotted along the y-axis. In \(\hat{y} = 5 - 2x\), \(\hat{y}\) is the dependent variable, meaning its value depends on the value of \(x\).
Understanding these roles is crucial because the model is used to predict how changes in the independent variable \(x\) will affect the dependent variable \(\hat{y}\). In practical situations, knowing which variable influences the other helps in making informed decisions and analyses based on the model.

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Most popular questions from this chapter

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

For a fixed confidence level, how does the length of the confidence interval for predicted values of \(y\) change as the corresponding \(x\) values become further away from \(\bar{x}\) ?

In Section 5.1, we studied linear combinations of independent random variables. What happens if the variables are not independent? A lot of mathematics can be used to prove the following: Let \(x\) and \(y\) be random variables with means \(\mu_{x}\) and \(\mu_{y}\), variances \(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\), and population correlation coefficient \(\rho\) (the Greek letter rho). Let \(a\) and \(b\) be any constants and let \(w=a x+b y .\) Then, In this formula, \(\rho\) is the population correlation coefficient, theoretically computed using the population of all \((x, y)\) data pairs. The expression \(\sigma_{x} \sigma_{y} \rho\) is called the covariance of \(x\) and \(y .\) If \(x\) and \(y\) are independent, then \(\rho=0\) and the formula for \(\sigma_{w}^{2}\) reduces to the appropriate formula for independent variables (see Section 5.1). In most real-world applications, the population parameters are not known, so we use sample estimates with the understanding that our conclusions are also estimates. Do you have to be rich to invest in bonds and real estate? No, mutual fund shares are available to you even if you aren't rich. Let \(x\) represent annual percentage return (after expenses) on the Vanguard Total Bond Index Fund, and let \(y\) represent annual percentage return on the Fidelity Real Estate Investment Fund. Over a long period of time, we have the following population estimates (based on Morningstar Mutual Fund Report). $$ \mu_{x} \approx 7.32 \quad \sigma_{x} \approx 6.59 \quad \mu_{y} \approx 13.19 \quad \sigma_{y} \approx 18.56 \quad \rho \approx 0.424 $$ (a) Do you think the variables \(x\) and \(y\) are independent? Explain. (b) Suppose you decide to put \(60 \%\) of your investment in bonds and \(40 \%\) in real estate. This means you will use a weighted average \(w=0.6 x+0.4 y\). Estimate your expected percentage return \(\mu_{w}\) and risk \(\sigma_{w}\) (c) Repeat part (b) if \(w=0.4 x+0.6 y\). (d) Compare your results in parts (b) and (c). Which investment has the higher expected return? Which has the greater risk as measured by \(\sigma_{w} ?\)

Do people who spend more time on social networking sites spend more time using Twitter? Megan conducted a study and found that the correlation between the times spent on the two activities was \(0.8\). What does this result say about the relationship between times spent on the two activities? If someone spends more time than average on a social networking site, can you automatically conclude that he or she spends more time than average using Twitter? Explain.

There are several extensions of linear regression that apply to exponential growth and power law models. Problems \(22-25\) will outline some of these extensions. First of all, recall that a variable grows linearly over time if it adds a fixed increment during each equal time period. Exponential growth occurs when a variable is multiplied by a fixed number during each time period. This means that exponential growth increases by a fixed multiple or percentage of the previous amount. College algebra can be used to show that if a variable grows exponentially, then its logarithm grows linearly. The exponential growth model is \(y=\alpha \beta^{x}\), where \(\alpha\) and \(\beta\) are fixed constants to be estimated from data. How do we know when we are dealing with exponential growth, and how can we estimate \(\alpha\) and \(\beta\) ? Please read on. Populations of living things such as bacteria, locusts, fish, panda bears, and so on, tend to grow (or decline) exponentially. However, these populations can be restricted by outside limitations such as food, space, pollution, disease, hunting, and so on. Suppose we have data pairs \((x, y)\) for which there is reason to believe the scatter plot is not linear, but rather exponential, as described above. This means the increase in \(y\) values begins rather slowly but then seems to explode. Note: For exponential growth models, we assume all \(y>0\). Consider the following data, where \(x=\) time in hours and \(y=\) number of bacteria in a laboratory culture at the end of \(x\) hours. $$ \begin{array}{l|rrrrr} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline y & 3 & 12 & 22 & 51 & 145 \\ \hline \end{array} $$ (a) Look at the Excel graph of the scatter diagram of the \((x, y)\) data pairs. Do you think a straight line will be a good fit to these data? Do the \(y\) values seem almost to explode as time goes on? (b) Now consider a transformation \(y^{\prime}=\log y .\) We are using common logarithms of base 10 (however, natural logarithms of base \(e\) would work just as well). $$ \begin{array}{l|lllll} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline y^{\prime}=\log y & 0.477 & 1.079 & 1.342 & 1.748 & 2.161 \\ \hline \end{array} $$
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