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Chapter 9: Problem 2

In the least squares line \(\hat{y}=5+3 x\), what is the marginal change in \(\hat{y}\) for each unit change in \(x\) ?

Short Answer

Expert verified
The marginal change in \(\hat{y}\) for each unit change in \(x\) is 3.

Step by step solution

01

Understand the Linear Equation

The given equation is the least squares line \(\hat{y} = 5 + 3x\), where \(\hat{y}\) is the dependent variable and \(x\) is the independent variable. The equation represents a straight line that best fits a set of data points.
02

Identify the Slope

In the equation \(\hat{y} = 5 + 3x\), the coefficient of \(x\) is 3. This coefficient is known as the slope of the line.
03

Interpret the Slope

The slope of a line in a least squares equation represents the rate of change of \(\hat{y}\) with respect to \(x\). Therefore, the slope of 3 implies that for every 1 unit increase in \(x\), \(\hat{y}\) increases by 3 units.

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Key Concepts

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

Slope Interpretation
In a linear regression equation like \( \hat{y} = 5 + 3x \), the slope is a fundamental concept. The slope is the coefficient of the independent variable \( x \). It tells us how much the dependent variable \( \hat{y} \) changes when \( x \) changes by one unit. In this equation, the slope is 3. This means for every increase of 1 unit in \( x \), \( \hat{y} \) increases by 3 units.
This can be thought of as the responsiveness of \( \hat{y} \) to changes in \( x \). If the slope were negative, it would indicate that \( \hat{y} \) decreases as \( x \) increases. Understanding the slope is crucial for making predictions based on the regression line.
Least Squares Method
The least squares method is a standard approach in linear regression to find the best-fitting line through a set of data points. The goal is to minimize the sum of the squares of the vertical distances between the data points and the line. These vertical distances are known as residuals.
The formula for a least squares line is generally represented as \( \hat{y} = a + bx \), where \( a \) is the intercept and \( b \) is the slope. The coefficients \( a \) and \( b \) are calculated in such a way that the overall error, measured by the squares of the residuals, is minimized.
This method gives the most accurate line fitting for the data, so it's widely used in statistical analyses to predict trends, make forecasts, or understand relationships between variables.
Dependent and Independent Variables
In regression analysis, understanding the roles of dependent and independent variables is key. The dependent variable, often denoted as \( \hat{y} \), is what you are trying to predict or explain. On the other hand, the independent variable, denoted as \( x \), is the predictor or the factor that might influence changes in the dependent variable.
Let's consider \( \hat{y} = 5 + 3x \), here \( \hat{y} \) is dependent on \( x \). This implies that variations in \( x \) could potentially bring about changes in \( \hat{y} \).
In research and data analysis, identifying which variable is independent and which is dependent helps in understanding and interpreting the direction and strength of relationships between data points.

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Most popular questions from this chapter

How much should a healthy Shetland pony weigh? Let \(x\) be the age of the pony (in months), and let \(y\) be the average weight of the pony (in kilograms). The following information is based on data taken from The Merck Veterinary Manual (a reference used in most veterinary colleges). $$ \begin{array}{r|rrrrr} \hline x & 3 & 6 & 12 & 18 & 24 \\ \hline y & 60 & 95 & 140 & 170 & 185 \\ \hline \end{array} $$ (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 strong? positive or negative? (c) Use a calculator to verify that \(\Sigma x=63, \quad \Sigma x^{2}=1089, \quad \Sigma y=650\) \(\Sigma y^{2}=95,350\), and \(\Sigma x y=9930 .\) Compute \(r .\) As \(x\) increases from 3 to 24 months, does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.

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 \(\bar{x}\) ?

What is the optimal amount of time for a scuba diver to be on the bottom of the ocean? That depends on the depth of the dive. The U.S. Navy has done a lot of research on this topic. The Navy defines the "optimal time" to be the time at each depth for the best balance between length of work period and decompression time after surfacing. Let \(x=\) depth of dive in meters, and let \(y=\) optimal time in hours. A random sample of divers gave the following data (based on information taken from Medical Physiology by A. C. Guyton, M.D.). $$ \begin{array}{c|ccccccc} \hline x & 14.1 & 24.3 & 30.2 & 38.3 & 51.3 & 20.5 & 22.7 \\ \hline y & 2.58 & 2.08 & 1.58 & 1.03 & 0.75 & 2.38 & 2.20 \\ \hline \end{array} $$ (a) Verify that \(\Sigma x=201.4, \quad \Sigma y=12.6, \quad \Sigma x^{2}=6734.46, \quad \Sigma y^{2}=25.607\), \(\Sigma x y=311.292\), and \(r \approx-0.976\). (b) Use a \(1 \%\) level of significance to test the claim that \(\rho<0\). (c) Verify that \(S_{e} \approx 0.1660, a \approx 3.366\), and \(b \approx-0.0544\). (d) Find the predicted optimal time in hours for a dive depth of \(x=18\) meters. (e) Find an \(80 \%\) confidence interval for \(y\) when \(x=18\) meters. (f) Use a \(1 \%\) level of significance to test the claim that \(\beta<0\). (g) Find a \(90 \%\) confidence interval for \(\beta\) and its meaning.

Let \(x=\) day of observation and \(y=\) number of locusts per square meter during a locust infestation in a region of North Africa. $$ \begin{array}{l|llrrr} \hline x & 2 & 3 & 5 & 8 & 10 \\ \hline y & 2 & 3 & 12 & 125 & 630 \\ \hline \end{array} $$ (a) Draw a 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 almost seem to explode as time goes on? (b) Now consider a transformation \(y^{\prime}=\log y .\) We are using common logarithms of base \(10 .\) Draw a scatter diagram of the \(\left(x, y^{\prime}\right)\) data pairs and compare this diagram with the diagram of part (a). Which graph appears to better fit a straight line? (c) Use a calculator with regression keys to find the linear regression equation for the data pairs \(\left(x, y^{\prime}\right) .\) What is the correlation coefficient? (d) The exponential growth model is \(y=\alpha \beta^{x}\). Estimate \(\alpha\) and \(\beta\) and write the exponential growth equation. Hint: See Problem 22 .

Over the past 50 years, there has been a strong negative correlation between average annual income and the record time to run 1 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 increase in one or both of the variables? Explain.
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