/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 43 The table below gives a small se... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 43

The table below gives a small set of data. Which of the following two lines fits the data better: \(\hat{y}=1-x\) or \(\hat{y}=3-2 x ?\) Use the least squares criterion to justify your answer. (Note: Neither of these two lines is the least-squares regression line for these data. $$\begin{array}{lccccc}\hline x: & -1 & 1 & 1 & 3 & 5 \\ y: & 2 & 0 & 1 & -1 & -5 \\\\\hline\end{array}$$

Short Answer

Expert verified
\( \hat{y} = 1 - x \) fits better as it has a lower sum of squared errors (3 vs. 18).

Step by step solution

01

Understand the Problem

We need to compare two functions \( \hat{y}=1-x \) and \( \hat{y}=3-2x \) to the given data set \((x, y)\) and determine which one better fits the data using the least squares criterion. The criterion aims to minimize the sum of squared differences between the observed \( y \) values and the predicted \( \hat{y} \) values.
02

Calculate Errors for \(\hat{y} = 1-x\)

For each \( x \) value in the data set, calculate the predicted \( \hat{y} \) from the line \( \hat{y} = 1-x \), then find the error \( e = y - \hat{y} \). Finally, calculate \( e^2 \).- For \( x = -1 \), \( \hat{y} = 1 - (-1) = 2 \). Error, \( e = 2 - 2 = 0 \), so \( e^2 = 0 \).- For \( x = 1 \), \( \hat{y} = 1 - 1 = 0 \). Error, \( e = 0 - 0 = 0 \), so \( e^2 = 0 \).- For \( x = 1 \), \( \hat{y} = 1 - 1 = 0 \). Error, \( e = 1 - 0 = 1 \), so \( e^2 = 1 \).- For \( x = 3 \), \( \hat{y} = 1 - 3 = -2 \). Error, \( e = -1 + 2 = 1 \), so \( e^2 = 1 \).- For \( x = 5 \), \( \hat{y} = 1 - 5 = -4 \). Error, \( e = -5 + 4 = -1 \), so \( e^2 = 1 \).Total \( \sum e^2 = 0 + 0 + 1 + 1 + 1 = 3 \).
03

Calculate Errors for \(\hat{y} = 3-2x\)

For each \( x \) value in the data set, calculate the predicted \( \hat{y} \) from the line \( \hat{y} = 3-2x \), then find the error \( e = y - \hat{y} \). Finally, calculate \( e^2 \).- For \( x = -1 \), \( \hat{y} = 3 - 2(-1) = 5 \). Error, \( e = 2 - 5 = -3 \), so \( e^2 = 9 \).- For \( x = 1 \), \( \hat{y} = 3 - 2(1) = 1 \). Error, \( e = 0 - 1 = -1 \), so \( e^2 = 1 \).- For \( x = 1 \), \( \hat{y} = 3 - 2(1) = 1 \). Error, \( e = 1 - 1 = 0 \), so \( e^2 = 0 \).- For \( x = 3 \), \( \hat{y} = 3 - 2(3) = -3 \). Error, \( e = -1 + 3 = 2 \), so \( e^2 = 4 \).- For \( x = 5 \), \( \hat{y} = 3 - 2(5) = -7 \). Error, \( e = -5 + 7 = 2 \), so \( e^2 = 4 \).Total \( \sum e^2 = 9 + 1 + 0 + 4 + 4 = 18 \).
04

Compare the Sum of Squares

The sum of squared errors for \( \hat{y} = 1 - x \) is 3, while for \( \hat{y} = 3 - 2x \) it is 18. According to the least squares criterion, the line that minimizes the sum of squared errors better fits the data. Therefore, \( \hat{y} = 1 - x \) provides a better fit than \( \hat{y} = 3 - 2x \).

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Key Concepts

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

Sum of Squared Errors
The Sum of Squared Errors (SSE) is a key concept in statistics, specifically in the realm of regression analysis. It is used to measure how well a model's predicted values match the actual observed data. The process begins by calculating the difference between the actual and predicted values, known as the error. Once the errors are found for each data point, they are squared and summed up to provide the SSE. Squaring the errors ensures that all differences are positive and amplifies larger errors more than smaller ones.

In our specific problem, the goal was to determine which line, among the options given, fits the dataset better by using the SSE. The formula for the sum of squared errors is:
  • For each observed value (\(y\)) and predicted value (\(\hat{y}\)), find the error: \(e = y - \hat{y}\).
  • Square each error: \(e^2\).
  • Compute the total: \(\sum e^2 = \sum (y - \hat{y})^2\).
The smallest SSE indicates the best fitting model since it means the predicted values are closest to the observed ones.
Regression Line
A Regression Line represents the best estimate of the relationship between the independent variable (in this case, \( x \)) and the dependent variable (\(y\)). It is a line drawn through the data points in a scatterplot to show the overall trend.

In this exercise, two candidate lines are proposed: \(\hat{y} = 1 - x\) and \(\hat{y} = 3 - 2x\). These lines are not true least squares regression lines but are instead being tested to find which is closer to being that by using the SSE.
  • The slope of the line decides if the line rises, falls, or stays constant as \(x\) moves forward.
  • The intercept determines the point where the line crosses the y-axis.
A regression line helps us to make predictions for \(y\) given new \(x\) values. The line鈥檚 goodness of fit can be visually checked by how broadly data points scatter around it.
Data Fitting
Data Fitting is an essential concept in data analysis where we aim to model the observed data points accurately using a mathematical function. The purpose of fitting data is to describe the general trend exhibited by the data and make predictions.

In the given exercise, the aim was to fit two proposed lines to the dataset provided. Using the least squares method enables us to quantify how well each line fits, by determining how much the predicted values from the line deviate from actual observed values
Key aspects of Data Fitting include:
  • Choosing a model: In this example, a linear model.
  • Estimating parameters: Use the given formulas to plot lines and adjust them based on the data.
  • Evaluating fit quality: Done by calculating measures like the SSE.
Good data fitting helps enhance the reliability of predictions made using the regression line.

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Most popular questions from this chapter

The percent of an animal species in the wild that survives to breed again is often lower following a successful breeding season. A study of merlins (small falcons) in northern Sweden observed the number of breeding pairs in an isolated area and the percent of males (banded for identification) that returned the next breeding season. Here are data for seven years: $$\begin{array}{llllllll}\hline \text { Breeding pairs: } & 28 & 29 & 29 & 29 & 30 & 32 & 33 \\\\\text { Percent return: } & 82 & 83 & 70 & 61 & 69 & 58 & 43 \\\\\hline \end{array}$$ Make a scatterplot to display the relationship between breeding pairs and percent return. Describe what you see.

Here are some hypothetical data: $$\begin{array}{lllllll}\hline x & 1 & 2 & 3 & 4 & 10 & 10 \\\y: & 1 & 3 & 3 & 5 & 1 & 11 \\\\\hline\end{array}$$ (a) Make a scatterplot to show the relationship between \(x\) and \(y\) (b) Calculate the correlation for these data by hand or using technology. (c) What is responsible for reducing the correlation to the value in part (b) despite a strong straight-line relationship between \(x\) and \(y\) in most of the observations?

In its recent Fuel Economy Guide, the Environmental Protection Agency gives data on 1152 vehicles. There are a number of outliers, mainly vehicles with very poor gas mileage. If we ignore the outliers, however, the combined city and highway gas mileage of the other 1120 or so vehicles is approximately Normal with mean 18.7 miles per gallon (mpg) and standard deviation 4.3 mpg. The Chevrolet Malibu with a four-cylinder engine has a combined gas mileage of 25 mpg. What percent of all vehicles have worse gas mileage than the Malibu?

You use the same bar of soap to shower each morning. The bar weighs 80 grams when it is new. Its weight goes down by 6 grams per day on average. What is the equation of the regression line for predicting weight from days of use?

Each year, students in an elementary school take a standardized math test at the end of the school year. For a class of fourth-graders, the average score was 55.1 with a standard deviation of \(12.3 .\) In the third grade, these same students had an average score of 61.7 with a standard deviation of \(14.0 .\) The correlation between the two sets of scores is \(r=0.95\). Calculate the equation of the least-squares regression line for predicting a fourth-grade score from a third-grade score. (a) \(\hat{y}=3.60+0.835 x\) (b) \(\hat{y}=15.69+0.835 x\) (c) \(\hat{y}=2.19+1.08 x\) (d) \(\hat{y}=-11.54+1.08 x\) (e) Cannot be calculated without the data.
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