Chapter 7, Problem 1E

True or false If false, explain briefly.

1. a) We choose the linear model that passes through the most data points on the scatterplot.
2. b) The residuals are the observed y-values minus the y-values predicted by the linear model.
3. c) Least squares means that the square of the largest residual is as small as it could possibly be.

To determine

Find whether the given statement is true or false.

Answer to Problem 1E

The given statement is false.

Explanation of Solution

Given info:

The linear model, which passes through the most data points on the scatterplot, are chosen.

Justification:

If the line passes through the most data points on the then that line won’t be straight line. The line touches some of the points or none of the points, but it minimize the sum of least squares.

Thus, the given statement is false.

Statistics Concept Introduction

The conditions for a scatterplot that is well fitted for the data are,

• Straight Enough Condition: The relationship between y and x straight enough to proceed with a linear regression model.
• Outlier Condition: No outlier must be there which influences the fit of the least square line.
• Thickness Condition: The spread of the data around the generally straight relationship seem to be consistent for all values of predictor variable.

To determine

Find whether the given statement is true or false.

Answer to Problem 1E

The given statement is true.

Explanation of Solution

Given info:

The residuals are the observed y-values minus the predicted y-values by the linear model.

Justification:

The residual gives the difference between the observed value of the response variable and that of the predicted value.

Thus, the given statement is true.

Statistics Concept Introduction

Residual:

The residual is defined as $\text{Residual = Observed value }-\text{ Predicted value}$ or $e=y-\widehat{y}$ where $\widehat{y}$ be the predicted value of the response variable and y be the actual value of the response variable.

To determine

Find whether the given statement is true or false.

Answer to Problem 1E

The given statement is false.

Explanation of Solution

Given info:

Least square defines the square of the largest residual is as small as it could possibly be.

Justification:

A least square criterion defines the unique line that minimize the variance of residuals or sum of squared of the residuals. It does not imply to minimize the largest residual is as small as it could possibly be.

Thus, the given statement is false.

Statistics Concept Introduction

Least-square criterion:

The least-square criterion is of a regression line that the best fits of a set of data points having the smallest possible sum of squares.