Q: Given the regression equation $\hat{y} = -15.4 + 2.3x$, what is the predicted value of $y$ when $x = 60$?

$\hat{y} = -15.4 + 2.3 \times 60 = 122.6$

Q: Which of the following is the correct formula for the coefficient of determination, $r^2$?

$r^2 = 1 - \dfrac{\text{Unexplained Variation}}{\text{Total Variation}}$

Question 1

Which of the following best describes the coefficient of determination, $r^2$, in the context of regression analysis?

Accepted Answer

It measures the proportion of the total variation in the response variable that is explained by the regression model.

Question 2

Suppose the correlation coefficient between $x$ and $y$ is $r = 0.7$. What percentage of the variation in $y$ is explained by the regression of $y$ on $x$?

Accepted Answer

$49\%$

Question 3

Given the regression equation $\hat{y} = -15.4 + 2.3x$, what is the predicted value of $y$ when $x = 60$?

Accepted Answer

$\hat{y} = -15.4 + 2.3 	imes 60 = 122.6$

Question 4

Which of the following residual plots indicates that a linear regression model may NOT be appropriate?

Accepted Answer

A plot where residuals show a clear curved pattern.

Question 5

If the residuals from a regression model show increasing spread as $x$ increases, which assumption of linear regression is violated?

Accepted Answer

Constant error variance (homoskedasticity)

Question 6

A data point has a large residual and is far from the mean of $x$. What is this point likely to be?

Accepted Answer

An influential observation

Question 7

Which formula correctly expresses the relationship between total, explained, and unexplained variation in regression?

Accepted Answer

$\sum (y - \overline{y})^2 = \sum (y - \hat{y})^2 + \sum (\hat{y} - \overline{y})^2$

Question 8

Which of the following is the correct formula for the coefficient of determination, $r^2$?

Accepted Answer

$r^2 = 1 - \dfrac{	ext{Unexplained Variation}}{	ext{Total Variation}}$

Question 9

A regression analysis yields $r = -0.8$. What is the correct interpretation of $r^2$?

Accepted Answer

$r^2 = 0.64$, so 64\% of the variation in $y$ is explained by $x$.

Question 10

Which of the following best describes an outlier in the context of regression analysis?

Accepted Answer

An observation whose response variable is inconsistent with the overall pattern of the data.

Question 11

Why is it important to plot data and examine residual plots, even when $r$ and $r^2$ are high?

Accepted Answer

Because high $r$ and $r^2$ do not guarantee that the relationship is linear or that the regression model is appropriate.

Question 12

Which of the following is NOT a recommended action when an influential observation is found and its removal is not justified?

Accepted Answer

Ignore the observation and proceed as if it does not exist.

Question 13

In the context of regression, what is leverage?

Accepted Answer

A measure of how much an observation's $x$ value differs from the mean of $x$.

Question 14

A residual plot shows a 'fan shape' (residuals spread out as $x$ increases). What does this indicate?

Accepted Answer

The variance of the errors is not constant (heteroskedasticity).

Question 15

Which of the following statements about the least-squares regression line is TRUE?

Accepted Answer

It minimizes the sum of the squares of the residuals.

Question 16

Suppose you remove an observation from your dataset and the slope of your regression line changes dramatically. What is the most likely explanation?

Accepted Answer

The observation was influential.

Question 17

Which of the following is the correct definition of a residual for a value $x$ in the dataset?

Accepted Answer

$y - \hat{y}$

Question 18

Francis Galton coined the term 'regression' in the context of which real-world phenomenon?

Accepted Answer

The tendency for sons' heights to regress toward the mean height of the population, regardless of their fathers' heights.

Question 19

Which of the following scenarios would most likely result in a high-leverage point?

Accepted Answer

A data point with an $x$ value far from the mean of $x$.

Diagnostics on Regression: Coefficient of Determination, Residual Analysis, and Influential Observations

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