Question 1

A fitted regression Profit = 262 + 1.51 Sales (all variables in thousands of dollars) was estimated from a random sample of 15 small coffee kiosks. We can say that:

Accepted Answer

The intercept in a regression equation represents the expected value of the dependent variable (in this case, Profit) when all independent variables (in this case, Sales) are equal to zero. Saying the intercept "does not seem reasonable" implies a judgment about the practicality or realism of the intercept value in the context of the problem. In this scenario, an intercept of 262 (thousands of dollars) as a base profit without any sales might not seem reasonable because it suggests a significant profit even without any sales, which is unlikely for small coffee kiosks. The other options (A, C, D) cannot be definitively assessed as correct or incorrect without additional statistical output (e.g., p-values for significance, information on the size of the residuals for underestimation, or details on the model's fit for the comment on R-squared) or context (e.g., the economic environment of the kiosks).

Question 2

A variable transformation in a regression (e.g., replacing Y with log(Y)):

Accepted Answer

Variable transformation, such as using the log transformation of Y, may reduce heteroscedasticity by stabilizing the variance of the residuals. This can improve the accuracy and reliability of the regression model. The other options are not necessarily true for all cases of variable transformation.

Question 3

A fitted regression Profit = -570 + 30 Sales (all variables in thousands of dollars) was estimated from a random sample of pharmacies. From this regression, in order to break even (Profit ≥ 0), a pharmacy's Sales would have to be at least:

Accepted Answer

To break even, set Profit = 0 and solve for Sales: 0 = -570 + 30 Sales. Solving for Sales gives Sales = 570 / 30 = 19.

Question 4

A fitted regression Profit = -570 + 30 Sales (all variables in thousands of dollars) was estimated from a random sample of 20 pharmacies. For a pharmacy with Sales = 10, we predict that Profit will be:

Accepted Answer

The predicted Profit is calculated by substituting Sales = 10 into the regression equation: Profit = -570 + 30(10) = -570 + 300 = -270.

Question 5

In the population model y_i = β₀ + β₁ x + ε, the β_j are:

Accepted Answer

The β_j are fixed, unknown constants that describe the relationship between the variables in the model. Therefore, they are considered parameters of the model.

Question 6

If the residuals violate the assumption of normality, we expect that:

Accepted Answer

If the residuals violate the assumption of normality, it means that the errors in the linear regression model are not normally distributed, leading to unreliable confidence intervals. This can be due to outliers, skewed distributions, or other factors. Therefore, the best choice is D - confidence intervals may be unreliable. It does not necessarily imply that the sample was not taken randomly, or that the residuals will not sum to zero or that least squares formulas will fail.

Question 7

Which is not an assumed characteristic of ε_i in the population model y_i = β₀ + β₁ x_i + ε_i?

Accepted Answer

ε_i is not a statistic, but rather a term used to represent the error of the model.

Question 8

The slope of a proposed population regression model yS1U1B1iS1U1B0 = ?S1U1B10 S1U1B0+ ?S1U1B11S1U1B0 xS1U1B1i S1U1B0+ ?S1U1B1iS1U1B0 is assumed to be:

Accepted Answer

The slope βS1U1B11 is a parameter, which represents a fixed, unknown value that describes the true relationship between the predictor variable x and the response variable y. It is not a random variable, a statistic, or assumed to be distributed normally. The error term εS1U1B1iS1U1B0 may be a random variable with certain assumptions, but it is not the slope itself.

Question 9

Which is not an assumed property of the errors in a population regression model?

Accepted Answer

The assumption that errors have zero variance is not plausible because if they had zero variance, it would imply that all predicted values have no variability around the actual values. This assumption is not only unrealistic but also contradicts the very nature of regression models, where some deviation from the expected value is expected to be present due to random variation.

Question 10

Pedro became interested in vehicle fuel efficiency, so he performed a simple regression using 93 cars to estimate the model CityMPG = β₀ + β₁ Weight, where Weight is the weight of the vehicle in pounds. His results are shown below. Write a brief analysis of these results, using what you have learned in this chapter. Is the intercept meaningful in this regression? Make a prediction of CityMPG when Weight = 3000 and also when Weight = 4000. Do these predictions seem believable? If you could make a car 1000 pounds lighter, what change would you predict in its CityMPG? Regression Analysis $\begin{array} { l l r l } r ^ { 2 } 0.711 & \text { n } 93 \ r - 0.843 & k 1 \ \text { Std. Error } & 3.038 & \text { Dep. Var. CityMPG } \end{array}$ $\text { ANOVA table }$ $\begin{array}{lrrrrr} \text { Source } & \text { SS } & \text { df } & \text { MS } & F & p \text {-value } \ \hline \text { Regression } & 2,065.5191 & 1 & 2,065.5191 & 223.75 & 2.97 \mathrm{E}-26 \ \text { Residual } & 840.0508 & 91 & 9.2313 & & \ \hline \text { Total } & 2,905.5699 & 92 & & & \ \hline \end{array}$ $\text { Rearession outout }$ $ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $ $\text { confidence interval }$ $\begin{array}{lrrrrrr} \hline \text { variables } & \text { coefficients } & \text { std. error } & t(d f=91) & p \text {-value } & 95 \% \text { lower } & 95 \% \text { upper } \ \text { Intercept } & 47.0484 & 1.6799 & 28.006 & 1.63 \mathrm{E}-46 & 43.7114 & 50.3853 \ \text { Weight } & -0.0080 & 0.0005 & -14.958 & 2.97 \mathrm{E}-26 & -0.0091 & -0.0070 \ \hline \end{array}$

Accepted Answer

The answer of Pedro became interested in vehicle fuel efficiency,...

Question 11

A financial regression yielded a standard error of 12 dollars, so a residual of 23 dollars would be:

Accepted Answer

The answer of A financial regression yielded a standard error...

Question 12

If the residuals violate the assumption of autocorrelation, we know that they:

Accepted Answer

The answer of If the residuals violate the assumption of...

Question 13

Mary noticed that old coins are smoother and more worn. She weighed 31 nickels and recorded their ages, and then performed a simple regression to estimate the model Weight = β₀ + β₁ Age, where weight is the weight of the coin in grams and Age is the age of the coin in years. Her results are shown below. Write a brief analysis of these results, using what you have learned in this chapter. Make a prediction of Weight when Age = 10 and also when Age = 20. What does this tell you? Is the intercept meaningful in this regression? Regression Analysis $$\begin{array} { c c r l } r ^ { 2 } & 0.442 & n & 31 \ r & - 0.665 & k & 1 \ \text { or } & 0.050 & \text { Dep. Var. Weight (gm) } \end{array}$$ ANOVA table $\begin{array}{rcccrr} \hline \text { Source } & \text { SS } & \text { df } & M S & F & p \text {-value } \ \hline \text { Regression } & 0.0582 & 1 & 0.0582 & 22.99 & 4.48 \mathrm{E}-05 \ \text { Residual } & 0.0734 & 29 & 0.0025 & & \ \hline \text { Total } & 0.1316 & 30 & & \end{array}$ Regression output confidence interval $\begin{array}{crrrrrr} \hline \text { variables } & \text { coefficients } & \text { std. emor } & f(d f=29) & \text { p-value } & 95 \% \text { lower } & 95 \% \text { upper } \ \hline \text { Intercept } & 5.0210 & 0.0137 & 367.740 & 9.33 \mathrm{E}-55 & 4.9931 & 5.0489 \ \text { Age (Yrs) } & -0.0040 & 0.0008 & -4.795 & 4.48 \mathrm{E}-05 & -0.0057 & -0.0023 \ \hline \end{array}$

Accepted Answer

The answer of Mary noticed that old coins are smoother...

Question 14

If the residuals from a fitted regression violate the assumption of homoscedasticity, we know that:

Accepted Answer

Violation of the assumption of homoscedasticity means that the variance of the residuals is not constant across different values of the predictor variable(s). This can cause biased regression coefficients and inaccurate hypothesis testing.

Question 15

To calculate a residual for the ith observation, we do not need the:

Accepted Answer

To calculate a residual for the ith observation, the standard error is not needed. Residuals are calculated as the difference between the actual values and the predicted values, which are determined using the estimated slope and intercept, not the standard error.

Quiz 12: Simple Regression

A fitted regression Profit = 262 + 1.51 Sales (all variables in thousands of dollars) was estimated from a random sample of 15 small coffee kiosks. We can say that:

A variable transformation in a regression (e.g., replacing Y with log(Y) ) :

A fitted regression Profit = -570 + 30 Sales (all variables in thousands of dollars) was estimated from a random sample of pharmacies. From this regression, in order to break even (Profit ≥ 0) , a pharmacy's Sales would have to be at least:

A fitted regression Profit = -570 + 30 Sales (all variables in thousands of dollars) was estimated from a random sample of 20 pharmacies. For a pharmacy with Sales = 10, we predict that Profit will be:

In the population model y_i = β₀ + β₁ x + ε, the β_j are:

If the residuals violate the assumption of normality, we expect that:

Which is not an assumed characteristic of ε_i in the population model y_i = β₀ + β₁ x_i + ε_i?

The slope of a proposed population regression model y_i = ?₀+ ?₁ x_i+ ?_i is assumed to be:

Which is not an assumed property of the errors in a population regression model?

A financial regression yielded a standard error of 12 dollars, so a residual of 23 dollars would be:

If the residuals violate the assumption of autocorrelation, we know that they:

If the residuals from a fitted regression violate the assumption of homoscedasticity, we know that:

To calculate a residual for the ith observation, we do not need the:

Quiz 12: Simple Regression

A fitted regression Profit = 262 + 1.51 Sales (all variables in thousands of dollars) was estimated from a random sample of 15 small coffee kiosks. We can say that:

A variable transformation in a regression (e.g., replacing Y with log(Y) ) :

A fitted regression Profit = -570 + 30 Sales (all variables in thousands of dollars) was estimated from a random sample of pharmacies. From this regression, in order to break even (Profit ≥ 0) , a pharmacy's Sales would have to be at least:

A fitted regression Profit = -570 + 30 Sales (all variables in thousands of dollars) was estimated from a random sample of 20 pharmacies. For a pharmacy with Sales = 10, we predict that Profit will be:

In the population model yi = β0 + β1 x + ε, the βj are:

If the residuals violate the assumption of normality, we expect that:

Which is not an assumed characteristic of εi in the population model yi = β0 + β1 xi + εi?

The slope of a proposed population regression model yi = ?0 + ?1 xi + ?i is assumed to be:

Which is not an assumed property of the errors in a population regression model?

A financial regression yielded a standard error of 12 dollars, so a residual of 23 dollars would be:

If the residuals violate the assumption of autocorrelation, we know that they:

If the residuals from a fitted regression violate the assumption of homoscedasticity, we know that:

To calculate a residual for the ith observation, we do not need the:

In the population model y_i = β₀ + β₁ x + ε, the β_j are:

Which is not an assumed characteristic of ε_i in the population model y_i = β₀ + β₁ x_i + ε_i?

The slope of a proposed population regression model y_i = ?₀+ ?₁ x_i+ ?_i is assumed to be: