Deck 25: Multiple and Logistic Regression

Question 1

Tail-feather length is a sexually dimorphic trait in long-tailed finches; that is, the trait differs substantially for males and for females of the same species. A bird's size and environment might also influence tail-feather length. Researchers studied the relationship between tail-feather length (measuring the R1 central tail feather) and weight in a sample of 20 male long-tailed finches raised in an aviary and 5 male long-tailed finches caught in the wild. The data are displayed in the scatterplot below.
​
The following model is proposed for predicting the length of the R1 feather from the bird's weight and origin:
R1Length_i = β₀ + β₁ (BodyWeight_i) + β₂ (Aviary0Wild1_i) + ε_i
Where the deviations ε_i were assumed to be independent and Normally distributed with mean 0 and standard deviation σ . This model was fit to the sample of 25 male finches. The following results summarize the least-squares regression fit of this model:
$\begin{array}{l}\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 35.79 & 16.95 \\\text { BodyWeight } & 2.826 & 1.030 \\\text { Aviary0Wild1 } & -10.453 & 4.996\end{array}\\\mathrm{S}=9.62172 \quad \mathrm{R}-\mathrm{Sq}=30.2 \%\end{array}$

$\begin{array}{lrrr}\text { Analysis of Variance } & & \\\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 2 & 879.79 & 439.90 \\\text { Residual Error } & 22 & 2036.71 & 92.58 \\\text { Total } & 24 & 2916.50 &\end{array}$ Based on the software output provided, what is the equation for the multiple regression model?

A)R1Length = 35.79 + 2.83 BodyWeight - 10.45 Aviary0Wild1
B)R1Length = 16.95 + 1.03 BodyWeight + 5.00 Aviary0Wild1
C)R1Length = 35.79 BodyWeight + 2.83 Aviary0Wild1 - 10.45
D)R1Length = 16.95 BodyWeight + 1.03 Aviary0Wild1 + 5.00

Answer 1

A

Question 2

Tail-feather length is a sexually dimorphic trait in long-tailed finches; that is, the trait differs substantially for males and for females of the same species. A bird's size and environment might also influence tail-feather length. Researchers studied the relationship between tail-feather length (measuring the R1 central tail feather) and weight in a sample of 20 male long-tailed finches raised in an aviary and 5 male long-tailed finches caught in the wild. The data are displayed in the scatterplot below.
​
The following model is proposed for predicting the length of the R1 feather from the bird's weight and origin:
R1Length_i = β₀ + β₁ (BodyWeight_i) + β₂ (Aviary0Wild1_i) + ε_i
Where the deviations ε_i were assumed to be independent and Normally distributed with mean 0 and standard deviation σ . This model was fit to the sample of 25 male finches. The following results summarize the least-squares regression fit of this model:
$\begin{array}{l}\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 35.79 & 16.95 \\\text { BodyWeight } & 2.826 & 1.030 \\\text { Aviary0Wild1 }-10.453 & 4.996\end{array}\\\mathrm{S}=9.62172 \quad \mathrm{R}-\mathrm{Sq}=30.2 \%\end{array}$

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 2 & 879.79 & 439.90 \\\text { Residual Error } & 22 & 2036.71 & 92.58 \\\text { Total } & 24 & 2916.50 &\end{array}\end{array}$ Suppose we wish use the ANOVA F test to test the following hypotheses:
H₀: β₁ = β₂ = 0
H_α: At least one of β₁ or β₂ is not 0
What is the value of the F statistic for these hypotheses?

A)1.43
B)2.31
C)3.31
D)4.75

Answer 2

D

Question 3

Tail-feather length is a sexually dimorphic trait in long-tailed finches; that is, the trait differs substantially for males and for females of the same species. A bird's size and environment might also influence tail-feather length. Researchers studied the relationship between tail-feather length (measuring the R1 central tail feather) and weight in a sample of 20 male long-tailed finches raised in an aviary and 5 male long-tailed finches caught in the wild. The data are displayed in the scatterplot below.
​
The following model is proposed for predicting the length of the R1 feather from the bird's weight and origin:
R1Length_i = β₀ + β₁ (BodyWeight_i) + β₂ (Aviary0Wild1_i) + ε_i
Where the deviations ε_i were assumed to be independent and Normally distributed with mean 0 and standard deviation σ . This model was fit to the sample of 25 male finches. The following results summarize the least-squares regression fit of this model:
$\begin{array}{l}\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 35.79 & 16.95 \\\text { BodyWeight } & 2.826 & 1.030 \\\text { Aviary0Wild1 }-10.453 & 4.996\end{array}\\\mathrm{S}=9.62172 \quad \mathrm{R}-\mathrm{Sq}=30.2 \%\end{array}$

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 2 & 879.79 & 439.90 \\\text { Residual Error } & 22 & 2036.71 & 92.58 \\\text { Total } & 24 & 2916.50 &\end{array}\end{array}$ Suppose we wish use the ANOVA F test to test the following hypotheses:
H₀: β₁ = β₂ = 0
H_α: At least one of β₁ or β₂ is not 0
What is the P-value of this test?

A)Greater than 0.10
B)Between 0.05 and 0.10
C)Between 0.01 and 0.05
D)Less than 0.01

Answer 3

C

Question 4

Tail-feather length is a sexually dimorphic trait in long-tailed finches; that is, the trait differs substantially for males and for females of the same species. A bird's size and environment might also influence tail-feather length. Researchers studied the relationship between tail-feather length (measuring the R1 central tail feather) and weight in a sample of 20 male long-tailed finches raised in an aviary and 5 male long-tailed finches caught in the wild. The data are displayed in the scatterplot below.
​
The following model is proposed for predicting the length of the R1 feather from the bird's weight and origin:
R1Length_i = β₀ + β₁ (BodyWeight_i) + β₂ (Aviary0Wild1_i) + ε_i
Where the deviations ε_i were assumed to be independent and Normally distributed with mean 0 and standard deviation σ . This model was fit to the sample of 25 male finches. The following results summarize the least-squares regression fit of this model:
$\begin{array}{l}\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 35.79 & 16.95 \\\text { BodyWeight } & 2.826 & 1.030 \\\text { Aviary0Wild1 }-10.453 & 4.996\end{array}\\\mathrm{S}=9.62172 \quad \mathrm{R}-\mathrm{Sq}=30.2 \%\end{array}$

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 2 & 879.79 & 439.90 \\\text { Residual Error } & 22 & 2036.71 & 92.58 \\\text { Total } & 24 & 2916.50 &\end{array}\end{array}$ What percent of variation in tail-feather length can be explained by the regression model?

A)9%
B)30%
C)55%
D)93%

Question 5

Tail-feather length is a sexually dimorphic trait in long-tailed finches; that is, the trait differs substantially for males and for females of the same species. A bird's size and environment might also influence tail-feather length. Researchers studied the relationship between tail-feather length (measuring the R1 central tail feather) and weight in a sample of 20 male long-tailed finches raised in an aviary and 5 male long-tailed finches caught in the wild. The data are displayed in the scatterplot below.
​
The following model is proposed for predicting the length of the R1 feather from the bird's weight and origin:
R1Length_i = β₀ + β₁ (BodyWeight_i) + β₂ (Aviary0Wild1_i) + ε_i
Where the deviations ε_i were assumed to be independent and Normally distributed with mean 0 and standard deviation σ . This model was fit to the sample of 25 male finches. The following results summarize the least-squares regression fit of this model:
$\begin{array}{l}\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 35.79 & 16.95 \\\text { BodyWeight } & 2.826 & 1.030 \\\text { Aviary0Wild1 } & -10.453 & 4.996\end{array}\\\mathrm{S}=9.62172 \quad \mathrm{R}-\mathrm{Sq}=30.2 \%\end{array}$

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 2 & 879.79 & 439.90 \\\text { Residual Error } & 22 & 2036.71 & 92.58 \\\text { Total } & 24 & 2916.50 &\end{array}\end{array}$ What is a 95% confidence interval for 1, the coefficient of BodyWeight in this model?

A)2.826 ± 1.030
B)2.826 ± 2.019
C)2.826 ± 2.126
D)2.826 ± 2.136

Question 6

Tail-feather length is a sexually dimorphic trait in long-tailed finches; that is, the trait differs substantially for males and for females of the same species. A bird's size and environment might also influence tail-feather length. Researchers studied the relationship between tail-feather length (measuring the R1 central tail feather) and weight in a sample of 20 male long-tailed finches raised in an aviary and 5 male long-tailed finches caught in the wild. The data are displayed in the scatterplot below.
​
The following model is proposed for predicting the length of the R1 feather from the bird's weight and origin:
R1Length_i = β₀ + β₁ (BodyWeight_i) + β₂ (Aviary0Wild1_i) + ε_i
Where the deviations ε_i were assumed to be independent and Normally distributed with mean 0 and standard deviation σ . This model was fit to the sample of 25 male finches. The following results summarize the least-squares regression fit of this model:
$\begin{array}{l}\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 35.79 & 16.95 \\\text { BodyWeight } & 2.826 & 1.030 \\\text { Aviary0Wild1 } & -10.453 & 4.996\end{array}\\\mathrm{S}=9.62172 \quad \mathrm{R}-\mathrm{Sq}=30.2 \%\end{array}$

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 2 & 879.79 & 439.90 \\\text { Residual Error } & 22 & 2036.71 & 92.58 \\\text { Total } & 24 & 2916.50 &\end{array}\end{array}$ To determine whether the bird's origin affects tail-feather length in this model, we test the hypotheses
H₀: β₂ = 0, H_α: β₂ ≠ 0 in this particular model. Using the estimates provided, what is the P-value for this test?

A)Greater than 0.10
B)Between 0.05 and 0.10
C)Between 0.01 and 0.05
D)Less than 0.01

Question 7

Tail-feather length is a sexually dimorphic trait in long-tailed finches; that is, the trait differs substantially for males and for females of the same species. A bird's size and environment might also influence tail-feather length. Researchers studied the relationship between tail-feather length (measuring the R1 central tail feather) and weight in a sample of 20 male long-tailed finches raised in an aviary and 5 male long-tailed finches caught in the wild. The data are displayed in the scatterplot below.
​
The following model is proposed for predicting the length of the R1 feather from the bird's weight and origin:
R1Length_i = β₀ + β₁ (BodyWeight_i) + β₂ (Aviary0Wild1_i) + ε_i
Where the deviations ε_i were assumed to be independent and Normally distributed with mean 0 and standard deviation σ . This model was fit to the sample of 25 male finches. The following results summarize the least-squares regression fit of this model:
$\begin{array}{l}\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 35.79 & 16.95 \\\text { BodyWeight } & 2.826 & 1.030 \\\text { Aviary0Wild1 } & -10.453 & 4.996\end{array}\\\mathrm{S}=9.62172 \quad \mathrm{R}-\mathrm{Sq}=30.2 \%\end{array}$

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 2 & 879.79 & 439.90 \\\text { Residual Error } & 22 & 2036.71 & 92.58 \\\text { Total } & 24 & 2916.50 &\end{array}\end{array}$ Using a significance level of 0.05, what should you conclude about long-tailed finches, based on your calculations for a confidence interval for β₁ and for a test of hypothesis on β₂?

A)A bird's body weight and origin both significantly explain tail-feather length.
B)A bird's body weight significantly helps explain tail-feather length, even after accounting for the bird's origin, and a bird's origin significantly helps explain tail-feather length, even after accounting for the bird's body weight.
C)A bird's body weight significantly helps explain tail-feather length, even after accounting for the bird's origin, but a bird's origin does not significantly help explain tail-feather length after accounting for the bird's body weight.
D)A bird's origin significantly helps explain tail-feather length even after accounting for the bird's body weight, but a bird's body weight does not significantly help explain tail-feather length after accounting for the bird's origin.

Question 8

Tail-feather length is a sexually dimorphic trait in long-tailed finches; that is, the trait differs substantially for males and for females of the same species. A bird's size and environment might also influence tail-feather length. Researchers studied the relationship between tail-feather length (measuring the R1 central tail feather) and weight in a sample of 20 male long-tailed finches raised in an aviary and 5 male long-tailed finches caught in the wild. The data are displayed in the scatterplot below.
​
The following model is proposed for predicting the length of the R1 feather from the bird's weight and origin:
R1Length_i = β₀ + β₁ (BodyWeight_i) + β₂ (Aviary0Wild1_i) + ε_i
Where the deviations ε_i were assumed to be independent and Normally distributed with mean 0 and standard deviation σ . This model was fit to the sample of 25 male finches. The following results summarize the least-squares regression fit of this model:
$\begin{array}{l}\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 35.79 & 16.95 \\\text { BodyWeight } & 2.826 & 1.030 \\\text { Aviary0Wild1 }-10.453 & 4.996\end{array}\\\mathrm{S}=9.62172 \quad \mathrm{R}-\mathrm{Sq}=30.2 \%\end{array}$

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 2 & 879.79 & 439.90 \\\text { Residual Error } & 22 & 2036.71 & 92.58 \\\text { Total } & 24 & 2916.50 &\end{array}\end{array}$ Given this regression model, what is an estimate of the average difference in tail-feather length between 18-g male finches either raised in an aviary or caught in the wild?

A)2.8 g
B)5.0 g
C)9.6 g
D)10.5 g

Question 9

Researchers investigated the effects of acute stress on emotional picture processing and recall by randomly assigning 40 adult males to receive either a stressful cold stimulus or a neutral warm stimulus before viewing pictures. The researchers computed the relative brainwave amplitude (in microvolts, μV) for each subject viewing unpleasant versus neutral pictures. They also recorded how many pictures the subjects were able to recall 24 hours later. The data are displayed in the following scatterplot.
μ
The following model is proposed for predicting the brainwave relative amplitude ("amplitude") from the number of images recalled ("recall"), the indicator variable reflecting the nature of the stimulus ("stress"), and an interaction term ("recall*stress"):
Amplitude_i = β₀ + β₁ (recall_i) + β₂ (stress_i) + β₃ (recall*stress_i) + ε_i
Where the deviations ε_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the sample of 40 adult males. The following results summarize the least-squares regression fit of this model:
​
$\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 4.795 & 1.329 \\\text { recall } & -0.1139 & 0.1069 \\\text { stress } & -4.262 & 2.205 \\\text { recall* } \text { stress } & 0.4337 & 0.1677 \\\mathrm{~S}=2.13946 & \text { R-Sq }=22.8 \% &\end{array}$

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 3 & 48.788 & 16.263 \\\text { Residual Error } & 36 & 164.783 & 4.577 \\\text { Total } & 39 & 213.571 &\end{array}\end{array}$ Based on the software output provided, what is the equation for the multiple regression model?

A)amplitude = 4.795 - 0.114 recall - 4.262 stress + 0.434 recall*stress
B)amplitude = 1.329 + 0.107 recall + 2.205 stress + 0.168 recall*stress
C)amplitude = 4.795 recall - 0.114 stress - 4.262 recall*stress + 0.434
D)amplitude = 1.329 recall + 0.107 stress + 2.205 recall*stress + 0.168

Question 10

Researchers investigated the effects of acute stress on emotional picture processing and recall by randomly assigning 40 adult males to receive either a stressful cold stimulus or a neutral warm stimulus before viewing pictures. The researchers computed the relative brainwave amplitude (in microvolts, μV) for each subject viewing unpleasant versus neutral pictures. They also recorded how many pictures the subjects were able to recall 24 hours later. The data are displayed in the following scatterplot.
μ The following model is proposed for predicting the brainwave relative amplitude ("amplitude") from the number of images recalled ("recall"), the indicator variable reflecting the nature of the stimulus ("stress"), and an interaction term ("recall*stress"):
Amplitude_i = β₀ + β₁ (recall_i) + β₂ (stress_i) + β₃ (recall*stress_i) + ε_i
Where the deviations ε_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the sample of 40 adult males. The following results summarize the least-squares regression fit of this model:
​
$\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 4.795 & 1.329 \\\text { recall } & -0.1139 & 0.1069 \\\text { stress } & -4.262 & 2.205 \\\text { recall } & 0.4337 & 0.1677 \\\mathrm{~S}=2.13946 & \text { R-Sq }=22.8 \% &\end{array}$

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 3 & 48.788 & 16.263 \\\text { Residual Error } & 36 & 164.783 & 4.577 \\\text { Total } & 39 & 213.571 &\end{array}\end{array}$ Suppose we wish use the ANOVA F test to test the following hypotheses:
H₀: β₁ = β₂ = β₃ = 0
H_α: At least one of β₁ or β₂ or β₃ is not 0
What is the value of the F statistic for these hypotheses?

A)2.14
B)3.55
C)4.58
D)5.42

Question 11

Researchers investigated the effects of acute stress on emotional picture processing and recall by randomly assigning 40 adult males to receive either a stressful cold stimulus or a neutral warm stimulus before viewing pictures. The researchers computed the relative brainwave amplitude (in microvolts, μV) for each subject viewing unpleasant versus neutral pictures. They also recorded how many pictures the subjects were able to recall 24 hours later. The data are displayed in the following scatterplot.
μ The following model is proposed for predicting the brainwave relative amplitude ("amplitude") from the number of images recalled ("recall"), the indicator variable reflecting the nature of the stimulus ("stress"), and an interaction term ("recall*stress"):
Amplitude_i = β₀ + β₁ (recall_i) + β₂ (stress_i) + β₃ (recall*stress_i) + ε_i
Where the deviations ε_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the sample of 40 adult males. The following results summarize the least-squares regression fit of this model:
​
$\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 4.795 & 1.329 \\\text { recall } & -0.1139 & 0.1069 \\\text { stress } & -4.262 & 2.205 \\\text { recall }{ }^{\star} \text { stress } & 0.4337 & 0.1677 \\\mathrm{~S}=2.13946 & \text { R-Sq }=22.8 \% &\end{array}$

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 3 & 48.788 & 16.263 \\\text { Residual Error } & 36 & 164.783 & 4.577 \\\text { Total } & 39 & 213.571 &\end{array}\end{array}$ Suppose we wish use the ANOVA F test to test the following hypotheses:
H₀: β₁ = β₂ = β₃ = 0
H_α: At least one of β₁ or β₂ or β₃ is not 0
What is the P-value of the test?

A)Greater than 0.10
B)Between 0.05 and 0.10
C)Between 0.01 and 0.05
D)Less than 0.01

Question 12

Researchers investigated the effects of acute stress on emotional picture processing and recall by randomly assigning 40 adult males to receive either a stressful cold stimulus or a neutral warm stimulus before viewing pictures. The researchers computed the relative brainwave amplitude (in microvolts, μV) for each subject viewing unpleasant versus neutral pictures. They also recorded how many pictures the subjects were able to recall 24 hours later. The data are displayed in the following scatterplot.
μ The following model is proposed for predicting the brainwave relative amplitude ("amplitude") from the number of images recalled ("recall"), the indicator variable reflecting the nature of the stimulus ("stress"), and an interaction term ("recall*stress"):
Amplitude_i = β₀ + β₁ (recall_i) + β₂ (stress_i) + β₃ (recall*stress_i) + ε_i
Where the deviations ε_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the sample of 40 adult males. The following results summarize the least-squares regression fit of this model:
​
$\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 4.795 & 1.329 \\\text { recall } & -0.1139 & 0.1069 \\\text { stress } & -4.262 & 2.205 \\\text { recall } \text { stress } & 0.4337 & 0.1677 \\\mathrm{~S}=2.13946 & \mathrm{R}-\mathrm{Sq}=22.8 \% &\end{array}$

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 3 & 48.788 & 16.263 \\\text { Residual Error } & 36 & 164.783 & 4.577 \\\text { Total } & 39 & 213.571 &\end{array}\end{array}$ What is the percent of variation in brainwave relative amplitude that can be explained by the regression model?

A)23%
B)48%
C)77%
D)95%

Question 13

Researchers investigated the effects of acute stress on emotional picture processing and recall by randomly assigning 40 adult males to receive either a stressful cold stimulus or a neutral warm stimulus before viewing pictures. The researchers computed the relative brainwave amplitude (in microvolts, μV) for each subject viewing unpleasant versus neutral pictures. They also recorded how many pictures the subjects were able to recall 24 hours later. The data are displayed in the following scatterplot.
μ The following model is proposed for predicting the brainwave relative amplitude ("amplitude") from the number of images recalled ("recall"), the indicator variable reflecting the nature of the stimulus ("stress"), and an interaction term ("recall*stress"):
Amplitude_i = β₀ + β₁ (recall_i) + β₂ (stress_i) + β₃ (recall*stress_i) + ε_i
Where the deviations ε_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the sample of 40 adult males. The following results summarize the least-squares regression fit of this model:
​
$\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 4.795 & 1.329 \\\text { recall } & -0.1139 & 0.1069 \\\text { stress } & -4.262 & 2.205 \\\text { recall }{ }^{*} \text { stress } & 0.4337 & 0.1677 \\\mathrm{~S}=2.13946 & \text { R-Sq }=22.8 \% &\end{array}$

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 3 & 48.788 & 16.263 \\\text { Residual Error } & 36 & 164.783 & 4.577 \\\text { Total } & 39 & 213.571 &\end{array}\end{array}$ Here are the residual plots for this regression model:
Based on the study description, the scatterplot, and the residual plots, what should you conclude?

A)The assumption of Normality is not met but the other assumptions are met.
B)The assumption of constant variance is not met but the other assumptions are met.
C)The assumption of independence is not met but the other assumptions are met.
D)All of the assumptions for regression inference are met.

Question 14

Researchers investigated the effects of acute stress on emotional picture processing and recall by randomly assigning 40 adult males to receive either a stressful cold stimulus or a neutral warm stimulus before viewing pictures. The researchers computed the relative brainwave amplitude (in microvolts, μV) for each subject viewing unpleasant versus neutral pictures. They also recorded how many pictures the subjects were able to recall 24 hours later. The data are displayed in the following scatterplot.
The following model is proposed for predicting the brainwave relative amplitude ("amplitude") from the number of images recalled ("recall"), the indicator variable reflecting the nature of the stimulus ("stress"), and an interaction term ("recall*stress"):
Amplitude_i = β₀ + β₁ (recall_i) + β₂ (stress_i) + β₃ (recall*stress_i) + ε_i
Where the deviations ε_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the sample of 40 adult males. The following results summarize the least-squares regression fit of this model:
​
$\begin{array}{l}\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 4.795 & 1.329 \\\text { recall } & -0.1139 & 0.1069 \\\text { stress } & -4.262 & 2.205 \\\text { recall }{ }^{*} \text { stress } & 0.4337 & 0.1677 \\\mathrm{~S}=2.13946 & \mathrm{R}-\mathrm{Sq}=22.8 \% &\end{array}\\\\\text { Analysis of Variance }\end{array}$

$\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 3 & 48.788 & 16.263 \\\text { Residual Error } & 36 & 164.783 & 4.577 \\\text { Total } & 39 & 213.571 &\end{array}$ To determine whether the number of pictures recalled affects the brainwave relative amplitude in this model, we test the hypotheses H₀: β₁ = 0, H_a: β₁ ≠ 0 in this particular model. Using the estimates provided, what is the P-value for this test?

A)Greater than 0.10
B)Between 0.05 and 0.10
C)Between 0.01 and 0.05
D)Less than 0.01

Question 15

Researchers investigated the effects of acute stress on emotional picture processing and recall by randomly assigning 40 adult males to receive either a stressful cold stimulus or a neutral warm stimulus before viewing pictures. The researchers computed the relative brainwave amplitude (in microvolts, μV) for each subject viewing unpleasant versus neutral pictures. They also recorded how many pictures the subjects were able to recall 24 hours later. The data are displayed in the following scatterplot.
The following model is proposed for predicting the brainwave relative amplitude ("amplitude") from the number of images recalled ("recall"), the indicator variable reflecting the nature of the stimulus ("stress"), and an interaction term ("recall*stress"):
Amplitude_i = β₀ + β₁ (recall_i) + β₂ (stress_i) + β₃ (recall*stress_i) + ε_i
Where the deviations ε_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the sample of 40 adult males. The following results summarize the least-squares regression fit of this model:
​
$\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 4.795 & 1.329 \\\text { recall } & -0.1139 & 0.1069 \\\text { stress } & -4.262 & 2.205 \\\text { recall }{ }^{*} \text { stress } & 0.4337 & 0.1677 \\\mathrm{~S}=2.13946 & \text { R-Sq }=22.8 \% &\end{array}$

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 3 & 48.788 & 16.263 \\\text { Residual Error } & 36 & 164.783 & 4.577 \\\text { Total } & 39 & 213.571 &\end{array}\end{array}$ What is a 95% confidence interval for β₂, the coefficient of the indicator variable stress in this model?

A)-4.26 ± 2.21
B)-4.26 ± 4.47
C)2.21 ± 1.93
D)2.21 ± 4.47

Question 16

Researchers investigated the effects of acute stress on emotional picture processing and recall by randomly assigning 40 adult males to receive either a stressful cold stimulus or a neutral warm stimulus before viewing pictures. The researchers computed the relative brainwave amplitude (in microvolts, μV) for each subject viewing unpleasant versus neutral pictures. They also recorded how many pictures the subjects were able to recall 24 hours later. The data are displayed in the following scatterplot.
The following model is proposed for predicting the brainwave relative amplitude ("amplitude") from the number of images recalled ("recall"), the indicator variable reflecting the nature of the stimulus ("stress"), and an interaction term ("recall*stress"):
Amplitude_i = β₀ + β₁ (recall_i) + β₂ (stress_i) + β₃ (recall*stress_i) + ε_i
Where the deviations ε_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the sample of 40 adult males. The following results summarize the least-squares regression fit of this model:
​
$\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 4.795 & 1.329 \\\text { recall } & -0.1139 & 0.1069 \\\text { stress } & -4.262 & 2.205 \\\text { recall }{ }^{*} \text { stress } & 0.4337 & 0.1677 \\\mathrm{~S}=2.13946 & \text { R-Sq }=22.8 \% &\end{array}$

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 3 & 48.788 & 16.263 \\\text { Residual Error } & 36 & 164.783 & 4.577 \\\text { Total } & 39 & 213.571 &\end{array}\end{array}$ The interaction term in this model is statistically significant at the 0.05 level. Given this and the scatterplot provided, what should you conclude?

A)The effect of the number of pictures recalled on brainwave relative amplitude is substantially different when subjects are exposed to a stressful stimulus versus a neutral stimulus.
B)The effect of the number of pictures recalled on brainwave relative amplitude is essentially the same when subjects are exposed to a stressful stimulus or a neutral stimulus.
C)The number of pictures recalled does not affect brainwave relative amplitude at all.
D)The type of stimulus received does not affect brainwave relative amplitude at all.

Question 17

Researchers investigated the effects of acute stress on emotional picture processing and recall by randomly assigning 40 adult males to receive either a stressful cold stimulus or a neutral warm stimulus before viewing pictures. The researchers computed the relative brainwave amplitude (in microvolts, μV) for each subject viewing unpleasant versus neutral pictures. They also recorded how many pictures the subjects were able to recall 24 hours later. The data are displayed in the following scatterplot.
The following model is proposed for predicting the brainwave relative amplitude ("amplitude") from the number of images recalled ("recall"), the indicator variable reflecting the nature of the stimulus ("stress"), and an interaction term ("recall*stress"):
Amplitude_i = β₀ + β₁ (recall_i) + β₂ (stress_i) + β₃ (recall*stress_i) + ε_i
Where the deviations ε_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the sample of 40 adult males. The following results summarize the least-squares regression fit of this model:
​
$\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 4.795 & 1.329 \\\text { recall } & -0.1139 & 0.1069 \\\text { stress } & -4.262 & 2.205 \\\text { recall }^{*} \text { stress } & 0.4337 & 0.1677 \\\mathrm{~S}=2.13946 & \text { R-Sq }=22.8 \% &\end{array}$

$\begin{array}{l}\text { Analysis of Variance }\\\begin{array}{lrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 3 & 48.788 & 16.263 \\\text { Residual Error } & 36 & 164.783 & 4.577 \\\text { Total } & 39 & 213.571 &\end{array}\end{array}$ Based on the software output provided, what is the equation of the regression model when a stressful stimulus is received (indicator variable stress = 1)?

A)amplitude = 0.533 - 0.114 recall
B)amplitude = 0.533 + 0.320 recall
C)amplitude = 4.795 - 0.114 recall
D)amplitude = 4.795 + 0.320 recall

Question 18

Researchers investigated the effects of acute stress on emotional picture processing and recall by randomly assigning 40 adult males to receive either a stressful cold stimulus or a neutral warm stimulus before viewing pictures. The researchers computed the relative brainwave amplitude (in microvolts, μV) for each subject viewing unpleasant versus neutral pictures. They also recorded how many pictures the subjects were able to recall 24 hours later. The following model is proposed for predicting the brainwave relative amplitude ("amplitude") from the number of images recalled ("recall"), the indicator variable reflecting the nature of the stimulus ("stress"), and an interaction term ("recall*stress"):
Amplitude_i = β₀ + β₁ (recall_i) + β₂ (stress_i) + β₃ (recall*stress_i) + ε_i
Where the deviations ε_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the sample of 40 adult males. Using software, we obtained the following output:
$\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 4.795 & 1.329 \\\text { recall } & -0.1139 & 0.1069 \\\text { stress } & -4.262 & 2.205 \\\text { recall* stress } & 0.4337 & 0.1677\end{array}$


Which of the following statements best describes the interval (2.465, 4.996)?

A)A confidence interval for the mean brainwave relative amplitude of all adult men in a similar experiment exposed to a neutral stimulus and who remember 10 pictures
B)A confidence interval for the mean brainwave relative amplitude of all adult men in a similar experiment exposed to a stressful stimulus and who remember 10 pictures
C)A prediction interval for the brainwave relative amplitude of one adult man in a similar experiment exposed to a neutral stimulus and who remembers 10 pictures
D)A prediction interval for the brainwave relative amplitude of one adult man in a similar experiment exposed to a stressful stimulus and who remembers 10 pictures

Question 19

Researchers investigated the effects of acute stress on emotional picture processing and recall by randomly assigning 40 adult males to receive either a stressful cold stimulus or a neutral warm stimulus before viewing pictures. The researchers computed the relative brainwave amplitude (in microvolts, μV) for each subject viewing unpleasant versus neutral pictures. They also recorded how many pictures the subjects were able to recall 24 hours later. The following model is proposed for predicting the brainwave relative amplitude ("amplitude") from the number of images recalled ("recall"), the indicator variable reflecting the nature of the stimulus ("stress"), and an interaction term ("recall*stress"):
Amplitude_i = β₀ + β₁ (recall_i) + β₂ (stress_i) + β₃ (recall*stress_i) + ε_i
Where the deviations ε_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the sample of 40 adult males. Using software, we obtained the following output:

$\begin{array} { l r r } \text { Predictor } & \text { Coef } & \text { SE Coef } \\ \text { Constant } & 4.795 & 1.329 \\ \text { recall } & - 0.1139 & 0.1069 \\ \text { stress } & - 4.262 & 2.205 \\ \text { recall* } \text { stress } & 0.4337 & 0.1677 \end{array}$


Which of the following statements best describes the interval (-0.804, 8.115)?

A)A confidence interval for the mean brainwave relative amplitude of all adult men in a similar experiment exposed to a neutral stimulus and who remember 10 pictures
B)A confidence interval for the mean brainwave relative amplitude of all adult men in a similar experiment exposed to a stressful stimulus and who remember 10 pictures
C)A prediction interval for the brainwave relative amplitude of one adult man in a similar experiment exposed to a neutral stimulus and who remembers 10 pictures
D)A prediction interval for the brainwave relative amplitude of one adult man in a similar experiment exposed to a stressful stimulus and who remembers 10 pictures

Question 20

A study examined the effect of body weight and origin (aviary or wild) on the length of the R1 central tail feather in male finches. The following model is proposed for predicting the length of the R1 feather from the bird's weight (in grams) and origin (aviary = 0, wild = 1):
R1Length_i = β₀ + β₁ (BodyWeight_i) + β₂ (Aviary0Wild1_i) + ε_i
Where the deviations ε_i were assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to a sample of 25 male finches. Using software, we obtained the following output:
$\begin{array}{lrr}\text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & 35.79 & 16.95 \\\text { BodyWeight } & 2.826 & 1.030 \\\text { Aviary0Wild1 } & -10.453 & 4.996\end{array}$

$\begin{array}{l}\text { Predicted Values for New Observations }\\\begin{array}{rrrcc}\text { NewObs } & \text { Fit } & \text { SE Fit } & 95 \% \mathrm{CI} & 95 \% \text { PI } \\1 & 76.22 & 4.32 & (67.26,85.17) & (54.35,98.09) \\2 & 86.67 & 2.75 & (80.96,92.38) & (65.91,107.43)\end{array}\end{array}$

$\begin{array}{l}\text { Values of Predictors for New Observations }\\\begin{array}{crr}\text { NewObs } & \text { BodyWeight } & \text { Aviary0Wild } \\1 & 18.0 & 1.0 \\2 & 18.0 & 0.0\end{array}\end{array}$ What is the best description of the interval (67.26, 85.17)?

A)A confidence interval for the mean tail-feather length of all male finches that weigh 18 g and are raised in an aviary
B)A confidence interval for the mean tail-feather length of all male finches that weigh 18 g and are caught in the wild
C)A prediction interval for the tail-feather length of one male finch that weighs 18 g and was raised in an aviary
D)A prediction interval for the tail-feather length of one male finch that weighs 18 g and was caught in the wild

Question 21

A study examined the effect of body weight and origin (aviary or wild) on the length of the R1 central tail feather in male finches. The following model is proposed for predicting the length of the R1 feather from the bird's weight (in grams) and origin (aviary = 0, wild = 1):
R1Length_i = β₀ + β₁ (BodyWeight_i) + β₂ (Aviary0Wild1_i) + ε_i
Where the deviations ε_i were assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to a sample of 25 male finches. Using software, we obtained the following output:
$\begin{array} { l r r } \text { Predictor } & \text { Coef } & \text { SE Coef } \\ \text { Constant } & 35.79 & 16.95 \\ \text { BodyWeight } & 2.826 & 1.030 \\ \text { Aviary0Wild1 } & - 10.453 & 4.996 \end{array}$

Predicted Values for New Observations

$\begin{array} { r r r c c } \text { NewObs } & \text { Fit } & \text { SE Fit } & 95 \% \text { CI } & 95 \% \text { PI } \\ 1 & 76.22 & 4.32 & ( 67.26,85.17 ) & ( 54.35,98.09 ) \\ 2 & 86.67 & 2.75 & ( 80.96,92.38 ) & ( 65.91,107.43 ) \end{array}$

Values of Predictors for New Observations

$\begin{array} { c r r } \text { NewObs } & \text { BodyWeight } & \text { Aviary0Wild1 } \\ 1 & 18.0 & 1.0 \\ 2 & 18.0 & 0.0 \end{array}$ What is the best description of the interval (65.91, 107.43)?

A)A confidence interval for the mean tail-feather length of all male finches that weigh 18 g and are raised in an aviary
B)A confidence interval for the mean tail-feather length of all male finches that weigh 18 g and are caught in the wild
C)A prediction interval for the tail-feather length of one male finch that weighs 18 g and was raised in an aviary
D)A prediction interval for the tail-feather length of one male finch that weighs 18 g and was caught in the wild

Question 22

A study looked for a protein marker that might help predict active flare-ups in a particular chronic disease. A sample of 32 individuals with the disease had some blood taken to determine their disease status (active flare-up or inactive) and the relative abundance of the protein marker in their blood. We use logistic regression to model disease status as a function of relative abundance. Software gives the following information:
For Log Odds of Active/Inactive
$$\begin{array} { c c c } \text { Term } & \text { Estimate } & \text { Std Error } \\\text { Intercept } & - 6.186 & 2.416 \\\text { Relative abundance } & 6.229 & 2.651\end{array}$$ What is the probability p that a patient who has a protein marker relative abundance of 0.8 has an active flare-up?

A)0.231
B)0.300
C)0.511
D)0.770

Question 23

A study looked for a protein marker that might help predict active flare-ups in a particular chronic disease. A sample of 32 individuals with the disease had some blood taken to determine their disease status (active flare-up or inactive) and the relative abundance of the protein marker in their blood. We use logistic regression to model disease status as a function of relative abundance. Software gives the following information:
For Log Odds of Active/Inactive
$$\begin{array} { c c c } \text { Term } & \text { Estimate } & \text { Std Error } \\\text { Intercept } & - 6.186 & 2.416 \\\text { Relative abundance } & 6.229 & 2.651\end{array}$$ We want to know if the slope of the logistic regression model is significant. What is the value of the test statistic?

A)t = 2.35
B)t = 2.56
C)z = 2.35
D)z = 2.56

Question 24

A study looked for a protein marker that might help predict active flare-ups in a particular chronic disease. A sample of 32 individuals with the diseases had some blood taken to determine their disease status (active flare-up or inactive) and the relative abundance of the protein marker in their blood. We use logistic regression to model disease status as a function of relative abundance. Software gives the following information:
For Log Odds of Active/Inactive
$$\begin{array} { c c c } \text { Term } & \text { Estimate } & \text { Std Error } \\\text { Intercept } & - 6.186 & 2.416 \\\text { tive abundance } & 6.229 & 2.651\end{array}$$ We want to know if the slope of the logistic regression model is significant. What is the P-value for the two-sided test?

A)Greater than 0.10
B)Between 0.05 and 0.10
C)Between 0.01 and 0.05
D)Less than 0.01

Question 25

A study looked for a protein marker that might help predict active flare-ups in a particular chronic disease. A sample of 32 individuals with the disease had some blood taken to determine their disease status (active flare-up or inactive) and the relative abundance of the protein marker in their blood. We use logistic regression to model disease status as a function of relative abundance. Software gives the following information:
For Log Odds of Active/Inactive
$$\begin{array} { c c c } \text { Term } & \text { Estimate } & \text { Std Error } \\\text { Intercept } & - 6.186 & 2.416 \\\text { ive abundance } & 6.229 & 2.651\end{array}$$ What is a 95% confidence interval for the odds ratio e^β₁?

A)(0.002, 507.2)
B)(0.096, 10.4)
C)(2.8, 91579.3)
D)(35.8, 7186.8)

Question 26

A case study enrolled case individuals with glioma (a type of brain cancer) and similar control individuals without glioma. Participants were asked whether they had been regular cell phone users for at least a year. We use logistic regression to model glioma status (1 = glioma, 0 = no glioma) as a function of regular cell phone use (1 = yes, 0 = no). Software gives the following information:
For log odds of glioma/no glioma
$$\begin{array} { l c c } \text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & - 0.08074 & 0.04987 \\\text { Regular use } & - 0.02565 & 0.07275\end{array}$$ We want to know if the slope of the logistic regression model is significant. What is the value of the test statistic?

A)t = -1.62
B)t = -0.35
C)z = -1.62
D)z = -0.35

Question 27

A case study enrolled case individuals with glioma (a type of brain cancer) and similar control individuals without glioma. Participants were asked whether they had been regular cell phone users for at least a year. We use logistic regression to model glioma status (1 = glioma, 0 = no glioma) as a function of regular cell phone use (1 = yes, 0 = no). Software gives the following information:
For log odds of glioma/no glioma
$$\begin{array} { l c c } \text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & - 0.08074 & 0.04987 \\\text { Regular use } & - 0.02565 & 0.07275\end{array}$$ We want to know if the slope of the logistic regression model is significant. What is the P-value for the two-sided test?

A)Greater than 0.10
B)Between 0.05 and 0.10
C)Between 0.01 and 0.05
D)Less than 0.01

Question 28

A case study enrolled case individuals with glioma (a type of brain cancer) and similar control individuals without glioma. Participants were asked whether they had been regular cell phone users for at least a year. We use logistic regression to model glioma status (1 = glioma, 0 = no glioma) as a function of regular cell phone use (1 = yes, 0 = no). Software gives the following information:
For log odds of glioma/no glioma
$$\begin{array} { l c c } \text { Predictor } & \text { Coef } & \text { SE Coef } \\\text { Constant } & - 0.08074 & 0.04987 \\\text { Regular use } & - 0.02565 & 0.07275\end{array}$$ What is a 95% confidence interval for the odds ratio e^β₁?

A)(-0.17, 0.12)
B)(0.12, 0.35)
C)(0.85, 1.12)
D)(0.97, 1.26)