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?

Accepted Answer

The equation for the multiple regression model is:
R1Length = 35.79 + 2.826 BodyWeight - 10.453 Aviary0Wild1. This can be obtained by simply plugging in the coefficients obtained from the software output into the proposed model equation.

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?

Accepted Answer

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?

Accepted Answer

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?

Accepted Answer

The R-squared value of the regression model is 30.2%, which indicates that 30.2% of the variation in tail-feather length (R1Length) can be explained by the linear relationship between body weight (BodyWeight) and origin (Aviary0Wild1). Therefore, the best choice is B.

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?

Accepted Answer

The answer of Tail-feather length is a sexually dimorphic trait...

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?

Accepted Answer

We can use the t-test to test the hypotheses. The t-statistic is calculated as (estimate of the parameter - hypothesized value)/(standard error of the estimate) = (-10.453 - 0) / 4.996 = -2.09. The P-value for a two-tailed test with 22 degrees of freedom is between 0.01 and 0.05. Therefore, we reject the null hypothesis and conclude that the bird's origin affects tail-feather length in this model.

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?

Accepted Answer

The answer of Tail-feather length is a sexually dimorphic trait...

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?

Accepted Answer

The answer of Tail-feather length is a sexually dimorphic trait...

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?

Accepted Answer

The equation for the multiple regression model is in the following form:
amplitude = β0 + β1(recall) + β2(stress) + β3(recall*stress) + ε
From the given software output, we can see that the coefficient estimates (β0, β1, β2, and β3) are:
β0 = 4.795
β1 = -0.1139
β2 = -4.262
β3 = 0.4337
Therefore, the equation for the multiple regression model is:
amplitude = 4.795 - 0.114 recall - 4.262 stress + 0.434 recall*stress. Hence, the correct choice is (A).

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?

Accepted Answer

The answer of Researchers investigated the effects of acute stress...

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?

Accepted Answer

The answer of Researchers investigated the effects of acute stress...

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?

Accepted Answer

The answer of Researchers investigated the effects of acute stress...

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?

Accepted Answer

The answer of Researchers investigated the effects of acute stress...

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?

Accepted Answer

The answer of Researchers investigated the effects of acute stress...

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?

Accepted Answer

The 95% confidence interval for β₂, the coefficient of the indicator variable stress in this model is given by: [β₂ - t(0.025, 36)*SE(β₂), β₂ + t(0.025, 36)*SE(β₂)] = [-4.26 - 2.222*2.205, -4.26 + 2.222*2.205] = [-4.26 - 4.89, -4.26 + 4.89] = [-9.15, -0.37] Thus, the answer is option B, -4.26 ± 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?

Accepted Answer

The statistically significant interaction term between recall and stress suggests that the effect of the number of pictures recalled on brainwave relative amplitude is different depending on whether the subject was exposed to a stressful stimulus or a neutral one. This means that the relationship between recall and brainwave amplitude is not the same for both groups. Specifically, the effect of recall on brainwave amplitude is stronger for those who were exposed to a stressful stimulus than for those who received a neutral one.

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)?

Accepted Answer

The answer of Researchers investigated the effects of acute stress...

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

Predicted Values for New Observations
New					Obs	Fit	SEFit	95%CI	95%
PI
	1	3.731	0.624	(2.465,4.996)
	2	3.656	0.508	(2.625,4.686)

Values of Predictors for New Observations New
New					Obs	recall	stress	recall*stress
	1	10.0	1.00	10.00
	2	10.0	0.00	0.00

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

Accepted Answer

The interval (2.465, 4.996) is a confidence interval for the mean brainwave relative amplitude for all adult men exposed to a stressful stimulus who recall 10 pictures, as indicated by the stress value of 1.00 in the predictors table.

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

Predicted Values for New Observations
New					Obs	Fit	SEFit	95%CI	95%
PI
	1	3.731	0.624	(2.465,4.996)
	2	3.656	0.508	(2.625,4.686)

Values of Predictors for New Observations New
New					Obs	recall	stress	recall*stress
	1	10.0	1.00	10.00
	2	10.0	0.00	0.00

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

Accepted Answer

The interval (-0.804, 8.115) is 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.

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)?

Accepted Answer

The interval (67.26, 85.17) is associated with NewObs 1, which has a body weight of 18.0 and Aviary0Wild value of 1.0, indicating it is a wild bird. This interval is a 95% confidence interval for the mean tail-feather length of all male finches with these characteristics, not a prediction interval for a single bird.

Quiz 25: Multiple and Logistic Regression