FiltersDone

Question type

Essay

Multiple Choice

Short Answer

True False

Matching

Exam 21: Inference for Regression

Show Answer

Share

---

All types

Filters

Study Flashcards

Question 1

Multiple Choice

Review LaterAdd Note

Review Later

A fisheries biologist has been studying horseshoe crabs using categorical variables, but she has decided that reporting the data as continuous variables would be more useful. She has sampled 100 horseshoe crabs and recorded their weight (in kilograms) and width (in centimeters) . The proposed regression equation is Weight_i = α+β × width_i + Ɛ_i Where the deviations Ɛ_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the data using the method of least squares. The following results were obtained from statistical software: $\begin{array} { | l | c | c | } \hline{ \text { Variable } } & \text { Estimate } & \begin{array} { c } \text { Standard Error of } \\\text { Estimate }\end{array} \\\hline \text { Constant } & 2.3013 & 0.9788 \\\text { Width } & 0.7963 & 0.0939 \\\hline\end{array}$ r² = 0.423, s = 2.2018. What is a 95% confidence interval for the slope in the simple linear regression model?

A) 2.3013 ± 1.9419
B) 2.3013 ± 0.1863
C) 0.7963 ± 1.9419
D) 0.7963 ± 0.1863

Correct Answer

verified

Show Answer

Question 2

Multiple Choice

Review LaterAdd Note

Review Later

Researchers found a linear relationship between tail-feather length (in mm) and weight (in g) in a random sample of 20 male long-tailed finches raised in an aviary (with weights ranging from about 13 to 21 g) . Using software, we obtain the following output for predicting feather length when weight = 20 g: $\begin{array} { r r r c c } \text { New Obs } & \text { Fit } & \text { SE Fit } & 95 \% \text { CI } & 95 \% \text { PI } \\20.0 & 92.34 & 5.26 & ( 81.29,103.38 ) & ( 67.61,117.06 ) \end{array}$ What is a 95% prediction interval for the weight of one male long-tailed finch weighing 20 g?

A) 92.34 ± 5.26
B) 92.34 ± 11.04
C) 92.34 ± 12.36
D) 92.34 ± 24.72

Correct Answer

verified

Show Answer

Question 3

Multiple Choice

Review LaterAdd Note

Data on the water quality in the eastern United States were obtained by a researcher who wanted to ascertain whether the amount of particulates in water (ppm) could be used to accurately predict the water quality score. Suppose we use the following simple linear regression model: ​ Quality_i = α+ β× particulates_i + Ɛ_i Where the deviations Ɛ_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the data using the method of least squares. The following results were obtained from statistical software based on a sample of size 61: $\begin{array} { | l | c | c | } \hline \text { Variable } & \text { Estimate } & \begin{array} { c } \text { Standard Error of } \\\text { Estimate }\end{array} \\\hline \text { Constant } & 6.214 & 1.003 \\\text { Particulates } & - 0.009 & 0.020 \\\hline\end{array}$ r² = 0.005, s = 0.7896. What is the intercept of the least-squares regression line?

Researchers want to know if reading skills can explain IQ test scores in children with dyslexia. The following scatterplot examines the relationship between reading skill score and IQ test score for 22 dyslexic children. The least-squares regression line is displayed on the plot, along with the value of r². We want to test a hypothesis of no relationship between the two scores, H₀: p= 0 versus H_a: p≠0. What is the P-value for this test?

A random sample of 79 patients from a walk-in clinic records the low- and high-density cholesterol levels of the patients, labeled LDL and HDL, respectively. LDL is often referred to as "bad" cholesterol, while HDL is often referred to as "good" cholesterol. A researcher is interested in fitting the following linear regression curve to LDL and HDL levels: ​ LDL_i = α+ β× HDL_i + Ɛ_i Where the deviations Ɛ_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the data using the method of least squares. The following results were obtained from statistical software: $\begin{array} { |c | c | c | } \hline \text { Variable } & \text { Estimate } & \begin{array} { c } \text { Standard Error of } \\\text { Estimate }\end{array} \\\hline \text { Constant } & - 25 & 3.000 \\\text { HDL } & 0.75 & 0.025 \\\hline\end{array}$ r² = 0.810, s = 50. Is there strong evidence (and if so, why?) of a straight-line relationship between LDL and HDL?

Tail-feather length in birds is sometimes a sexually dimorphic trait; that is, the trait differs substantially for males and for females. Researchers studied the relationship between weight (x) and tail-feather length (y) in a sample of 5 wild male long-tailed finches. Here are the data: $\begin{array} { c c } \text { weight } ( \mathrm { g } ) & \text { tail length } ( \mathrm { mm } ) \\20.8 & 82.5 \\19.1 & 82.5 \\15.9 & 67.0 \\16.7 & 70.5 \\15.7 & 73.5\end{array}$ We want to test the hypotheses H₀: β= 0, versus H_a: β> 0, where βis the true value of the population slope. Using appropriate software, what is the P-value for this test?

Tail-feather length in birds is sometimes a sexually dimorphic trait; that is, the trait differs substantially for males and for females. Researchers studied the relationship between weight (x) and tail-feather length (y) in a sample of 5 wild male long-tailed finches. Here are the data: $\begin{array} { c c } \text { weight } ( \mathrm { g } ) & \text { tail length } ( \mathrm { mm } ) \\20.8 & 82.5 \\19.1 & 82.5 \\15.9 & 67.0 \\16.7 & 70.5 \\15.7 & 73.5\end{array}$ Using appropriate software, what percentage of variation in tail-feather length can be explained by weight among male long-tailed finches in the wild?

Researchers found a linear relationship between tail-feather length (in mm) and weight (in g) in a random sample of 20 male long-tailed finches raised in an aviary (with weights ranging from about 13 to 21 g) . Using software, we obtain the following output for predicting feather length when weight = 20 g: $\begin{array} { r r r c c } \text { New Obs } & \text { Fit } & \text { SE Fit } & 95 \% \text { CI } & 95 \% \text { PI } \\20.0 & 92.34 & 5.26 & ( 81.29,103.38 ) & ( 67.61,117.06 ) \end{array}$ What is a 95% confidence interval for the mean tail-feather length of all male long-tailed finches that weigh 20 g?

A fisheries biologist has been studying horseshoe crabs using categorical variables, but she has decided that reporting the data as continuous variables would be more useful. She has sampled 100 horseshoe crabs and recorded their weight (in kilograms) and width (in centimeters) . The proposed regression equation is Weight_i = α+β × width_i + Ɛ_i Where the deviations Ɛ_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the data using the method of least squares. The following results were obtained from statistical software: $\begin{array} { | l | c | c | } \hline{ \text { Variable } } & \text { Estimate } & \begin{array} { c } \text { Standard Error of } \\\text { Estimate }\end{array} \\\hline \text { Constant } & 2.3013 & 0.9788 \\\text { Width } & 0.7963 & 0.0939 \\\hline\end{array}$ r² = 0.423, s = 2.2018. Suppose the researchers test the following hypotheses: H⁰: = β=0, H^α: β> 0 What is the value of the t statistic for this test?

A fisheries biologist has been studying horseshoe crabs using categorical variables, but she has decided that reporting the data as continuous variables would be more useful. She has sampled 100 horseshoe crabs and recorded their weight (in kilograms) and width (in centimeters) . The proposed regression equation is Weight_i = α+β × width_i + Ɛ_i Where the deviations Ɛ_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the data using the method of least squares. The following results were obtained from statistical software: $\begin{array} { | l | c | c | } \hline { \text { Variable } } & \text { Estimate } & \begin{array} { c } \text { Standard Error of } \\\text { Estimate }\end{array} \\\hline \text { Constant } & 2.3013 & 0.9788 \\\text { Width } & 0.7963 & 0.0939 \\\hline\end{array}$ r² = 0.423, s = 2.2018. Based on the sample data, what is the correlation between the width and weight of horseshoe crabs?

Tail-feather length in birds is sometimes a sexually dimorphic trait; that is, the trait differs substantially for males and for females of the same species. 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. The data are displayed in the scatterplot below, followed with software output about the least-squares regression model of feather length as a function of weight. ​ ​ ​ \begin{array}{l}\begin{array} { l l r } \hline &\text {Coefficients }&\text {Standard Error }\\\hline\text {Intercept }&35.7379 & 21.0523 \\\text { Bird-weight}&2.8299 & 1.2811\\\hline\text {R Square }&0.2133\\\text {Standard Error }&\ 10.5270\\end{array}\end{array} What is the value of the slope for the least-squares regression line?

A fisheries biologist has been studying horseshoe crabs using categorical variables, but she has decided that reporting the data as continuous variables would be more useful. She has sampled 100 horseshoe crabs and recorded their weight (in kilograms) and width (in centimeters) . The proposed regression equation is weight_i = α+β × width_i + Ɛ_i Where the deviations Ɛ_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ . This model was fit to the data using the method of least squares. The following results were obtained from statistical software: $\begin{array} { | l | c | c | } \hline \text { Variable } & \text { Estimate } & \begin{array} { c } \text { Standard Error of } \\\text { Estimate }\end{array} \\\hline \text { Constant } & 2.3013 & 0.9788 \\\text { Width } & 0.7963 & 0.0939 \\\hline\end{array}$ r² = 0.423, s = 2.2018. Suppose we use statistical software to predict the weight for a horseshoe crab with width 2.25 centimeters, which yields the following: ​ $\begin{array} { | c | c | c | c | } \hline \text { Predicted Weight } & \begin{array} { c } \text { Std Error } \\\text { Predict }\end{array} & \mathbf { 9 5 . 0 \% } \text { CI } & \mathbf { 9 5 . 0 \% } \text { PI } \\\hline 4.093 & 0.774 & ( 2.556,5.630 ) & ( - 0.539,8.725 ) \\\hline\end{array}$ ​ What is a 95% confidence interval for this prediction according to this output?

Tail-feather length in birds is sometimes a sexually dimorphic trait; that is, the trait differs substantially for males and for females of the same species. 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. The data are displayed in the scatterplot below, followed with software output about the least-squares regression model of feather length as a function of weight. ​ ​ \begin{array}{l}\begin{array} { l l r } \hline &\text {Coefficients }&\text {Standard Error }\\\hline\text {Intercept }&35.7379 & 21.0523 \\\text { Bird-weight}&2.8299 & 1.2811\\\hline\text {R Square }&0.2133\\\text {Standard Error }&\ 10.5270\\end{array}\end{array} Which of the following conclusions best describes the findings?

A fisheries biologist has been studying horseshoe crabs using categorical variables, but she has decided that reporting the data as continuous variables would be more useful. She has sampled 100 horseshoe crabs and recorded their weight (in kilograms) and width (in centimeters) . The proposed regression equation is Weight_i = α+β × width_i + Ɛ_i Where the deviations Ɛ_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the data using the method of least squares. The following results were obtained from statistical software: $\begin{array} { | l | c | c | } \hline { \text { Variable } } & \text { Estimate } & \begin{array} { c } \text { Standard Error of } \\\text { Estimate }\end{array} \\\hline \text { Constant } & 2.3013 & 0.9788 \\\text { Width } & 0.7963 & 0.0939 \\\hline\end{array}$ r² = 0.423, s = 2.2018. What is the explanatory variable in this study?

To estimate the mean response, we use a prediction interval.

Tail-feather length in birds is sometimes a sexually dimorphic trait; that is, the trait differs substantially for males and for females. Researchers studied the relationship between weight (x) and tail-feather length (y) in a sample of 5 wild male long-tailed finches. Here are the data: $\begin{array} { c c } \text { weight } ( \mathrm { g } ) & \text { tail length } ( \mathrm { mm } ) \\20.8 & 82.5 \\19.1 & 82.5 \\15.9 & 67.0 \\16.7 & 70.5 \\15.7 & 73.5\end{array}$ We want to test the hypotheses H₀: β= 0, versus H_a: β> 0, where βis the true value of the population slope. Using appropriate software, what is the value of the t statistic for this test?

Researchers examined hormonal changes in 39 women during the menopausal years. They found a weak linear relationship between age (in years) and the six-month percent change in plasma level of the reproductive hormone LH (luteinizing hormone). Software gives the following output for a least-squares regression analysis: ​ $\begin{array} { l c r r r } \text { Predictor } & \text { Coef } & \text { SE Coef } & \mathrm { T } & \mathrm { P } \\\text { Constant } & 101.46 & 79.05 & 1.28 & 0.207 \\\text { age } & - 1.627 & 1.559 & - 1.04 & 0.303\end{array}$ ​ S = 41.5236 R-Sq = 2.9% R-Sq(adj) = 0.2% Based on this output, the P-value for testing the hypotheses H₀: β₁ = 0 versus H_a: β₁ ≠ 0 is ______________.

Tail-feather length in birds is sometimes a sexually dimorphic trait; that is, the trait differs substantially for males and for females. Researchers studied the relationship between weight (x) and tail-feather length (y) in a sample of 5 wild male long-tailed finches. Here are the data: $\begin{array} { c c } \text { weight } ( \mathrm { g } ) & \text { tail length } ( \mathrm { mm } ) \\20.8 & 82.5 \\19.1 & 82.5 \\15.9 & 67.0 \\16.7 & 70.5 \\15.7 & 73.5\end{array}$ Using appropriate software to perform the necessary calculations, what does the value of the slope for the least-squares regression line tells us?

Data on the water quality in the eastern United States were obtained by a researcher who wanted to ascertain whether the amount of particulates in water (ppm) could be used to accurately predict the water quality score. Suppose we use the following simple linear regression model: ​ Quality_i = α+ β× particulates_i + Ɛ_i Where the deviations Ɛ_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. This model was fit to the data using the method of least squares. The following results were obtained from statistical software based on a sample of size 61: $\begin{array} { | l | c | c | } \hline \text { Variable } & \text { Estimate } & \begin{array} { c } \text { Standard Error of } \\\text { Estimate }\end{array} \\\hline \text { Constant } & 6.214 & 1.003 \\\text { Particulates } & - 0.009 & 0.020 \\\hline\end{array}$ r² = 0.005, s = 0.7896. Is there strong evidence (and if so, why?) that there is a straight-line dependence between the amount of particulates and water quality?

A random sample of 79 patients from a walk-in clinic records the low- and high-density cholesterol levels of the patients, labeled LDL and HDL, respectively. LDL is often referred to as "bad" cholesterol, while HDL is often referred to as "good" cholesterol. A researcher is interested in fitting the following linear regression curve to LDL and HDL levels: LDL_i = α+ β× HDL_i + Ɛ_i Where the deviations _i are assumed to be independent and Normally distributed with mean 0 and standard deviation . This model was fit to the data using the method of least squares. The following results were obtained from statistical software: $\begin{array} { | c | c | c | } \hline \text { Variable } & \text { Estimate } & \begin{array} { c } \text { Standard Error of } \\\text { Estimate }\end{array} \\\hline \text { Constant } & - 25 & 3.000 \\\text { HDL } & 0.75 & 0.025 \\\hline\end{array}$ r² = 0.810, s = 50. What is the intercept of the least-squares regression line?