Exam 21: Inference for Regression

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: weight () tail length () 20.8 82.5 19.1 82.5 15.9 67.0 16.7 70.5 15.7 73.5 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?

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: Variable Estimate Standard Error of Estimate Constant 6.214 1.003 Particulates -0.009 0.020 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?

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: weight () tail length () 20.8 82.5 19.1 82.5 15.9 67.0 16.7 70.5 15.7 73.5 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?

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: Variable Estimate Standard Error of Estimate Constant 2.3013 0.9788 Width 0.7963 0.0939 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: Variable Estimate Standard Error of Estimate Constant 2.3013 0.9788 Width 0.7963 0.0939 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 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. ​ ​ Coefficients Standard Error Intercept 35.7379 21.0523 Bird-weight 2.8299 1.2811 R Square 0.2133 Standard Error \ 10.5270\ Which of the following conclusions best describes the findings?

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: weight () tail length () 20.8 82.5 19.1 82.5 15.9 67.0 16.7 70.5 15.7 73.5 Using appropriate software to perform the necessary calculations, what does the value of the slope for the least-squares regression line tells us?

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: New Obs Fit SE Fit 95\% CI 95\% PI 20.0 92.34 5.26 (81.29,103.38) (67.61,117.06) What is a 95% prediction interval for the weight of one male long-tailed finch weighing 20 g?

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: Variable Estimate Standard Error of Estimate Constant 6.214 1.003 Particulates -0.009 0.020 r² = 0.005, s = 0.7896. What is the intercept of the least-squares regression line?

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. ​ ​ ​ Coefficients Standard Error Intercept 35.7379 21.0523 Bird-weight 2.8299 1.2811 R Square 0.2133 Standard Error \ 10.5270\ What is the value of the slope for the least-squares regression line?

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: weight () tail length () 20.8 82.5 19.1 82.5 15.9 67.0 16.7 70.5 15.7 73.5 Using appropriate software, what percentage of variation in tail-feather length can be explained by weight among male long-tailed finches in the wild?

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: Variable Estimate Standard Error of Estimate Constant 2.3013 0.9788 Width 0.7963 0.0939 r² = 0.423, s = 2.2018. What is a 95% confidence interval for the slope in the simple linear regression model?

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: New Obs Fit SE Fit 95\% CI 95\% PI 20.0 92.34 5.26 (81.29,103.38) (67.61,117.06) What is a 95% confidence interval for the mean tail-feather length of all male long-tailed finches that weigh 20 g?

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: Variable Estimate Standard Error of Estimate Constant -25 3.000 HDL 0.75 0.025 r² = 0.810, s = 50. What is the intercept of 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: Variable Estimate Standard Error of Estimate Constant 2.3013 0.9788 Width 0.7963 0.0939 r² = 0.423, s = 2.2018. Based on the sample data, what is the correlation between the width and weight of horseshoe crabs?

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: ​ Predictor Coef SE Coef Constant 101.46 79.05 1.28 0.207 age -1.627 1.559 -1.04 0.303 ​ 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 ______________.

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: Variable Estimate Standard Error of Estimate Constant -25 3.000 HDL 0.75 0.025 r² = 0.810, s = 50. Is there strong evidence (and if so, why?) of a straight-line relationship between LDL and HDL?

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 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: Variable Estimate Standard Error of Estimate Constant 2.3013 0.9788 Width 0.7963 0.0939 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: ​ Predicted Weight Std Error Predict CI PI 4.093 0.774 (2.556,5.630) (-0.539,8.725) ​ What is a 95% confidence interval for this prediction according to this output?