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?

Question

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?

Quizplus · Accepted Answer

The 95% confidence interval for the slope (β) in a simple linear regression model is calculated using the formula: estimate ± (critical value) * (standard error of estimate). Given the estimate of the slope is 0.7963 and the standard error is 0.0939, and assuming the critical value from the t-distribution for a 95% confidence interval (which is not directly provided but typically around 2 for large samples), the calculation would closely resemble the values in option D, indicating a confidence interval centered around the slope estimate with a margin calculated from the standard error.

A Fisheries Biologist Has Been Studying Horseshoe Crabs Using Categorical