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?

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.
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?

Quizplus · Accepted Answer

The t statistic is calculated by dividing the estimate of the parameter (in this case, β for width) by its standard error. Thus, $ t = \frac{0.7963}{0.0939} \approx 8.48 $.

A Fisheries Biologist Has Been Studying Horseshoe Crabs Using Categorical