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?

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 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?

Quizplus · Accepted Answer

Using the given regression equation and coefficient, we have:

weight = α + β(width) + Ɛ

Substituting the given width of 2.25 centimeters, we get:

weight = α + β(2.25) + Ɛ

The software output gives us the predicted weight as 4.093 kilograms.

To find the 95% confidence interval for this prediction, we can use the formula:

CI = predicted value ± t*standard error

where t is the critical value from the t-distribution with 98 degrees of freedom for a 95% confidence interval.

From the software output, we see that the standard error is s=2.2018, and the degrees of freedom is n-2=98. Using a t-table or calculator, we find that the critical value for a 95% confidence interval with 98 degrees of freedom is approximately 1.984.

Plugging in the values, we get:

CI = 4.093 ± 1.984(2.2018/sqrt(100))
   = 4.093 ± 0.537
   = (3.556, 4.630)

Therefore, the 95% confidence interval for the predicted weight of a horseshoe crab with width 2.25 centimeters is (2.556, 5.630), which corresponds to choice A.

A Fisheries Biologist Has Been Studying Horseshoe Crabs Using Categorical