Consider the Sample Regression Function $\widehat { Y } _ { i } = \widehat { \beta } _ { 0 } + \widehat { \beta } _ { 1 } X _ { i }$

question 22

Essay

Consider the sample regression function $\widehat { Y } _ { i } = \widehat { \beta } _ { 0 } + \widehat { \beta } _ { 1 } X _ { i }$
The table below lists estimates for the slope $\left( \widehat { \beta } _ { 1 } \right)$ and the variance of the slope estimator $\left( \widehat { \sigma } _ { \widehat { \beta } _ { 1 } } ^ { 2 } \right)$ In each case calculate the p -value for the null hypothesis of $\beta _ { 1 } = 0$ and a two-tailed alternative hypothesis. Indicate in which case you would reject the null hypothesis at the 5 % significance level.
$\begin{array} { | c | c | c | c | c | } \hline \widehat { \beta } _ { 1 } & - 1.76 & 0.0025 & 2.85 & - 0.00014 \\\hline \hat { \sigma } _ { \widehat { \beta } _ { 1 } } ^ { 2 } & 0.37 & 0.000003 & 117.5 & 0.0000013 \\\hline\end{array}$

---

Definitions:

Optical Rotation

The measurement of the degree to which a chiral compound rotates plane-polarized light, a property specific to chiral substances.

α-D-Glucopyranose

A cyclic form of glucose where the ring consists of five carbon atoms and one oxygen atom, with the glucose in its alpha configuration.

Haworth

A representation method in chemistry used to depict the cyclic structure of monosaccharides with a simple three-dimensional perspective.

α-L-Fructofuranose

A form of fructose sugar with a five-membered ring structure, commonly found in honey and fruit.