Solve the Problem $y = \beta _ { 0 } + \beta _ { 1 } x$

question 166

Multiple Choice

Solve the problem.
-After obtaining a regression line $y = \beta _ { 0 } + \beta _ { 1 } x$ , a confidence interval for the mean of all values $y$ for which $x = x 0$ can be obtained as follows:
$( \hat { y } - \mathrm { E } , \hat { \mathrm { y } } + \mathrm { E } )$ where $E = \left( t _ { \alpha / 2 } \right) \operatorname { se } \sqrt { \frac { 1 } { n } + \frac { n \left( x _ { 0 } - \bar { x } \right) ^ { 2 } } { n \sum x ^ { 2 } - \left( \sum x \right) ^ { 2 } } }$
The critical value $t _ { \alpha / 2 }$ is found from the t-table using $n - 2$ degrees of freedom.
Use the data below to obtain a $95 \%$ confidence interval estimate of the mean test score of all students who study $10.5$ hours. Note that the equation of the regression line is $\hat { y } = 52.8804 + 3.405 \mathrm { x }$ and that $\mathrm { s } = 6.8359$ .
$\begin{array}{l|lllll}x \text { (hours studied) } & 2.5 & 4.5 & 5.1 & 7.9 & 11.6 \\\hline \mathrm{y} \text { (score on test) } & 66 & 70 & 60 & 83 & 93\end{array}$

• A) \(( 80.48,96.78 )\)
• B) \(( 77.11,100.15 )\)
• C) \(( 72.49,104.78 )\)
• D) \(( 61.55,115.71 )\)

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Definitions:

Comparative Method

A research approach in evolutionary biology that compares different species or populations to understand adaptation, phylogeny, and function.

Shared Ancestry

The common evolutionary origin of two or more species, which is reflected in their genetic material.

Ideal Free Distribution

A theory in ecology that predicts how animals distribute themselves among habitats to maximize their fitness, assuming individuals are free to choose and are perfectly informed.

Habitats

The natural environments or places where a particular species lives and grows; it is crucial for the survival of species.