A Confidence Interval for The $y$ -Intercept $\beta _ { 0 }$

question 113

Multiple Choice

A confidence interval for the $y$ -intercept $\beta _ { 0 }$ for a regression line $y = \beta _ { 0 } + \beta _ { 1 } x$ can be found by evaluating the limits in the interval below:
$$\mathrm { b } _ { 0 } - \mathrm { E } < \beta _ { 0 } < \mathrm { b } _ { 0 } + \mathrm { E } ,$$
where $\mathrm { E } = \left( \mathrm { t } _ { \alpha / 2 } \right) \mathrm { se } \sqrt { \frac { 1 } { \mathrm { n } } + \overline { \mathrm { x } } 2 / \left[ \sum \mathrm { x } ^ { 2 } - \left( \sum \mathrm { x } \right) ^ { 2 } / \mathrm { n } \right] }$ .
The critical value $t _ { \alpha / 2 }$ is found from the $t$ -table using $n - 2$ degrees of freedom and $b _ { 0 }$ is calculated in the usual v from the sample data.
Use the data below to obtain a $95 \%$ confidence interval estimate of $\beta _ { 0 }$ .
$\begin{array}{c|ccccc}\mathrm{x} \text { (hours studied) } & 2.5 & 4.5 & 5.1 & 7.9 & 11.6 \\\hline \mathrm{y} \text { (score on test) } & 66 & 70 & 60 & 83 & 93\end{array}$

• A) \(31.10 < \beta _ { 0 } < 74.66\)
• B) \(37.83 < \beta _ { 0 } < 67.93\)
• C) \(33.50 < \beta _ { 0 } < 72.26\)
• D) \(28.10 < \beta _ { 0 } < 77.66\)

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Definitions:

Asset Acquisition

Asset acquisition refers to the process of obtaining control over assets, typically through purchase or lease, for use in operations.

Straight-Line Depreciation

Straight-line depreciation is an accounting method wherein an asset's cost is evenly distributed over its estimated useful life, leading to a consistent depreciation expense each year.

Salvage Value

The calculated return value of an asset at the end of its functional life.

Useful Life

The period over which an asset is expected to be useful in the operations of a business.