Solve the Problem $y$ -Intercept $\beta _ { 0 }$ For a Regression Line

question 21

Multiple Choice

Solve the problem.
-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 $E = \left( t _ { \alpha / 2 } \right) \operatorname { se } \sqrt { \frac { 1 } { n } + x ^ { 2 } / \left[ \sum x ^ { 2 } - \left( \sum x \right) ^ { 2 } / n \right] }$ 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 way from the sample data.
Use the data below to obtain a $95 \%$ confidence interval estimate of $\beta _ { 0 }$ .
$\begin{array}{c|ccccc}x \text { (hours studied) } & 2.5 & 4.5 & 5.1 & 7.9 & 11.6 \\\hline y \text { (score on test) } & 66 & 70 & 60 & 83 & 93\end{array}$

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

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Definitions:

Call Premium

The additional amount that must be paid over the par value by the issuer to redeem a callable security before its maturity date.

Default Risk Premium (DRP)

The additional yield that investors demand for holding a bond that has a risk of default over a risk-free bond.

Liquidity Premium (LP)

Additional return that investors demand for holding an asset that is not easily convertible to cash without a loss in value.

Real Risk-free Rate

The return on an investment with no risk of financial loss, adjusted for inflation.