After Obtaining a Regression Line $y = \beta _ { 0 } + \beta _ { 1 } x$

question 5

Multiple Choice

After obtaining a regression line $y = \beta _ { 0 } + \beta _ { 1 } x$ , a confidence interval for the mean of all values $y$ for which $x =$ $\mathrm { x } _ { 0 }$ can be obtained as follows:
$$\left( \hat { y } - E _ { r } \hat { y } + E \right) ,$$
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 hours. Note that the equation of the regression line is $\hat { y } = 52.8804 + 3.405 \mathrm { x }$ and that $\mathrm { s } _ { \mathrm { e } } = 6.8359$ .
$\begin{array}{l|ccccc}\mathrm{x} \text { (hours studied) } & 2.5 & 4.5 & 5.1 & 7.9 & 11.6 \\\hline \mathrm{y} \text { (score on test) } & 66 & 70 & 60 & 83 & 93\end{array}$

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

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Definitions:

Chebyshev's Theorem

A statistical theorem stating that for any set of data, the proportion of results that fall within k standard deviations from the mean is at least 1-1/k^2 for k>1.

Empirical Rule

A rule that can be used to compute the percentage of data values that must be within one, two, and three standard deviations of the mean for data that exhibit a bell-shaped distribution.

Sample Space

The entire set of possible outcomes in a probability experiment or situation.

Sample Point

A single outcome from a statistical experiment.