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Q. 14

Expert-verifiedFound in: Page 790

Book edition
6th

Author(s)
Daren Starnes, Josh Tabor

Pages
837 pages

ISBN
9781319113339

The students in Mr. Shenk’s class measured the arm spans and heights (in inches) of a random sample of $18$ students from their large high school. Here is computer output from a least-squares regression analysis of these data. Construct and interpret a $90\%$ confidence interval for the slope of the population regression line. Assume that the conditions for performing inference are met.

$\mathrm{Predictor}\mathrm{Coef}\mathrm{Stdev}\mathrm{t}-\mathrm{ratio}\mathrm{P}\phantom{\rule{0ex}{0ex}}\mathrm{Constant}11.5475.6002.060.056\phantom{\rule{0ex}{0ex}}\mathrm{Armspan}0.840420.0809110.390.000\phantom{\rule{0ex}{0ex}}\mathrm{S}=1.613\mathrm{R}-\mathrm{Sq}=87.1\%\mathrm{R}-\mathrm{Sq}\left(\mathrm{adj}\right)=86.3\%$

We are $90\%$ confident that the slope of the true regression line is between $0.69915114\mathrm{and}0.98168886$.

We need to construct and interpret a $90\%$ confidence interval for the slope of the population regression line.

Consider:

$\mathrm{n}=18\phantom{\rule{0ex}{0ex}}\mathrm{b}=0.84042\phantom{\rule{0ex}{0ex}}{\mathrm{SE}}_{\mathrm{b}}=0.08091$

The degrees of freedom in sample size decreased by $2$:

$\mathrm{df}=\mathrm{n}-2=18-2=16$

The critical $\mathrm{t}$-value can be found in table B in the row of $\mathrm{df}=16$ and the column of $\mathrm{c}=90\%$

$\mathrm{t}\text{'}=1.746$

The boundaries of the confidence interval then become:

$b-t\text{'}\times S{E}_{b}=0.84042-1.746\times 0.08091=0.69915114$role="math" localid="1654161040846" $b+t\text{'}\times S{E}_{b}=0.84042+1.746\times 0.08091=0.98168886$

Exercises T12.4–T12.8 refer to the following setting. An old saying in golf is “You drive for show and you putt for dough.” The point is that good putting is more important than long driving for shooting low scores and hence winning money. To see if this is the case, data from a random sample of 69 of the nearly 1000 players on the PGA Tour’s world money list are examined. The average number of putts per hole (fewer is better) and the player’s total winnings for the previous season are recorded and a least-squares regression line was fitted to the data. Assume the conditions forinference about the slope are met. Here is computer output from the regression analysis:

T12.5 Suppose that the researchers test the hypotheses $H0:{\beta}_{1}=0$ versus

$Ha:{\beta}_{1}<0$. Which of the following is the value of the t statistic for this

a. 2.61

b. −2.44

c. 2.44

d. −20.24

e. 0.081

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