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I'm trying to solve the following problem:


Question


I don't quite get how to solve the hypotheses testing part.

Here is how I tried tackling the problem: $$H_0 : x_t = 0 $$ $$H_1 : x_t \ne 0 $$ I then checked my T distribution table and got that: $$\begin{align}&t_L = -1.3304\\&t_U = 1.3304\end{align}$$ Now the problem accures when I try to calculate $t$: $$t = \frac {0-\overline x}{se(x_t )}$$ as I don't know $\overline x$ and $se(x_t)$ I can't calculate it.

I'm thinking of doing the following: $$t = \frac {0-0.38}{0.084} = -4.524$$ But not sure if it would be correct or not. Thank you very much for your help.

(Just to be sure, is x important in determining y due to the fact that they are positively correlated)

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    $\begingroup$ You want to test for the coefficient value ($0.38$), rather than the regressor $x_{t}$ (ie. you're interested to know whether the slope of the line is non-zero). To get started, set up with the hypotheses $H_{0}: \beta_{1} = 0$ and $H_{0}: \beta_{1} \neq 0$, where your regression equation is given by $\hat{y}_{t} = \beta_{0} + \beta_{1}x_{t}$. $\endgroup$
    – rzch
    Jan 24 '19 at 13:56
  • $\begingroup$ Thank you very much for your reply. So t will be equall to -4.524? $$t = \frac {0-0.38}{0.084} = -4.524$$ @rzch $\endgroup$
    – Fozoro
    Jan 24 '19 at 14:03
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    $\begingroup$ The coefficient estimate should come first in the formula, so it's $t = \frac{0.38 - 0}{0.084}$. $\endgroup$
    – rzch
    Jan 24 '19 at 14:05
  • $\begingroup$ @rzch oops my bad, are the upper and lower bounds correct? and is my reason for the importance of x to determine y correct? Thanks a ton for your help! $\endgroup$
    – Fozoro
    Jan 24 '19 at 14:08
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    $\begingroup$ It's asking whether $x$ is 'important', but doesn't specify any preference for which direction. So you should take this to mean any correlation whatsoever (positive or negative slope). This gives you a two-tailed test, and looking up the t-table at 18 degrees of freedom for 10% two-tailed, (eg. sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf) it gives you an upper and lower bound of -1.734 and 1.734 respectively. $\endgroup$
    – rzch
    Jan 24 '19 at 14:15

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