# Test the hypothesis that $B_1=0$ at the $5 \%$ significance level

I am told to test the hypothesis and this is what I did:

$H_{0}:\beta_{1}=0$

$H_{a}:\beta_{1}\not=0$

So then I have $$t^{*}=\dfrac{\hat{\beta_{1}}-\beta_{1}}{\dfrac{s_{\beta_1}}{\sqrt n}}$$

$\beta_{1}$=.5 and $\hat{\beta_{1}}$=.283

n=10 and $s_{\beta_{1}}$=.26

So $t^{*}$ ended up being equal to -2.67

The degree's of freedom is $n-1=9$ and so $t_{1}$ at $\alpha=.05$ is 2.262

So since |$t^{*}$|>2.62, we can reject.

I am unsure if that is perfectly correct and it probably is not, and if that is the case can you explain why? Thanks.

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How come did you set $\beta_1=0.5$? The test statistic is in this case computed from the value given in the null hypothesis, so $\beta_1=0$. Otherwise, how can $\alpha$ be the probability of type I error? – Gene Arboit Feb 17 '13 at 18:43
Actually, I wonder if the title is correct: you say that you wish to test that $\beta_1=0$? Is $\beta_1=0$ really the null hypothesis? If $\beta_1=0$ represents the change from status quo, usually what "you would test for", then $\beta_1=0$ is the alternative hypothesis. I suppose that your title meant to say "Testing for the rejection of the null hypothesis $\beta_1=0$"? – Gene Arboit Feb 17 '13 at 20:02