a) i) Suppose H0: \muμ = \muμ 0 is rejected in favour of H1 : \muμ != \muμ 0 at \alphaα = 0.05 level of signifi cance. Would H0 necessarily be rejected at the \alphaα = 0.01 level of signi cance? Explain. ii) Suppose H0: \muμ = \muμ 0 is rejected in favour of H1 : \muμ != \muμ 0 at \alphaα = 0.01 level of signfi cance. Would H0 necessarily be rejected at the \alphaα = 0.05 level of signifi cance? Explain. iii) If H0: \muμ = \muμ 0 is rejected in favour of H0: \muμ > \muμ 0, will it necessarily be rejected in favour of H1 : \muμ != \muμ 0 ? Assume that \alphaα remains the same.

$a)$

No,

When $H_0$ is rejected at $\alpha=0.05$, it implies that $p-value\lt \alpha=0.05$. Changing the level of significance to $\alpha=0.01$ would not necessarily lead to rejection of $H_0$ since $0.01\lt p-value\lt 0.05$.

$b)$

Yes,

When $H_0$ is rejected at $\alpha=0.01$, it implies that $p-value\lt \alpha=0.01$. Changing the level of significance to $\alpha=0.05$ would lead to rejection of $H_0$ also since $p-value\lt 0.01\lt 0.05$.

$c)$

No,

Let $p-value 1$ be the p-value for an upper tailed test and $p-value2$ be the p-value of a two tailed test.

When $H_0$ is rejected at a given $\alpha$ for an upper tailed test it shows that $p-value 1\lt \alpha$.

The relationship between the p-value for these tests is,

$p-value1=1-{p-value 2\over 2}\implies p-value 2=2(1-(p-value 1))$

Clearly, the p-value of a two tailed test will be large since the upper tailed p-value was small. Thus, the null hypothesis would not be rejected.