a) i) Suppose H0: "\\mu" = "\\mu" 0 is rejected in favour of H1 : "\\mu" != "\\mu" 0 at "\\alpha" = 0.05 level of significance. Would H0 necessarily be rejected at the "\\alpha" = 0.01 level of signicance? Explain.
ii) Suppose H0: "\\mu" = "\\mu" 0 is rejected in favour of H1 : "\\mu" != "\\mu" 0 at "\\alpha" = 0.01 level of signficance. Would H0 necessarily be rejected at the "\\alpha" = 0.05 level of significance? Explain.
iii) If H0: "\\mu" = "\\mu" 0 is rejected in favour of H0: "\\mu" > "\\mu" 0, will it necessarily be rejected in favour of H1 : "\\mu" != "\\mu" 0 ? Assume that "\\alpha" 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.
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