Question

You conduct a statistical test of hypotheses and find that the null hypothesis is statistically significant at level ????=0.05 . You may conclude that:

a. the test would also be significant at level ????=0.10 .
b. neither options are true.
c. the test would also be significant at level ????=0.01 .
d. both options are true.

Answer 1

The right answer is Option c: the test would also be significant at level α=0.01.

Explanation:

We have a hypothesis where the null hypothesis is not rejected at a significance level of α=0.05.

That means that the P-value of the statistic is bigger than 0.05, and that is why its result is not statistically significant to reject the null hypothesis.

This result will stay the same for every significance level below 0.05 (for example α=0.01).

At significance level of 0.01, the test is more conservative about rejecting the null hypothesis than at a significance level of 0.05. The P-value is still too big to reject the null hypothesis.

At a significance level of 0.10, which is less conservative, we do not know if the P-value is below or above that level, so we can not say if the null hypothesis is rejected or not.

Answer 2

`p_v <\alpha`

`p_v <0.05`

For the first option we know that
`p_v <0.05` and automatically needs to be
`p_v <0.1`, because
`0.05<0.1` so then the claim for part a makes sense

For the claim in part c we know that
`p_v<0.05` and in order that the result would be significant at 1% of significance we need that
`p_v<0.01` and with the condition given we don't have information to ensure this because if for example
`p_v =0.02 <0.05` but is
`0.02>0.01`

So then the best answer for this case would be:

a. the test would also be significant at level
`\alpha=0.10` .

Explanation:

For this case we know that after conduct an hypothesis test for a parameter of interest let's say
`\theta` we got a significant result at a significance level of
`\alpha=0.05` and that means that the p value of the test is lower than the significance level:

`p_v <\alpha`

`p_v <0.05`

For the first option we know that
`p_v <0.05` and automatically needs to be
`p_v <0.1`, because
`0.05<0.1` so then the claim for part a makes sense

For the claim in part c we know that
`p_v<0.05` and in order that the result would be significant at 1% of significance we need that
`p_v<0.01` and with the condition given we don't have information to ensure this because if for example
`p_v =0.02 <0.05` but is
`0.02>0.01`

So then the best answer for this case would be:

a. the test would also be significant at level
`\alpha=0.10` .