Hypotheses Test: H0:\mu = 5 vs Ha \mu >/< 5

[needhelp83]

Let $$\mu$$ denote the true average radioactivity level (picocuries per liter). The value 5 pCi/L is considered the dividing line between safe and unsafe water. Would you recommend testing H0:$$\mu$$ = 5 versus Ha: $$\mu$$ > 5 or H0:$$\mu$$ = 5 versus Ha: $$\mu$$ < 5? Explain your reasoning. (Hint: Think about the consequences of a type I and type II error for each possibility)

Attempted solution:
I would think about running the test of H0:$$\mu$$ = 5 versus Ha: $$\mu$$ > 5 where my Type I error would reject my null hypothesis when it is actually true. With this approach, I am more on the safe side and this would make the water unsafe. If I encountered the Type II error the null hypothesis wouldn't be rejected when it is actually false. This again, would be playing it on the safe side if it wasn't greater than 5.

If I used H0:$$\mu$$ = 5 versus $$\mu$$ < 5, type I error would be say the water is not at level 5 when it is actually 5. This would then mean water is less than 5. Type II error would fail to reject null hypothesis when it is false and say that the water is less than 5 when it is truly 5.

 

[Billy Bob]

Just a thought here: One of the tests is "assume water is safe until we prove it unsafe." The other is "assume water is unsafe until we prove it safe." Which is which?

 

[needhelp83]

Just a thought here: One of the tests is "assume water is safe until we prove it unsafe." The other is "assume water is unsafe until we prove it safe." Which is which?

I would think about running the test of H0:LaTeX Code: \\mu = 5 versus Ha: LaTeX Code: \\mu > 5 where my Type I error would reject my null hypothesis when it is actually true. With this approach, I am more on the safe side because you would actually do something about the water if it was unsafe. Since Type I errors occurs more often than, the rejection of Ho would hopefully not happen.

It is no need to test if the water is Ho:LaTeX Code: \\mu = 5 versus Ha: LaTeX Code: \\mu < 5 because we are more concerned with figuring out if the water is safe. If we do this test, we aren't necessarily proving if the water is unsafe or not.

Does this approach sound better?