Today I was confronted with a question- " There is little confusion in concept of confidence interval like if our null hypothesis µ = 5 and alternative hypothesis says µ = 6 and our confidence interval is from (4.5 , 6.5) at 95 percent confidence because our confidence interval includes our null hypothesis so we will accept the null hypothesis but our confidence interval also includes the alternative hypothesis why we do not consider our alternative hypothesis as accepted"
Here is what I think about the above question:
Firstly Alternative hypothesis is always composite, that is one cannot take it at a point as mentioned µ=6. Its the compliment of the null. So,the approriate alternative should be µ>5(if right tailed, < or not equal if left or both tailed)
Secondly, if 95% Confidence Interval is giving a dubious result, check it with a shorter C.I as in 90% as it would definitely give, a ranger in further left to (4.5,6.5) via which one will thus "Fail to reject the null hypothesis".
My only confusion is:
If the null and alternative hypothhesis was taken correctly. as in;
Ho: µ=5 vs H1:µ>5 (say it is right tailed)
Then, is it possible to get such a dubious C.I as (4.5,6.5) which would include both the null and alternative hypothesis ?
If so, what will be our conclusion?
Please help with this concept.