Question regarding Confidence Interval

Question

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:

1. 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)

2. 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.

Thank you.

Answer 1

What you are saying about the alternative hypothesis is correct. In hypothesis testing, the null hypothesis always includes equality, while the alternative hypothesis is the remaining, the oposite to the null hypothesis. In this sense you can have three types of tests:

• $H_0: \mu = 5$ and then the alternative would be $H_1: \mu\neq5$ (two sided test)
• $H_0: \mu \geq 5$ and then the alternative would be $H_1: \mu<5$ (left sided test)
• $H_0: \mu \leq 5$ and then the alternative would be $H_1: \mu>5$ (right sided test)

When computing a confidence interval, you obtain an interval centered in $\bar{x}$ that will likely cover the true unknown parameter value $\mu$. How likely? It depends on the confidence level. If you compute a 95% CI you are saying that you are 95% sure that your confidence interval will cover the true unknown population parameter $\mu$.

On the other side, when you compute a hypothesis test, you calculate the test statistic value if $H_0$ was true and then you check if this test statistic value is a likely one. If it is an unlikely value, then the value suggested for $\mu$ in the null hypothesis is a strange value and the null hypothesis is rejected.

Now, if you compute a 95% confidence interval and this interval yields the result $(4.5,6.5)$, basically, you are saying that you are 95% sure that any value within this interval is a likely value for $\mu$, and so, if you performed a hypothesis test in which the null hypothesis was any value in the interval, you will not reject the null hypothesis, and in the same way, if the null hypothesis was any value outside the CI, you will reject the null hypothesis.

This relaltion between confidence intervals and hypothesis tests is held always under two conditions:

• The confidence level in both things must be the same: You can only compare a 95% CI with a 95% test
• If you have a two sided test, you can only compare it with a two sided CI. If you have a left sided test, you can only compare it with a left sided CI, and if you have a right sided test, you can only compare it with a right sided test.