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Can someone please explain the difference between confidence interval and hypothesis tests.

For example in relation to estimating the mean, it seems to me that the main difference is that in a one sample test (α=0.05) you find the 2.5th percentile and 97.5th percentile of the distribution centered around the null mean and see if the alternative mean lies within it. This is in contrast to confidence intervals where you instead find the percentiles of the alternative hypothesis and see if the null mean is present in it. Is this correct?, this seems a little redundant to me. Is there some benefit of one over the other?

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    $\begingroup$ You don't ask any specific questions; your post simply makes a series of statements. Please identify something you specifically want answered about this situation. $\endgroup$ – Glen_b Jun 11 '17 at 0:45
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    $\begingroup$ I think I made it a little clearer, I was just asking about the difference between CI and one sample test. They both seem very similar. $\endgroup$ – Rahim Ahmed Jun 11 '17 at 4:09
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Your simple explanation of a hypothesis test works, however for the CI, the interval is also constructed assuming the null hypothesis to be true, at least initially. Only when we have rejected the null hypothesis do we construct a CI around the effect. Smithson elaborates on this in the reference below.

Smithson, M. (2001). Correct Confidence Intervals for Various Regression Effect Sizes and Parameters: The Importance of Noncentral Distributions in Computing Intervals. Educational and Psychological Measurement, 61(4), 605–632. https://doi.org/10.1177/00131640121971392

Confidence intervals are another way to conduct hypothesis tests. In this instance, there is no "versus". Some researchers also use them to communicate the uncertainty about the effect obtained from their single study. In this instance, as Smithson elaborates, CI are used differently.

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The hypothesis tests that we conduct are never about the sample statistics but always about the true population statistics. I'll try to make it as clear as possible through this example: Suppose we have a huge population and we have to find out the average time one spends on the internet, lets call it u. But we only have a sample of size 50 of the same population from which we found out the fact that- the average time one person among these 50 people spends on the internet is x(our observed value).

So, my population average is u which is unknown to me. And my sample average is x which we just found out.

So now suppose we have to perform a hypothesis test if my population average is suppose N.(this is my hypothesis since it is has been assumed and N is an arbitrary number).

i.e [Ho: u = N] and so my alternative shall be [Ha: u != N].

So u can see that it is always about the true population parameter(i.e average in our case) and never ever about the sample parameter or our observed value.

On the other hand, creating a confidence interval of this sample, suppose of 95%, what we really mean to say is this- If we take repeated samples of size 50 from this population and create their respective confidence intervals, we are 100% sure that 95% of these intervals will contain my true population mean i.e u .

CI = (x + Margin Of Error, x - Margin Of Error)

You can also see this: if my null value i.e N (check the null hypothesis) doesn't fall within this interval, we can reject our null hypothesis right away.

This is the concrete idea behind these 2 methods.Don't mix them up. I hope this was helpful.

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All values that fall into 95% confidence interval are considered to be plausible values for the parameter being estimated. If the value of the parameter lies within this interval, the null hypothesis cannot be rejected but if the values lies outside then we can reject the null-hypithesis. Lets say our 95% confidence interval lies between 10 and 20. The value of the parameter of the null-hypothesis is 21. We reject the null-hypothesis at the significance level of 5%.

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  • $\begingroup$ Why is it though you have a CI centered around our estimated parameter and we see if null hypothesis falls within in it. In the one sample test isn't it the opposite, a CI around the null hypothesis and testing if the estimated parameter falls within in. $\endgroup$ – Rahim Ahmed Jun 11 '17 at 14:22
  • $\begingroup$ @RahimAhmed One sample test: Null-hypothesis is that loading time of a page is 5s. You have a sample with the mean of 9s and 95% confidence interval of [7,10]. 5 does not fall into the CI, meaning we can reject the Null-hypothesis, the loading time for this sample is not 5s. As Saurav mentioned in his answer, CI is always about your sample, whereas hypothesis testing is about the sample and population. $\endgroup$ – Jekaterina Kokatjuhha Jun 17 '17 at 19:30
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I think the key here is the effect. A hypothesis test should simply be an up/down vote, but a confidence interval gives us an idea of how different (or not) our results are from what we expected under the null hypothesis.

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