 Now, the right hand side 1 minus alpha this is coming because of the choice you have made z alpha by 2 and this twice has nothing to do with what is the theta naught you are choosing. So, now is this expression true for any theta naught or is it true for only some specific theta naught? Yeah I mean theta naught maybe like let us say theta naught is some your parameter space some parameter space. Does it depend on any particular theta naught you are choosing or this holds for any theta that is coming from that space theta? Only. Only? Yeah I am saying that if instead of let us say this theta naught like this theta naught I have given you as 2 and this expression holds. And tomorrow I change this to theta equals to 3 does this relation holds or not? No. No in that case even the samples are coming with the new theta naught and now I am looking this probability under that true parameter only in that case it holds right it does not matter which theta naught we are looking as long as the underlying samples are generated with respect to that parameter and we are also computing this probability with respect to that parameter this relation holds. So this is true not necessarily for any particular theta naught so I will just write this this is my probability under any theta such that z alpha by 2 and sigma square by. Note that when I say theta naught I have to this theta naught and this theta are the same. So now if I change this theta now this theta also has to change ok. Now what is good thing about this now I have defined a set let us say for a given theta probability that theta that theta minus z alpha by 2 sigma square by n this guy is less than equals to theta plus z alpha by 2 sigma square by n this is equals to 1 minus alpha ok. Now can I say that if I just invert it in terms of my parameter I am just going to invert this. Now this is like z alpha by 2 sigma square by n I want to write now this in terms of theta. So ok let us now take one upper bound I am going to get through this. So theta is upper bounded by x bar plus z alpha by 2 sigma square by n and the lower bound for theta I am going to get from this. This is from these two I am going to get I can write it as plus plus x bar. Now see what is happening now you have an interval on the theta parameter that is now you are saying that theta belonging to in this range z alpha by 2 sigma square by n plus x bar this is one and the other end is z alpha by 2 sigma square by n plus x bar and what we are saying about this probability that theta belonging to this interval what this probability is exactly. So this I am now going to call this this is for a given x now I can treat it as for a given x this thing I can treat it as my confidence set on my parameter and now what is the probability that theta belongs to c of x c of x is this interval this is 1 minus f. Now can you tell me the test we have what is the test we have the test we have is in LRT in which we have set c to be equals to that lambda x less than or equals to c right in that c what is the value has set we have set exponential minus z square alpha by 2 2. If we set it like this I know that this is going to happen and this 1 minus alpha here does not depend what is that theta you are looking into or what is the c we are looking into. So now can you tell me we have a confidence interval right now so here you can take this to be your L of x and this to be your u of x and now you have guaranteed that probability that theta belongs to L of x and u of x is equals to 1 minus alpha for this confidence interval sorry for this let us say you have an interval estimator here what is the what is the term we used confidence coefficient what is the confidence coefficient of this test or like what is the confidence coefficient of this interval estimator can you recall the definition of a confidence coefficient. So it is simply ok we let it be like this here this is 1 minus alpha. So what is the definition of the confidence coefficient it is simply we say infimum over theta belongs to my theta space of this probability probability that theta belongs to L of x comma u of x. Now what is this infimum value this is going to be 1 minus alpha because your 1 minus alpha does not depend on what is your theta now did I come up with a confidence interval which has a confidence coefficient alpha right when this happens we are going to call this at 1 minus alpha confidence interval. So did you study this confidence interval before in any other course ok. So I said last time we said we defined confidence estimators and we associated a confidence coefficient with that and we said that if I want confidence coefficient to be let us say 1 minus alpha then we and we have a confidence interval which gives me that confidence coefficient then we are going to call it as 1 minus alpha confidence interval ok. Now obvious one obvious thing you can note from these two things I have defined a of theta and c of x here ok. So what is a of theta? a of theta is set up all those points which I will accept to be coming from parameter theta right. And now what is c of x here ok we will come to that but we have defined c of x to be this interval. Now suppose let us say a of theta is the set of all samples whereas c of x is the set of parameters right. So this is now set of points and this is now set of parameters. Suppose some x belongs to c of theta and I have constructed c of x. If x belongs to a of theta can I say that then this theta belongs to c of x yes because the way we are constructed automatically guarantees that this is one is the inversion of honor. So it is not only one implication this is also another implication. If x belongs to a of this this is true if and only if theta belongs to c of x ok. So now because of this inversion you will be able to construct nice confidence sets from your test ok. Now quickly understand pictorially how does this work ok. Suppose I want to do a now I am basically mapping my looking relation between my parameter space and sample space. Now notice that in the test we have covered always my samples the random sample is always coming as sample mean right. Everywhere wherever I dealt mostly I dealt with x bar that is a sample mean right. I do not need to look into the sample itself I have to just worry about the sample mean. So let us say because of that that is easier the vector has become one scalar for me right that entire random vector is now represented with sample mean. So let us represent that on the y axis ok. Maybe I need to get a and on the x axis is my parameter space for time being assumed that this theta is one dimensional different possible values ok. Now let us plot this what is the relation we get we get x bar is less than or equals to theta for a given theta let us say z alpha by 2 sigma square n and theta minus z alpha by 2 sigma square n. So x bar is going to lie in this interval. Let us start with let us put theta equals to 0 theta equals to 0 what is the value of x bar. So x bar is going to be either this value or the negative of this value. So let us say something this is the positive value and this is the associated negative value this is like a z alpha by 2 square root of sigma square by n and this is minus z alpha by 2 sigma square by n. Now as I increase theta and let us look into this this is like one linear function in theta with offset this much. So let us do this and similarly the other one sorry I do not know they are looking parallel or not but they are supposed to be parallel ok. Now let us see I have let us say let us take a particular parameter theta naught and I want to construct the set A theta naught from this can you tell me what is going to be A theta naught it will be all the points let us say if I take this thing. So this value of x bar is going to be A of theta naught for me ok. And now let us see I have given a sample and that sample is this something let us say let us take some x bar 0 some sample is given for which the mean value is x bar 0. Now how to construct a confidence set for that? So you do the same thing you do go like this here and then you do like this and then you will get some value here and then some value here and this is going to be what? C of x bar x bar 0 C of this entire thing is going to be C of x bar 0 ok. So with this can you see that it is possible to get from your test to construct a confidence interval in this fashion and what is happening is basically the test sorry the test and the confidence interval they are actually looking at the similar things but from a slightly different perspective what does this mean? What is the set A theta is looking into? A theta is looking into all possible so A theta is the acceptance right this is looking into all possible samples that could generated under parameter theta naught or possible that could be likely generated under the parameter theta naught. Now what is C of x bar 0 is looking into? This is looking into set of all potential parameters that could under those parameters this sample could have been generated ok all possible parameters that could have likely generated the sample x bar under the parameter theta naught. So obviously this correspondence leads to a theorem which I am going to write now and with that we will conclude this. Let us say for each theta naught belonging to your your theta space let A of theta naught be acceptance region of a level alpha test corresponding to corresponding to H naught which is going to test the hypothesis that theta is equals to theta naught. Now if you are going to now define C of x in terms of this A theta you are now going to define C of x to be set of all theta or theta naught such that x belongs to A theta naught. Then C of x is a what can I can claim about C of x? I know that A of theta is a level alpha test and using that I have been able to construct a confidence interval sorry confidence set or confidence interval what I can say further about that confidence interval. I can say that this is going to be 1 minus alpha confidence interval ok what we just said is if somebody gives you acceptance set of a level alpha test. So A theta is A theta of A theta is given to you and you have been told that it has a level alpha. Then using that A alpha you can construct a parameter space which is now going to be a confidence set with confidence coefficient of 1 minus alpha. So that means I readily have 1 minus alpha confidence interval. Now on the other hand now you have given a acceptance region and from this you are able to construct a confidence interval. Now if somebody has given you a confidence interval or 1 minus alpha confidence interval you should be able to construct an acceptance region with what level with alpha level. So let us let formally write that ok. Now conversely let C of x so I should be writing this like capital X the R 1 minus alpha confidence interval define A of theta naught equals to set of all theta such that x belongs to sorry I have been given so I this should be a set of x right. So let x such that your theta belongs to C of x ok. Then A of theta is a acceptance region of a level alpha test for H naught which is going to test whether theta is equals to theta ok. So now this is if you now construct A of theta naught which is a set given by this then this A of theta naught is acceptance region of a level alpha test for this test which is where the null hypothesis is whether to test whether the parameter is theta equals to theta naught ok. So with this we will stop I think proof is straightforward I will leave you to look into the proof ok.