When can we know that a natural population follows a Normal distribution. In fact, never. Moreover, how can we assure that a sample of the population is a simple random sampling with replecement of the population. I think that never, except with artificial Normal population simulated in a computer. How is it possible in the natural world to select a simple random sample from an infinite population. I think that never. My conclusion is this: Classical inference lacks of objetivity for practical natural purposes.

The certain interval you are looking for is [-Inf, +Inf]. Any smaller interval is not certain.

If you sample the entire population (possible only if the population is finite), you get to know the true mean (that's no estimate anymore; it is the actual mean). This is a particular certain value, not an interval.

@ Mariano Ruiz Espejo , I have been convinced that there is no way to know that a natural population follows a normal distribution. Also, there may not exist any population which perfectly follows a normal distribution. Moreover, it is not possible in reality to assure that a sample drawn from some population is a random sample. The conclusion

" classical inference lacks of objectivity for practical natural purposes "

is true as I understand.

However, my thrust is to know whether theoretically it is possible to obtain certain interval value of the mean of a population based on random (theoretically) sample (s) drawn from it.

@

Jochen Wilhelm

It is true that If we sample the entire population then we can get the true/actual value of the mean which is a point certain value, but not an interval value.

However, my thrust is to determine certain interval value based on the data obtained from random sample or random samples drawn from the population but not based on the data obtained from the entire population.

I agree with Dr. Jochen Wilhelm.

@ Salvatore S. Mangiafico ,

My question is clear. I want to know how can we obtain the certain interval value (not the confidence interval which is probabilistic) of the mean of a population if the population follows normal probability distribution based on random sample or random samples drawn from it. I need the method of obtaining the certain interval

value of the population mean.

@ Sahar Qazi , I too agree with Dr. Jochen Wilhelm.

But, my thrust is to know whether theoretically it is possible to obtain certain interval value of the mean of a population based on random (theoretically) sample (s) drawn from it.

The answer given by Dr. Jochen Wilhelm is silent on my core question that how can we obtain this certain interval value of the mean of a population following normal distribution.

Dhritikesh Chakrabarty,

my anwer is not silent to your core question "is it possible to obtain certain interval value of the mean of a population based on random (theoretically) sample (s) drawn from it.". My answer is:

"The certain interval you are looking for is [-Inf, +Inf]."

So, yes, you can construct an interval the will with certainty contain the population mean from a sample mean. But sadly, such an intervall will neccesarily be infinite. There is no finite interval that will surely contain the population mean. This is what I meant when writing "Any smaller interval is not certain."

This refers to the normal probability model, as you stated in your question. If, in contrast, you have a model that is bounded, the certain interval will just extend to the boundaries of the probability model. For instance. In this case, finite intervals that surely contain the population mean are possible (e.g. the beta-distribution: the interval surely containing the mean will be [0,1]; or you have some truncated model in a Bayesian analysis giving finite 100% credible intervals).

@ Salvatore S. Mangiafico ,

" [-Inf, +Inf] " is no doubt is a certain interval. However, it is trivial. In any situation, it is certain interval.

But, by the term "certain interval" we usually mean an interval with finite upper limit and finite lower limit such that it is certain that the value of the population mean will lie within the interval. My thrust is to obtain such an interval.

You will get such an interval only when the domain of the probability model is finite. The domain of the normal distribution is infinite. So for the normal distribution such an interval (with finite limits and 100% confidence) does not exist.

@ Dhritikesh Chakrabarty,

All you need is to take measurements of all the population, which is supposed to be finite. Then, in this case, you don't need to talk about certain intervals, because any interval containing the true mean is a certain interval.

Mohamed I. Riffi

Yes :) Hence, µ ± d is a certain interval, for any d ≠ 0.

If you know µ, you can easily construct an infinite amount of finite certain intervals...

Since I was mentioned in a couple of comments, let me try to sum up this thread, as far as I understand.

We are defining the “certain interval of the mean” as the interval in which there is 100% probability of the true mean of the population being located within this interval.

For the conditions of this question --- a normally-distributed population, and a finite sample ---, this interval ranges from negative to positive infinity ‡.

Cases in which this interval would be smaller are:

1) If the population distribution is bounded. For example, for a beta distribution bound by the 0 and 1, the certain interval of the mean ranges from 0 to 1 ‡.

2) For any distribution, if the population is finite, and the entire population is sampled, then the true population mean is known, and the interval contains only this value. (This, of course, ignores any imprecision in measurement, and so on.)

3) In theory, 2) would hold for an unbounded distribution and an infinite population if the sample were infinitely large.

4) For a bound and finite (but large) population, this interval could be lessened if we sample a reasonable portion of the population. I'm thinking for example, if we knew that the height of an adult american must be between 1 m and 3 m, and we sample 100,000 out of say 200,000,000 adult americans, then the certain interval would be narrowed by the force of simple arithmetic.

------------

‡ Note: I am not specifying whether the endpoints are included in these intervals or not.

The normal distribution of population is governed only by 2 parameters µ, mean and σ, standard deviation. One can be sure that µ±4σ will cover 100%, but generally people like shortest length of the interval and 2σ (or 1.96σ) is preferred. Another way is resampling methods, like bootstrap or jackknife, to get shorter CI with 100% level of confidence. It is used even for non-normal distributions too. Newer techniques/ methods like Bayesian or machine learning, to improve estimates.

@ Prasanthi: No, µ ± 4σ will not cover 100 %. Some disciplines use much wider intervals, and none of them claim a full 100 %.

µ ± 4σ covers approximatly 99,994 %. That's a lot, but that's still not 100 %.

As already said by several, with normal distribution, the one and only 100 % interval is ]-infinity, +infinity[.

@ Mohamed I. Riffi ,

How can we measure all the units of a population if the population is infinite ?

In this situation, how can we obtain the corresponding certain interval ?

@ Jochen Wilhelm ,

If certain interval can be obtained when the domain of the probability model is finite then is it possible to obtain certain interval of the population mean of a binomial distribution ? If so, how ?

@ Salvatore S. Mangiafico,

I have understood the explanation.

However, my thrust is to find out a certain interval value (i.e. interval value with 100% confidence)of the population mean of a normal distribution based on one sample or on more samples if necessary.

@ Prasanthi Sanjeevi ,

µ ± 4σ is not a certain interval of the parameter µ.

It is 99.994 % confidence interval of the parameter µ.

@ Emmanuel Curis ,

You are right.

I agree with you.

However, the 100 % interval namely ]-infinity, +infinity[ is a trivial one .

My thrust is to obtain a non-trivial certain interval.

@ Dhritikesh: there is no such interval.

For the binomial case you're asking: if the first parameter is n (number of trials), the certain interval is [0 ; n]. You can exclude at least one of the limits if you observe at least one success/one failure.

In general, for any distribution in support [a, b], the certain interval is [a, b] and I'm quite sure you can't do any narrower interval without acceptin a slight risk of error.

" is it possible to obtain certain interval of the population mean of a binomial distribution ? If so, how ? "

The domain of the binomial distribution is [0, N], with N = number of trials. The parameter of the distribution is p (probability of a "success" in a trial; note that 0 ≤ p ≤ 1). The population mean is the expectation: µ = sumk=0...N Pr(k), which is, as you can derive yourself or look up in google: µ = Np. Since 0 ≤ p ≤ 1, it follows that 0 ≤ µ ≤ N. Thus, the population mean must be in this interval, and so [0,N] is such a certain interval: it surely includes µ. If you know N (what should be clear from the design of the experiment) than you know a finite interval that does surely include µ. Thats so trivial that it is hard to explain.

@ Emmanuel Curis , For the binomial case , if the first parameter is n (number of trials) then no doubt the interval is [0 ; n] is a certain interval. However, this is the trivial certain interval. To obtain this certain interval, there is no necessity of applying estimation theory. In general, for any distribution in support [a, b], the interval is [a, b] is a certain interval which is again a trivial one.

However, my thrust is to obtain non-trivial certain interval.

@ Jochen Wilhelm ,

I have asked the same question in the case of a normal distribution namely

" is it possible to obtain certain interval of the population mean of a normal distribution ? If so, how ? "

In pursuance with the answer of my this question, given by Jochen Wilhelm , I mentioned the case of binomial distribution.

Dhritikesh Chakrabarty : let's be really clear — there is no non-trivial certain interval. Such intervals do not exist. Trivial intervals are the only certain intervals. And yes, they are useless.

@ Emmanuel Curis ,

I agree with you.

However, I like to add that I have not yet known whether it has been possible to determine non-trivial certain interval of the population mean of a distribution in general and of a normal distribution in particular. But, I guess that it may be possible to determine non-trivial certain interval from random sample (of course not from a single random sample but from sufficient number of random samples) drawn from the population. Since it is still beyond my capability, that is why I threw this question to the researchers.

Not, it is not possible, since no such interval exists. Only trivial intervals are certain. All other, narrower, intervals always have a slight probability of not containing the true value.

I except the very special case of the Bernoulli, were the trivial certain interval for the probability is [0,1] before any experiment, but after the first experiment you can remove either 0 or 1, and when repeating the experiment, if you observe at least one failure and one success, you can use the ]0;1[ interval instead of [0,1]. Which quite often does not add very much information... And even that may be seen as an overinterpretation, since events of null probability are not impossible, strictly speaking.

@ Emmanuel Curis ,

"It is not possible" means it has not yet been possible.

Therefore, it can be a challenging problem for the researcher to find out the non-trivial certain interval.

@ Dhritikesh : « It is not possible » means it is mathematically impossible, and is easy to demonstrate in the continuous case: if your variable has a law with support [a, b], assumed to be the minimal support (that is, there is no subset of [a, b] that has a null density of probability, then whatever the interval I included in [a, b], 0 < p( X in I ) < 1, the inequality is strict. Whatever the parameter of the law is. As a consequence, *all* possible values of the parameter can lead to the observation of any subset of [a, b], that is to any experimental results. So you are never certain that you can reject of value of the parameter.

I let you adapt for the discrete or more general cases, but the logic is the same.

The only certainty you have is if you observe a value that is outside [a, b]: then you can rule out your law, whatever it's parameter is.

An exception to that is the special case where the support depends on the parameter of interest, like in an uniform law. That's not the case for the normal distribution and most used experimental distributions. And even in that case, my guess is that adding observations can only extend the support, hence the possible parameter space, not reduce it. So there is neither « certain » interval in that case except R (or R+, eventually).

Here are the formulas for confidence intervals:

For one mean:

https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/PASS/Confidence_Intervals_for_One_Mean.pdf

Scroll down to the 'Confidence Interval' subsection for different situations (ex. mean paired difference, etc.)

https://www.ncss.com/software/pass/pass-documentation/

@ Matthew Robinson ,

The formula for certain interval is not found by me in the links (1) For one mean: https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/PASS/Confidence_Intervals_for_One_Mean.pdf & (2) Scroll down to the 'Confidence Interval' subsection for different situations (ex. mean paired difference, etc.) https://www.ncss.com/software/pass/pass-documentation/

as suggested.

@ Emmanuel Curis ,

"It is therefore a challenging problem for the researcher to find out the non-trivial certain interval.

It follows from logic that there is no such interval. If you don't want to re-define the entire set theory, probability and calculus, then this is a proven fact. So it is rather silly "for the researcher to (try to) find out the non-trivial certain interval".

@ Dhritikesh : until you point out a mistake in the mathematical demonstration that such an interval does not exist, trying to find one is not a challenge, it's a waste of time.

@ Emmanuel Curis ,

Every research, until its fruit is not obtained, seems to be wastage of time.

I have now got an idea that it can be possible to obtain certain interval value of the population mean of a normal distribution not from a single random sample but from a number of independent random samples. However, it is still a hypothesis only.

@ Dhritikesh: really curious to see the result and its demonstration...

@ Jochen Wilhelm , I have just got an idea that empirically It may be possible to establish that certain interval value of the population mean of a normal distribution obtained. However, it is still a hypothesis only.

@ Emmanuel Curis ,

I have been trying. I will certainly inform in right time.

@ Emmanuel Curis ,

I do suggest to go through my papers published in the journal " IJEAR ".

Go to the link

eses.net.in

and open the published volumes of the journal.

Complete references or direct link to the PDF would help. Don't have time to go through all journal issues to find your papers...

Thanks in advance,

Just read the paper in vol.1 issue 2. Since it begins with a false assumption... the result is wrong.

You cannot even be sure that your true parameter µ is between the first and the last observation of your dataset. After all, if X is a Gaussian with expectation µ, p(X > µ) = 1/2 and for a iid sample, p( all X > µ ) = (1/2)^n where n is the sample size, and is never 0. In other words, your equality (2.10) is wrong (even if you say it is trivial), and all your subsequent reasonning also.

Dhritikesh Chakrabarty your stamina is impressive. On the technical side, your logic is mathematically wrong the way you presented it. In vol. 1 issue 2 you re-invented the very old idea of a jackknife re-sampling but then confused the estimate of the mean (mu_bar) with the true mean (mu) which are not the same! Hence why Emmanuel Curis is correct. https://en.wikipedia.org/wiki/Jackknife_resampling

Dear Emmanuel Curis and Mathew Robinson ,

The objection raised in that paper will be studied/analyzed and attempt will be made on necessary modification/rectification.

Lot of thanks to you for your cooperation.