# Calculation of confidence interval of a population parameter

Consider that $P$ is the water pressure coming out from a valve A. Let $P_{dif}$ be defined as the difference between the maximum and the minimum pressure of valve A: $$P_{dif}:= P_{max} - P_{min}$$

Now, what I want to do is to estimate $P_{dif}$. In order to do that, I take a number of water pressure samples from valve A. Let $S$ be a set of 3 measured samples: $$S = \{X_1 = 5, X_2 = 7, X_3 = 1\}$$ That is, $S$ contains 3 random samples drawn from the population. Therefore, I then say that $\hat{P}_{dif} = 7 – 1 = 6$.

Fisrt question: Considering a Gaussian distribution of the population parameter, how can I find the pdf, cdf of the parameter?

Second question: Would a $95\%$ confidence interval (CI) of my estimation be defined as following?:

$$(pdf, mean) - 2\cdot (pdf, std) \leq (pdf, mean) \leq (pdf, mean) + 2\cdot (pdf, std)$$

where (pdf, mean) is the mean of the pdf and (pdf, std) its standard deviation.

If yes, how can I derive the (pdf, mean) and the (pdf, std)?

Edit:

As it seems quite difficult for me to find the CI of the range estimation, I am wondering whether I could use an alternative approach. whuber suggested that successive estimate should not depend strongly on the preceding ones. Say that I split my data samples into equal parts (e.g. 3) and I find one estimate for each of these sub-sets. Next, I find the 95% CI of the $\hat{P}_{dif}$ by that:

$$mean - 2\cdot std \leq mean \leq mean + 2\cdot std$$

where mean and std are the mean and std of the 3 different estimates, respectively. Would that be correct?

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## migrated from math.stackexchange.comFeb 25 '13 at 18:47

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Now that you got your Tumbleweed badge here, do you want your question migrated to stats.SE? If so, flag your post for moderator's attention. – 5pm Feb 25 '13 at 14:41
It's difficult to give specific advice without having more information about your "estimations." Are they based on independent sets of data or are they perhaps different ways to estimate the same parameter from the same data? And what specifically is the difference between "finding the ecdf ... for a normal distribution" and then computing its mean and SD, when usually one fits values to a Normal distribution by computing the mean and SD in the first place? – whuber Feb 25 '13 at 18:54
I am basically deriving as many estimations as possible from the same data set by considering each time a progressively bigger part of my data set. For example, consider that I have a data set of 10 numbers and each estimation requires at least 2 numbers. Then, estimation 1 is derived by the first 2 numbers, estimation 2 by the first 4 numbers, etc. I am finding the cdf and ecdf because I want to find the probability that a particular estimation will exceed. Therefore, the CI of the ecdf will have to give me an interval where the true value of the statistic lies. – limp Mar 1 '13 at 17:35
In that case, your procedure is invalid, because each successive estimate depends strongly on the preceding ones. Why don't you use that last estimate and its confidence limits based on all the data? – whuber Mar 1 '13 at 17:39
Ok, that sounds reasonable. Let’s say that my population parameter is defined as the $max – min$ of an infinite number of samples. Consequently, an estimation will require at least 2 numbers. How can I derive the pdf and cdf in this case if I only consider the last estimate? Because the population parameter is derived by an infinite range of samples, I thought it would make sense if each estimation considers a bit more of the (finite) range of samples. – limp Mar 1 '13 at 18:01