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I am a novice to Confidence intervals. To figure out the confidence interval for mean, one could either use the $Z$ distribution or t distribution depending on the sample size and population standard deviation. When the size is less than $30$ and standard deviation is unknown, we go for t distribution. On the other hand, when the standard deviation is known, we go for $Z$ distribution.

Confidence intervals for mean is used to quantify the uncertainty by providing a lower limit and upper limit that represent a range of values that will represent the true population mean with a specified level of confidence.

Now, in the case of $Z$ distribution, how is the population standard deviation alone known prior to the estimation of population mean? Or in other words, what are the cases when population standard deviation is known before estimating the population mean?

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We interpret your question as asking under what conditions is it reasonable to use a model in which the population standard deviation is known,

Let us suppose that we are using a high precision scientific instrument to determine say the mass of an object. We will do this by making a series of $n$ measurements of the mass of the object.

The behaviour of the instrument may be well-known, since it has been used for a long time. It is known that the result $Y_i$ produced by the instrument on the $i$-th measurement is $\mu+X_i$, where $\mu$ is the actual mass, and the $X_i$ are independent normally distributed "error" random variables, say with mean $0$ (the instrument is well-calibrated). From long experience with the instrument, the standard deviation $\sigma$ of the $X_i$ may be known with high accuracy. Then the standard deviation of $Y_i$ is $\sigma$, and may be assumed known.

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  • $\begingroup$ @Nicolas I haven't understood your example quite well, I guess. The standard deviation for the error random variables are known from experience. What is also known is its mean, i.e. zero. My doubt is on cases where you use the standard deviation to estimate the CI for mean. $\endgroup$
    – Raji
    Jan 6 '15 at 13:24
  • $\begingroup$ In the example, the mean of the $X_i$ is known. However, $\mu$ is not known, it is the actual weight of the object. So the mean of the $Y_i$ is not known. But the variance of the $Y_i$ is instrument-dependent only, and can in this situation be taken as known. $\endgroup$ Jan 6 '15 at 15:38
  • $\begingroup$ Reasonable enough. Thanks a lot. $\endgroup$
    – Raji
    Jan 6 '15 at 18:03
  • $\begingroup$ You are welcome. There are a number of situations where we take a quite small sample (make a small number of measurements), and assume standard deviation is known. Another instance is age determination using radioactive decay. A small number of tiny fragments are taken from the sample, and their mean age is determined. The errors in this process are well enough understood that we can assume we know $\sigma$. $\endgroup$ Jan 7 '15 at 1:46

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