Standard deviation used in confidence interval for meanI 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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Leon Webster · Accepted Answer

Step 1We 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.Step 2The 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   μ  +      X    i  , where   μ is the actual mass, and the       X    i   are independent normally distributed &quot;error&quot; random variables, say with mean 0 (the instrument is well-calibrated). From long experience with the instrument, the standard deviation   σ of the       X    i   may be known with high accuracy. Then the standard deviation of       Y    i   is   σ, and may be assumed known.

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