# Is Confidence Interval taken on one Random Sample or A Sampling Distribution

I am stuck at CI. Intuitively, a confidence interval (CI) is a type of interval estimate, computed from the statistics of the observed data ( random samples), that might contain the true value (mean) of an unknown population parameter. (Wikipedia)

CLT states that plotting means of sufficiently large no samples of sample size n will follow a normal distribution and the mean of sampling distribution will be approximately the same as the mean of a population. The standard deviation of sampling mean distribution (also called Standard Error) will be sigma/sqrt(n).

Assume Standard deviation of Population is known. Now, if we have sufficiently large no of samples, why do we need Confidence Intervals Is confidence interval used to tell the interval of the true mean from A SINGLE SAMPLE OF MEAN X_bar and Standard Error (sigma/sqrt(n)) or Is CI used to tell the Interval of the true population mean based on the X_bar of Sampling Distribution and the S.E (sigma/sqrt(n)).

Is the purpose of CI to tell us in a quantifiable manner how Sample Size (assuming large no of samples) impacts the finding of the true mean from sampling distribution?

Any help is highly appreciated.

TL:DR - While Calculating Confidence Interval (Population Std.Dev is known), whether SAMPLING DISTRIBUTION is used or a RANDOM SAMPLE is used?

## 1 Answer

The confidence interval is itself random, as it is based on a single sample mean. The idea is that with high probability, this interval will contain the true population mean. In this sense, it is an estimate of the true mean. The larger the sample, the narrower this interval is.

Another use of the confidence interval idea is in statistical control. Say you want to manufacture 500gm bags of flour. Your target mean is now 500gm and at regular intervals you sample n bags. You are now typically interested in this sample mean falling in the fixed interval $$(500-3\sigma /\sqrt{n}, 500+3\sigma /\sqrt{n})$$. In this interval your bags have acceptable variation, but a sample mean outside this interval indicates a problem with your manufacturing process, outside of natural variation.

• Thanks. But we are using z score to calculate the CI and I firmly believe that a Random Sample will be not normal distributed. Why do we treat it as normally distributed. – Ramneek Singh Kakkar Feb 25 at 13:19
• As you said, if the sample is large enough the distribution of the mean will be approximately normal via the CLT. In a manufacturing context (mass, length) the distribution will often be normal anyway. You can also create confidence intervals for non-normal distributions too - see any decent stats book like "Introduction to Probability Theory and Statistical Inference" by Larson for example. – Paul Feb 25 at 14:20