Confidence Interval for Estimating Population Mean

The point estimate of the population mean is often unreliable when obtained from a single sample.

So, we draw a confidence interval using some known variables obtained from that sample.

When the population standard deviation is known, the critical value at a specific confidence level is obtained from the z distribution.

However, when the population standard deviation is unknown, the critical value at a desired confidence level from the t distribution is used. 

As the population and sample can be assumed to be normally distributed, the confidence limits can be expressed as follows.

At a 95% confidence level, the calculated confidence intervals in both cases would cover the population mean.

In other words, if many confidence intervals are obtained through the identical sampling method, 95% of them will contain the true population mean.

Although the confidence intervals calculated using critical t values are wider than those calculated using critical z values, they are equally accurate in predicting the population mean.

In simpler terms, not knowing the population standard deviation does not make the estimates of mean inaccurate.