7.11:
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.
7.11:
Confidence Interval for Estimating Population Mean
A point estimate of the population mean is obtained from a single sample. Such a point estimate does not represent a population well because it needs to account for variability in the population. Single point estimate can also be biased despite the sample being selected randomly. Thus, a point estimate is often unreliable. A confidence interval is needed to reduce this unreliability.
A confidence interval for the mean is a range of values that provides an estimate of the population mean. As the population mean is an unknown but fixed value, it cannot be known from any number of samples or any method of sampling accurately, but it can be estimated. A confidence interval of a mean provides a range of values within which a true value of the population mean may be found.
The confidence interval calculation requires a margin of error and prior knowledge of population standard deviation (or variance). When the population standard deviation (variance) is known, the margin of error is calculated using z distribution as the normality of the samples is assumed. In this case, the sample size must be more than 30. In case when the population standard deviation (variance) is unknown, the margin of error is calculated using the t distribution. Although the t distribution is a non-normal, symmetric distribution, the estimation requires samples to be drawn from the normally distributed population, or the sample size should be greater than 30. The confidence interval calculated using the t distribution depends on the degrees of freedom (or on the sample size). They are wider than those computed using the z distribution for the given confidence level and sample size.
The confidence interval in both cases (i.e., population SD known or unknown) is estimated at a predecided confidence level, i.e., 90%. 95% or 99%.
When the confidence interval at 95% level is calculated, we are 95% confident that the true value of the population mean will fall between the lower and upper value of confidence limits. In other words, it may also mean that if we take several samples and calculate several confidence intervals, 95% of them will contain the population mean. As the population mean is a single fixed value, it is not appropriate to say that there is a 95% chance of finding the true value of the population parameter within the confidence interval. It is also incorrect to say that 95% of the sample means fall within the calculated range of confidence limits.