We want to derive a confidence interval for the unknown variance of a population under the following assumption:
- The expectation is unknown.
As we have seen above, an unbiased estimator of the unknown variance is given by: It has been shown (see distribution of the sample variance) that has a chi-square distribution with degrees of freedom.
We can now make probability statements of the following form: Here, is the - quantile and the - quantile of the chi-square distribution with degrees of freedom.
By algebraic manipulation we may isolate in the middle of the probability statement: The corresponding confidence interval is The interpretation is the same as before: a proportion of confidence intervals constructed in this fashion will contain the true parameter value .
- By construction, these confidence intervals assign equal probability mass to the tails:
- The confidence interval is not symmetric around the point estimate , since the chi-square distribution is not a symmetric distribution.
- The length of the confidence interval depends on the sampled values and is a random variable. The length of the interval also depends on the sample size and on the confidence level .
The following variables have been measured for a population of households: = monthly net income = monthly expenditure on rent = monthly expenditure on the automobile
The expectation and the variance of the variables are unknown. We assume the distributions to be normal.
Find point and interval estimates for the unknown variance .
One can examine the effect of the confidence level and sample size on the length of the confidence interval. We recommend that only one feature be altered at a time.
- the variable to be analyzed
- the sample size
- the confidence level (as a decimal e.g. 0.95)
Result:This interactive examples produces
- a boxplot
If you choose the same variable repeatedly, but enter different confidence levels/sample sizes, the previous results are displayed for comparison purposes.
Click on your mouse to start the interactive example.Launching can take several seconds.
Some time may elapse before the results are displayed.
For a population of households let denote net household income . We assume that is approximately normally distributed ; the two parameters and the variance are unknown.
Construction of confidence intervals for the unknown mean has been studied in chapter confidence intervals for the expectation.
Here, we want to focus on the unknown variance , for which we will construct a confidence interval with confidence level .
A random sample of size yields the following realizations (ordered by size):
||Household net income (DM)
||Household net income (DM)
Mean household income of the sample is Our point estimate for the unknown variance is given by Using chi-square tables we find Hence the confidence interval is given by