In practice, though a population may have a ‘true’ value for the variance, this is never know and the variance is always estimated from a sample using the formulaWe can use this to find a confidence interval for the unknown varianceof whichis an estimate.
We can do this using the fact thatthedistribution with degrees of freedom.
Denoting byandthe upper and lowerpoints of thedistribution withdegrees of freedom we have thatwith a certainty of
We can separate this into two inequalities:
We can combine these two into a single inequalitywith a certainty ofThe confidence interval is
Example: The standard deviation of a sample of 15 tomato plants is 5.8 cm. Find a 95% confidence interval for the variance of the tomato plant population.
The upper and lower 2.5% points of the %chi^2 distribution with (15-1)=14 degrees of freedom are 5.63 and 26.12 respectively. The confidence interval is