Estimating a Population Variance
We have seen how confidence intervals can be used to estimate the unknown value of a population mean or a proportion. We used the normal and student t distributions for developing these estimates. However, the variability of a population is also important. As we have learned, less variability is almost always better. We use the chisquare distribution (pronounce as kighsquare) to construct the confidence intervals (estimates) of variances or standard deviations. First we need to become acquainted with the χ^{2} distribution.
χ^{2} (chisquare) distribution
Suppose we take a random sample of size n from a normal population with mean µ and standard deviation σ. Then the sample statistic _{}follows a χ^{2} distribution with n1 degrees of freedom, where s^{2} represents the sample variance.
Properties of the χ^{2} (chisquare) distribution

The total area under χ^{2} curve equals 1.

The value of the χ^{2} random variable is never negative, so the χ^{2} curve starts at 0. However, it extends indefinitely to the right, with no upper bound.
(When a sample with variance s^{2} is close to the population variance σ^{2}, the value of χ^{2} will be close to the number of degrees of freedom n – 1, and n 1 is positive, so χ^{2}^{ }will be positive. This explains why the χ^{2} graph begins at 0)

Because of the characteristics just described, the χ^{2} curve is right skewed. In another word, the chisquare distribution is not symmetric.

There is a different curve for every different degrees of freedom, n1. As the number of degrees of freedom increases, the χ^{2} curve begins to look more symmetric.
Finding Critical Values for the χ^{2}
To construct the confidence intervals, we need to find the critical values of a chisquare distribution for the given confidence level 100 (1 – α)%. We can use either the chisquare table (table A4) or technology. Table A4 shows the degrees of freedom in the left column. The area to the right of the χ^{2} critical value is given across the top of the table.
Since chisquare distribution is not symmetric, we cannot construct the confidence interval for σ^{2} using the “point estimate _{} Margin of error” method. We must find two different chisquare critical values for each confidence interval for the given confidence level 100 (1 – α)%.

χ^{2}_{1 α/2} which represents the value of the chisquare distribution with area 1 α/2 to the right of it.

χ^{2}_{ α/2} which represents the value of chisquare distribution with area α/2 to the right of it.
The χ^{2} table is somewhat similar to the t table; both tables show the degrees of freedom in the left column. The area to the right of the χ^{2} critical value is given across the top of the table.
Ex(1)

Find χ^{2} critical values for a 90% confidence interval, where we have a sample size of n = 10.

With a sample size of n = 10 and a confidence level of 95%, find the critical value of chisquare separating an area of 0.025 in the left tail, and find the critical value of chisquare separating an area of 0.025 in the right tail.
Note: if the appropriate degrees of freedom are not given in the χ^{2} table, the conservative solution is to take the next row with the smaller df.
Constructing Confidence Intervals for the Population Variance and Standard Deviation
Suppose we take a sample of size n from a normal population with mean µ and standard deviation σ. Then a 100 (1 – α)% confidence interval for the population variance σ^{2} is given by
Lower bound = _{} , upper bound = _{}
Degrees of freedom is n – 1.
Confidence interval for the population standard deviation σ
A 100 (1 – α)% confidence interval for the population standard deviation σ is given by
Lower bound =_{}, upper bound = _{}
Ex 2)
The following table shows the city gas mileage for 6 hybrid cars.
Vehicle Mileage(mpg)
Honda Accord 30
Ford Escape 36
Toyota Highlander 33
Saturn VUE Green Line 27
Lexus RX 400h 31
Lexus GS 450h 25

Construct and interpret a 95% confidence interval for the population variance of hybrid gas mileage.

Construct and interpret a 95% confidence interval for the population standard deviation of hybrid gas mileage.
