|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 chi-square distribution (pronounce as kigh-square) to construct the confidence intervals (estimates) of variances or standard deviations. First we need to become acquainted with the χ2 distribution.
χ2 (chi-square) 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 χ2 distribution with n-1 degrees of freedom, where s2 represents the sample variance.
Properties of the χ2 (chi-square) distribution
The total area under χ2 curve equals 1.
The value of the χ2 random variable is never negative, so the χ2 curve starts at 0. However, it extends indefinitely to the right, with no upper bound.
(When a sample with variance s2 is close to the population variance σ2, the value of χ2 will be close to the number of degrees of freedom n – 1, and n -1 is positive, so χ2 will be positive. This explains why the χ2 graph begins at 0)
Because of the characteristics just described, the χ2 curve is right skewed. In another word, the chi-square distribution is not symmetric.
There is a different curve for every different degrees of freedom, n-1. As the number of degrees of freedom increases, the χ2 curve begins to look more symmetric.
Finding Critical Values for the χ2
To construct the confidence intervals, we need to find the critical values of a chi-square distribution for the given confidence level 100 (1 – α)%. We can use either the chi-square table (table A-4) or technology. Table A-4 shows the degrees of freedom in the left column. The area to the right of the χ2 critical value is given across the top of the table.
Since chi-square distribution is not symmetric, we cannot construct the confidence interval for σ2 using the “point estimate Margin of error” method. We must find two different chi-square critical values for each confidence interval for the given confidence level 100 (1 – α)%.
χ21- α/2 which represents the value of the chi-square distribution with area 1- α/2 to the right of it.
χ2 α/2 which represents the value of chi-square distribution with area α/2 to the right of it.
The χ2 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 χ2 critical value is given across the top of the table.
Find χ2 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 chi-square separating an area of 0.025 in the left tail, and find the critical value of chi-square separating an area of 0.025 in the right tail.
Note: if the appropriate degrees of freedom are not given in the χ2 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 σ2 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 =
The following table shows the city gas mileage for 6 hybrid cars.
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.