2uData ⮞Hypothesis Testing ⮞Comparing two variances

Comparing two variances

Consider two populations whose variances of random variable `X` are `sigma_1^2` and `sigma_2^2`. Drawing two sample with size `n_1` and `n_2`, whose variances are `s_1^2` and `s_2^2`.

In general, we would like to compare the variances of these populations with confidence level (`1-alpha`) or significance level `alpha`.

So the null hypothesis is   Ho : `sigma_1^2//sigma_2^2=1`(21)

In alternative hypothesis, depend on context, the ratio of variances can be differ from 1, or less than 1, or greater than 1.

Test statistic for comparing variance is:

`F=s_1^2/s_2^2`(22)

This quantity conforms to Fisher distribution with degree of freedom `nu_1=n_1-1` and `nu_2=n_2-1`

Example

The data show that the standard deviation of productivity of machine A in 30 days is 40 product/day. Meanwhile, this quantity of machine B in 25 days is 32 product/day. Compare the stability of productivity of two machines with confidence level of 95%.

To compare the stability of machines, we compare the their variances of productivity as follows

• Construct pair of hypotheses :
  • Ho : `sigma_A^2//sigma_B^2=1`
  • Ho : `sigma_A^2//sigma_B^2!=1`
• This test statistic is `F` (formula 6.22) conforms to Fisher distribution.
• This is a two-sided test with `nu_1=30-1=29` ; `nu_2=25-1=24` ; `alpha=0,05`.
• Therefore :   `F_2`*`=F_(0,025,29,24)=2,21`
  and :   `F_1"*"=1/F_(0,975,29,24)=1/(2,21)=0,45`
  
• From the data :   `F_o=1600/1024=1,5625`
• Because `F_1`*`< F_o< F_2`*, we cannot reject Ho.
• The stability of productivity of machines A and B are the same.