1. Bartlett’s Test

2. Test for the equality of several variances 𝜎 2 1 = 𝜎 2 2 =⋯= 𝜎 2 𝑘
Step 1:
𝐻 0 : 𝜎 2 1 = 𝜎 2 2 =⋯= 𝜎 2 𝑘
𝐻 𝑎 :𝑎𝑡 𝑙𝑒𝑎𝑡 𝑜𝑛𝑒 𝑝𝑎𝑖𝑟 𝑜𝑓 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝑠 𝑎𝑟𝑒 𝑢𝑛𝑒𝑞𝑢𝑎𝑙
Step 2: Level of Significance
Step 3: Critical Region
𝑁= 𝑖=1 𝑘 𝑛 𝑖
𝑏< 𝑏 𝑘 𝛼; 𝑛 1 , 𝑛 2 ,⋯, 𝑛 𝑘 ≈ 𝑖=1 𝑘 𝑛 𝑖 𝑏 𝑘 𝛼; 𝑛 𝑖 𝑁

3. Step 4: Test Statistic 𝑏= 𝑠 1 2 𝑛 1 −1 𝑠 2 2 𝑛 2 −1 ⋯ 𝑠 𝑘 2 𝑛 𝑘 −1 1 𝑁−𝑘 𝑆 𝑝 2 where 𝑠 𝑝 2 = 𝑖=1 𝑘 𝑛 𝑖 −1 𝑠 𝑖 2 𝑁−𝑘

4. Example`
It is suspected that higher-priced automobiles are assembled with greater care than lower-priced automobiles. To investigate whether there is any basis for this feeling, a large luxury model A, a medium-size sedan B, and a subcompact hatchback C were compared for defects when they arrived at the dealer’s showroom. All cars were manufactured by the same company. The number of defects for several of the three models are recorded.

5. Test the hypothesis at 0.05 level of significance that the average number of defects is the same for the three models.
MODEL A B C 4 5 8 7 1 6 3 9
TOTAL 23 21 36 80

6. 𝐻 𝑎 :𝑎𝑡 𝑙𝑒𝑎𝑡 𝑜𝑛𝑒 𝑝𝑎𝑖𝑟 𝑜𝑓 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝑠 𝑎𝑟𝑒 𝑢𝑛𝑒𝑞𝑢𝑎𝑙 STEP 2: 𝛼=0.05
STEP 3: Critical Region
𝑏< 𝑏 𝑘 𝛼; 𝑛 1 , 𝑛 2 ,⋯, 𝑛 𝑘 ≈ 𝑖=1 𝑘 𝑛 𝑖 𝑏 𝑘 𝛼; 𝑛 𝑖 𝑁
𝑏< 4 𝑏 ; 𝑏 ; 𝑏 ;
𝑏<0.5767
STEP 4: Test Statistic
𝑠 1 2 =1.583, 𝑠 2 2 =2.300, 𝑠 3 2 = 𝑠 𝑝 2 =2.254
𝑏= =0.9804