Consider the two independent random samples:

  • Set up H0 : σ21 = σ22
  • Set up H1 : σ21< σ22 or σ21> σ22, or σ21 ≠ σ22
  • Statistics to set up level of significance:

Statistics and Operational Research:

Here, F-distribution will be followed depending on H1, critical value. We have discussed it in the tablebelow:

H1Test StatisticsReject H0, if
σ21< σ22F =Fcal>Fα,n2 -1, n1 – 1
σ21> σ22F =Fcal>Fα,n1 -1, n2 – 1
σ21≠  σ22F =Fcal>Fα/ 2,nL -1, nl – 1


Example No. 13:

Test the null hypothesis of the following if the intensity of light of two bulbs manufactured by two companies A and B respectively has yield of S1 = 1.5 foot-candles, S2 = 1.75 foot-candles from a sample size of 16. Here, σ12 = σ22for alternative σ1222 at the level of significance as 0.01.


  • Set up H0 : σ12 = σ22
  • Set up H1 : σ1222
  • Statisticswhich follows the F-Distribution with degree of freedom as 15 and 15
  • α = 0.01. here, alternative hypothesis shows, it is left tailed test.


Critical Value, F0.01, 15, 15 = 3.52

  • Given:

S1 = 1.5 foot-candles

S2 = 1.75 foot-candles

  • Decision:

Fcal< 3.52


H0 is accepted and there will be no variability in intensity of light by these two bulbs.

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