Consider the two independent random samples:

- Set up H
_{0}: σ^{2}_{1}= σ^{2}_{2} - Set up H
_{1 : }σ^{2}_{1}< σ^{2}_{2}or σ^{2}_{1}> σ^{2}_{2}, or σ^{2}_{1}≠ σ^{2}_{2} - Statistics to set up level of significance:

Statistics and Operational Research:

Here, F-distribution will be followed depending on H_{1}, critical value. We have discussed it in the tablebelow:

H_{1} | Test Statistics | Reject H_{0}, if |

σ^{2}_{1}< σ^{2}_{2} | F = | F_{cal}>F_{α,n}_{2}_{ -1, n}_{1}_{ – 1} |

σ^{2}_{1}> σ^{2}_{2} | F = | F_{cal}>F_{α,n}_{1}_{ -1, n}_{2}_{ – 1} |

σ^{2}_{1}≠ σ^{2}_{2} | F = | F_{cal}>F_{α/ 2,n}_{L}_{ -1, n}_{l}_{ – 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 S_{1} = 1.5 foot-candles, S_{2} = 1.75 foot-candles from a sample size of 16. Here, σ_{1}^{2} = σ_{2}^{2}for alternative σ_{1}^{2}<σ_{2}^{2} at the level of significance as 0.01.

**Solution:**

- Set up H
_{0}: σ_{1}^{2}= σ_{2}^{2} - Set up H
_{1 : }σ_{1}^{2}<σ_{2}^{2} - 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.

Thus,

Critical Value, F_{0.01, 15, 15} = 3.52

- Given:

S_{1} = 1.5 foot-candles

S_{2} = 1.75 foot-candles

- Decision:

F_{cal}< 3.52

Hence,

H_{0} is accepted and there will be no variability in intensity of light by these two bulbs.