Consider two independent random samples from two normal populations.

- 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}.

- Set up the level of significance α.

STATISTICS & OPERATIONS RESEARCH

In this case the test statistic follows F-distribution and depending on HI' the critical value is obtained. We list in the following:

**Example **13. *It is desirable to determine whether there is less variability in the intensity of light by two bulbs made by company A and company B respectively in certain locations. lf the independent random samples of size 16 of the two bulbs yield S1 *= *1.5 foot-candles and S2 *= *1. 75 foot-candles, test the null hypothesis σ _{1}^{2 }*=

*σ*

_{2}^{2}*against the alternative σ*<

_{1}^{2 }*σ*

_{2}^{2}*at the 0.01 level of significance.*

** **

**Solution. **

- H
_{0}:*σ*=_{1}^{2 }*σ*_{2}^{2} - H
_{1}:*σ*<_{1}^{2 }*σ*_{2}^{2} - Test statistic:

which follows F-distribution with degrees of freedom 15 and 15.

4. α = 0.01. The alternative hypothesis shows that it is left tailed test.

Critical value F_{0.01, 15, 15} = 3.52.

5. Given Sl = 1.5 and S2 = 1.75

6. Decision : Since F_{cals} < 3.52, H_{0 }is accepted.

=> There is no variability in the intensity of light by two bulbs.