https://www.youtube.com/watch?v=m_AUjFI969c&list=PLsri7w6p16vtEz_J1G7HQG-Rm8vpZPtPS&index=6 ================================================================================ energy efficiency of car: 25km/l variance : 6.25 there is rumor that this energy efficiency is made up * Sample cars: 50 * Energy efficiency: 22.4km/l * Variance : 3.84 Question: * variance 6.25 is correct at p-value=0.05? ================================================================================ Answer $$$\sigma^2=6.25$$$ $$$H_0:\sigma^2=6.25$$$ $$$H_1:\sigma^2\ne 6.25$$$ $$$n=50$$$ $$$\bar{x}=22.4$$$ $$$s^2=3.84$$$ $$$\alpha=0.05$$$ test statistics $$$\chi^2_{(n-1)} \\ = \frac{(n-1)\times s^2}{\sigma^2} \\ = \frac{(50-1)\times 3.84}{6.25} \\ = 30.106$$$ Let's calculate following values $$$\chi^2_{n-1,1-\frac{\alpha}{2}} = 31.555$$$ $$$\chi^2_{n-1,\frac{\alpha}{2}} = 70.222$$$ ================================================================================ $$$H_0$$$ is rejected due to 30.106 Meaning: company insists the variance is 6.25 but it's not actually ================================================================================