"a_0=50\\\\\nn=20\\\\\n\\overline{x}=45\\\\\ns=9\\\\\n\\alpha=0.01\\\\\nH_0: \\mu=a_0=50, H_1: \\mu<a_0=50\\\\\n\\mu\\text{ --- the average length of time for students}\\\\\n\\text{to register under the new system}.\\\\\n\\text{We assume that the length of time for students}\\\\\n\\text{to register has normal distribution}.\\\\\n\\text{We will use the following random variable as a criterion:}\\\\\nT=\\frac{(\\overline{X}-a_0)\\sqrt{n}}{s}\\\\\nT\\text{ has t-distribution with } k=n\u22121 \\text{ degrees of freedom.}\\\\\n\\text{Observed value:}\\\\\nt_{obs}=\\frac{(45-50)\\sqrt{20}}{9}\\approx -2.48\\\\\n\\text{Critical value (one-sided):}\\\\\nt_{cr}=t_{cr}(\\alpha;k)=t_{cr}(0.01;19)\\approx 2.54\\\\\n(-\\infty, -2.54)\\text{ --- critical region}\\\\\nt_{obs}\\text{ does not fall into the critical region. So we accept } H_0.\\\\\n\\text{We cannot conclude that the new system is faster than the old}."
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