We have our hypotheses
"H_0: \\; \\mu=50 \\text{ vs. }H_1:\\;\\mu<50"
In order to test this hypothesis, we must constuct a test statistic. As we know the registration times follow a Normal distribution, we can create the following test statistic
"T=\\frac{\\hat{\\mu}-50}{s\/\\sqrt{n}}"
where, "s" is the sample standard deviation, and "\\hat{\\mu}" is the sample mean. This follows a t-distribution with degrees of freedom "n-1=11"
Thus, our critical region is "<t_{n-1,\\alpha}=-t_{n-1,1-\\alpha}" , which denotes the corresponding quantile of the t-distribution, as it is symmetric
We calculate "T=\\frac{-8\\sqrt{12}}{11.9}\\approx -2.3288" and from empirical tables, we have "t_{11,0.05}=1.795885"
Thus, we have "T<-1.795885" . Thus, there is enough evidence to reject the null hypothesis.
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