Question

The average length of time for students to register for summer classes at a certain college has been 50 minutes with a standard deviation of 10 minutes. A new registration procedure using modern computing machines is being tried. If a random sample of 12 students had an average registration time of 42 minutes with standard deviation of 11·9 minutes under the new system, test the hypothesis that the population mean has not changed.

Answer 1

$\mu=50$

$\sigma=10$

$\bar{X}=42$

$n=12$

$s=11.9$

Since a new machine is in play, population parameters might change. Thus, we will consider the case for unknown population standard deviation. Assume that the registration time is normally distributed. If this assumption is not made, nothing can be done since the sample size is small.

$H_0:\mu=50$

$H_a:\mu\ne50$

$t=\frac{\bar{X}-\mu}{\frac{s}{\sqrt{n}}}$

$=\frac{42-50}{\frac{11.9}{\sqrt{12}}}=-2.33$

Let $\alpha$ be 0.05

$Cv=t_{\frac{\alpha}{2},n-1}=t_{0.025,11}=2.201$

Since the absolute value of the test statistic t=2.33 is greater than the critical value 2.201, we reject the null hypothesis and conclude that at 95% confidence level there is sufficient evidence to conclude that the population mean has changed.