Answer to Question #151750 in Statistics and Probability for saadah

Question

1. Currently, total average time a patient spent time in a local health clinic is 47 minutes. A new procedure which is computer based system is being introduced by the management. Based on sample of 12 patient, the new mean spent time is 39 minutes with standard deviation of 6.9 minutes. Test whether the new system is better than the old system using 5% significance level.
2 Following a recent severe typhoon hit the National University of Malaysia, many buildings including lecture halls were badly affected. The management team must decide whether the campus is still safe for conducting a lecture or otherwise. If the null hypothesis is "safe to conduct lecture" and the alternative hypothesis is "not safe to conduct lecture", which error (type I error or type II error) is more important to avoid? Explain your answer

Answer 1

1) $\mu = 47$

$n = 12$

$\bar x = 39$

$s = 6.9$

$\alpha = 0.05$

$H_0: \mu = 47$

$H_a: \mu < 47$

The hypothesis test is left-tailed. Since the population standard deviation is unknown and the sample size is smaller than 30 we use t-test.

$df = n-1=11$

The critical value for $\alpha = 0.05$ and 11 degrees of freedom is –1.796.

The critical region is $t<-1.796$

Test staticstic:

$$t =\frac{\bar x-\mu}{\frac{s}{\sqrt n}}=\frac{39-47}{\frac{6.9}{\sqrt {12}}}=-4.01$$

Since –4.01 < –1.796 thus t falls in the rejection region we reject the null hypothesis.

At the 5% significance level the data do provide sufficient evidence to support the claim. We are 95% confident to conclude that the new system is better than the old system.

2) $H_0$ is "safe to conduct lecture"

$H_a$ is "not safe to conduct lecture"

Type I error is a rejection of the null hypothesis when it is true. Thus type I error is concluding that the campus is not safe to conduct lectures while it is actually safe.

Type II error is failing to reject the null hypothesis when it is false or rejection of alternate hypothesis when it is true. Thus type II error is concluding that the campus is safe to conduct lectures while it is actually unsafe.

Therefore this is more important to avoid the type II error because it is more dangerous.