A sprinkler system is being installed in a newly renovated building on campus. The average activation time is supposed to be at most 20 seconds. A series of 12 fire alarm/sprinkler system tests results in an average activation time of 21.5 seconds. Do these data indicate that the design specifications have not been met? The hypotheses to be tested are H0: μ = 20 versus Ha: μ > 20, where μ = the true average activation time of the sprinkler system. If the true average activation time of the sprinkler system is, in fact, greater than 20 seconds, and but you did not reject H0, what type of error would you have made?

To test this claim, a sample of 12 test results of the fire alarm/sprinkler was taken, the resulting sample mean is X[bar]= 21.5 seconds. Assume that activation times for this system are Normally distributed with s = 3 seconds.

The hypotheses are:

H₀: μ = 20

H₁: μ > 20

Using α: 0.05

The statistic for this test is a one sample t-test:

[tex]t_{H_0}= \frac{X[bar]-Mu}{\frac{S}{\sqrt{n} } } = \frac{21.5-20}{\frac{3}{\sqrt{12} } } = 1.73[/tex]

The p-value for this test is p-value: 0.0558

The p-value is greater than the level of significance so the decision is to not reject the null hypothesis.

If the average activation time is, in fact, 20 seconds then the null hypothesis is false.

The situation is that the null hypothesis was not rejected given that it is false. The error that was committed was a type II error.

I hope it helps!