Question

The times of first sprinkler activation for a series of tests with fire prevention sprinkler systems using an aqueous film-forming foam were (in sec)

27 41 22 27 23 35 30 33 24 27 28 22 24
(see "Use of AFFF in Sprinkler Systems," Fire Technology, 1976: 5). The system has been designed so that true average activation time is at most 25 sec under such conditions. Does the data strongly contradict the validity of this design specification? Test the relevant hypotheses at significance level .05 using the P-value approach.

Answer 1

We reject H₀, we have enough argument to explain that the production line is out of control. Is producing sprinkler out of specification

Explanation:

Data information:

27 41 22 27 23 35 30 33 24 27 28 22 24

sample size n = 13

sample mean x = 27.92

sample standard deviation s = 5.39

The manufacturing process under control will always produce an output with normal distribution, in this case as n < 30 we will use t-student distribution in our test.

Hypothesis Test:

Null Hypothesis H₀ x = 25

Alternative Hypothesis Hₐ x > 25

The Alternative hypothesis indicates that the test is a one-tail test.

Significance level is α = 0.05

From z-table and for α = 0.05 and df = n - 1 df = 13 - 1 df = 12

p-value = 1.782

t(s) = ( x - 25 ) / s/√n

t(s) = ( 27.92 - 25 )/ 5.39/√13

t(s) = 2.92*3.605/ 5.39

t(s) = 1.95

From t-table df = 12 we find that 1.95 corresponds to a p-value < 0.05

then as p-value < 0.05 we are in the rejection region for H₀ then we reject H₀. We can deduce that the production line for sprinkler is given products out of specification