The times of first sprinkler activation for a series of test

The times of first sprinkler activation for a series of tests with fire prevention sprinkler systems were (in seconds) :

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

The system has been designed so that true average activation time is at most 25 seconds 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.

Solution

Set Up Hypothesis
Null Hypothesis H0: U<=25
Alternate Hypothesis H1: U>25
Test Statistic
Population Mean(U)=25
Sample X(Mean)=27.923
Standard Deviation(S.D)=5.619
Number (n)=13
we use Test Statistic (t) = x-U/(s.d/Sqrt(n))
to =27.923-25/(5.619/Sqrt(12))
to =1.876
| to | =1.876
Critical Value
The Value of |t ?| with n-1 = 12 d.f is 1.782
We got |to| =1.876 & | t ? | =1.782
Make Decision
Hence Value of | to | > | t ?| and Here we Reject Ho
P-Value :Right Tail - Ha : ( P > 1.8756 ) = 0.04262
Hence Value of P0.05 > 0.04262,Here we Reject Ho

We don\'t have enough evvidence that the system has been designed so that true
average activation time is at most 25