[36] Let $X$ be the times of sprinkler activation for a certain fire prevention system (in sec). (Assume $X$ is normally distributed.) The system has been designed so that true average activation time is at most 25 seconds. As sample of five tests are done and the times are given below. Does the data contradict the validity of this design specification? Answer the following as part of a test of the relevant hypotheses at significance level .05 using the P-value approach. 29 44 24 30 23. a) Show that the sample mean $\bar{x} = 30$ and sample standard deviation $s = 8.40$. b) Identify the parameter of interest. c) Identify the null hypothesis. d) Identify the alternative (or research) hypothesis. e) Find the test statistic value. [Hint: Sample size is small.] f) Identify the rejection region. g) Decide whether to reject $H_0$ or not. h) Conclusion: Does the data contradict the validity of this design specification?
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The sample mean is: $\bar{x} = \frac{29 + 44 + 24 + 30 + 23}{5} = \frac{150}{5} = 30$. The sample standard deviation is calculated as follows: $s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}}$ $s = \sqrt{\frac{(29-30)^2 + (44-30)^2 + (24-30)^2 + (30-30)^2 + Show more…
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