Numerical example and its solution summary 

The average service time of a company in 2018 was 12.44 minutes. Management wants to know whether the arithmetic mean current is lower or equal to 12.44 minutes. A sample with 150 values ​​had an arithmetic mean of 13.71 minutes and a standard deviation of 2.65 minutes. Using α = 5%, can you conclude whether the time is currently higher?

Let's recall the development to solve the previously numerical example presented.

• Null hypothesis (Ho): The mean is lower or equal, i.e., μ ≤  12.44.

• Alternate hypothesis (Ha): Then, the mean is higher, i.e., μ >  12.44.

• From the table described in the step to choose a statistical test, the sign of the hypotheses Ho and Ha point that a Right-tailed test should be carried.

• Since the sample is larger than 30, then a normal distribution could be employed.

• The next table helps to understand the relation between confidence level, alpha (α), and the critical value z α.

One important aspect is that the test of the hypothesis is much more sensible to higher values obtained in the sample since the new critical value z α = 1.645 is lower than for a two-tailed test which was z α/2 = 1.96. The next Figure represents this idea in graphical terms.

The next step is to compute the Zobs, and comparing it with Zcrit, and make a decision

Finally, the next code help to visualize the critical regions of the test hypothesis with  Zcrit, and the Zobs, help to compare their values to make a decision.