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 different from 12.44 minutes. A sample with 150 values ​​had an arithmetic mean of 11.71 minutes and a standard deviation of 2.65 minutes. Using α = 5%, can you conclude whether the time is lower?

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

• Null hypothesis (Ho): The mean is higher, i.e., μ ≥  12.44.

• Alternate hypothesis (Ha): Then the mean is lower, 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 Left-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 (remember to employ negative values of  z α for a left-tailed test).

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.

import numpy as np

import matplotlib.pyplot as plt

from scipy.stats import norm

x1 = np.arange(-10, z, 0.001) # range of x in spec

x_all = np.arange(-10, 10, 0.001) # entire range of x, both in and out of spec

# mean = 0, stddev = 1, since Z-transform was calculated

y1 = norm.pdf(x1,0,1)

y_all = norm.pdf(x_all,0,1)

# build the plot

fig, ax = plt.subplots(figsize=(9,6))

plt.style.use('fivethirtyeight')

ax.plot(x_all,y_all)

ax.text(muz, 0.4, 'Ho', fontsize=14)

ax.text(muz-1.2*z, 0.1, 'Ha', fontsize=14)

ax.text(muz+z, 0.1, 'Ha', fontsize=14)

ax.fill_between(x1,y1,0, alpha=0.7, color='r')

ax.fill_between(x_all,y_all,0, alpha=0.1, color='g')

ax.set_xlim([-6,6])

ax.set_xlabel('# of Standard Deviations Outside the Mean')

ax.set_yticklabels([])

ax.set_title('Normal Gaussian Curve CL = '+str(round(p*100,0))+' %')

# drawing Zobs

ax.axvline(x = zobs, color = 'r', label = 'Zobs')

ax.text(0.85*zobs, 0.1, 'Zobs', fontsize=14)