Embedded Files
1. Concepts & Definitions
1.1. Defining statistical test of hypothesis
1.2. Numerical example of test of hypothesis for mean
1.3. Code for test of hypothesis for mean
1.4. Code for right tailed test of hypothesis for mean
1.5. Code for left tailed test of hypothesis for mean
1.6. Code for small sample hypothesis for mean
1.7. P-Value and test of hypothesis
1.8. Statistical power and power analysis
1.9. Shapiro Wilk for normality test
2. Problem & Solution
2.1. Shapiro Wilk to verify CLT Simulator
Recapitulating the concept of significance level
Recapitulating the concept of significance level
On the probability distribution plot, the significance level defines how far the sample value must be from the null value before we can reject the null. The percentage of the area under the curve that is shaded equals the probability that the sample value will fall in those regions if the null hypothesis is correct. To represent a significance level of 0.05, the next figure shade in a red color α = 5% of the distribution furthest from the null value [1].
The two shaded regions in the graph are equidistant from the central value of the null hypothesis. Each region has a probability of 0.025, which sums up our desired total of 0.05. These shaded areas are called the critical region for a two-tailed hypothesis test. For this specific shaded area the Zcrit = 1.96, and the Zobs = 5.87.
What is a P-value?
What is a P-value?
The first interpretation is P-values indicate that the null hypothesis is correct given the sample data. They gauge how consistent your sample statistics are with the null hypothesis. Specifically, if the null hypothesis is right, what is the probability of obtaining an effect at least as large as the one in your sample [1]?
High P-values: Your sample results are consistent with a true null hypothesis.
Low P-values: Your sample results are not consistent with a null hypothesis.
If your P-value is small enough, you can conclude that your sample is so incompatible with the null hypothesis that you can reject the null for the entire population. As a more clear decision rule:
If P-value ≥ α then cannot reject the null hypothesis
Else reject the null hypothesis
In a second interpretation, P-values tell you how consistent your sample data are with a true null hypothesis.
Suppose the hypothesis test generates a P-value of 0.03. You’d interpret this P-value as follows: although the null hypothesis holds on the population as a whole, 3% of individuals will obtain the effect observed in your sample, or larger, because of random sample error[2].
However, when your data are very inconsistent with the null hypothesis, P values can’t determine which of the following two possibilities is more probable:
The null hypothesis is true, but your sample is unusual due to random sampling error.
The null hypothesis is false.
This is the main reason why it is said that: “failed to reject the null hypothesis” or "cannot reject the null hypothesis" rather than accepted it.
How to obtain a P-Value from Zobs?
How to obtain a P-Value from Zobs?
For a certain value of Zobs it is possible to compute the corresponding:
α/2 for a two-tailed hypothesis test
α for a one-sided hypothesis test
The p-value will be precisely the value α related.
For example, suppose that for a certain sample data, Zobs = 2.50 is obtained. The Zobs = 2.50 will correspond to an area under the standard normal distribution equal to 0.0061. Since this region is defined from a two-tailed hypothesis test, this means α/2 = 0.0061, and then α = 0.0122. This is precisely, the P-value for this sample.
Numerical example
Numerical example
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 12.98 minutes and a standard deviation of 2.65 minutes. Using α = 5%, can you conclude whether the time is currently different?
Compute statistical test - Using P-Value
Compute statistical test - Using P-Value
To compute a statistical test is necessary to convert the observed value in the mean of the sample (x̄) to the scale of a standard normal distribution (Zobs). These could be done using the following equation:
zobs = (x̄ - μ)/(s/(n^0.5))
This equation will result in the following numbers:
zobs = (12.98 - 12.44)/(2.65/(150^0.5)) = (12.98-12.44)/0.2164 = 2.50
Applying Zobs = 2.50 to inverse standard normal corresponds to 0.9939, which means the region under the curve starting from Zobs = 2.50 is 1 - 0.9939 = 0.0061. This means αobs/2 = 0.0061, then αobs = 0.0122.
For this example, the P-value will be 0.0122. If the level of significance chosen is α = 1%, then the following rule can be applied:
If P-value ≥ α then cannot reject the null hypothesis
Compute statistical test - Using Zcrit and Zobs
Compute statistical test - Using Zcrit and Zobs
Another to see this question is to employ the Zcrit and Zobs values. Since, Zobs = 2.50, and for a level of significance α = 1%, which corresponds to Zcrit = 2.56, then Zobs is the region in which we cannot reject the null hypothesis.
The next Python code with figures summarizes both approaches.
Python code for using Zcrit and Zobs
Python code for using Zcrit and Zobs
First, let's compute Zcrit for the confidence level equal to 1%.
from scipy.stats import norm
muz = 0
sigmaz = 1
p = 0.99
alfa = 1 - p
pr = 1 - alfa/2
z = norm.ppf(pr,muz,sigmaz)
print("Given alpha = ",str(round(alfa,2)),", Zcrit = ",round(z,2))
Given alpha = 0.01 , Zcrit = 2.58
Now, use the sample data to compute Zobs.
mi = 12.44
H0 = "The value is equal to " + str(mi)
n = 150
xb = 12.98
s = 2.65
sx = s/(n**(0.5))
zobs = (xb - mi)/sx
print("Zobs = ",zobs,"and Zcrit = ",z)
if (zobs > z)|(-zobs < -z): # Zobs belongs to the critical region
print("Reject H0: ",H0)
else:
print("Do not reject H0: ",H0)
Zobs = 2.495706530382865 and Zcrit = 2.5758293035489004
Do not Reject H0: The value is equal to 12.44
To better visualize this situation, let's draw it.
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
x1 = np.arange(-z, 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='g')
ax.fill_between(x_all,y_all,0, alpha=0.1, color='r')
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, 0.2, 'Zobs', fontsize=14)
print("Zobs: ",zobs)
Python code for using P-Value and α
Python code for using P-Value and α
Let's first compute P-Value and use the chosen confidence level to make a decision.
muz = 0
sigmaz = 1
P = norm.cdf(zobs,muz,sigmaz)
Pvalue = 2*(1-P)
print("Pvalue = ",Pvalue)
print("alpha = ",alfa)
if (Pvalue >= alfa):
print("We cannot reject H0")
else:
print("Reject H0")
Pvalue = 0.012570655321079371
alpha = 0.010000000000000009
We cannot reject H0
Finally, let's draw in a graphic the P-value metric.
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
x1 = np.arange(-z, 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)
stralfa = str(round(alfa*100,2))
ax.text(muz-1.2*z, 0.1, 'Ha|alpha = ' + stralfa, fontsize=14)
ax.text(muz+z, 0.1, 'Ha|alpha = ' + stralfa, fontsize=14)
ax.fill_between(x1,y1,0, alpha=0.7, color='g')
ax.fill_between(x_all,y_all,0, alpha=0.1, color='r')
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 = 'P-Value')
ax.text(0.85*zobs, 0.2, 'P-value = '+str(round(Pvalue*100,2)), fontsize=14)
print('P-value',Pvalue)
P-value 0.012570655321079371
The previous complete code is available in the following link:
https://colab.research.google.com/drive/1P7DRHjbNfrVRrJqe2RL4w4uuvfgW5Osm?usp=sharing
References and additional material:
[1] https://statisticsbyjim.com/hypothesis-testing/hypothesis-tests-significance-levels-alpha-p-values/
[2] https://statisticsbyjim.com/hypothesis-testing/interpreting-p-values/
[3] https://www.scribbr.com/statistics/type-i-and-type-ii-errors/
[4] https://statisticsbyjim.com/hypothesis-testing/types-errors-hypothesis-testing/
[5] https://medium.com/@dhruvb30/practical-statistics-with-python-2-hypothesis-testing-51118b77463e
[6] https://www.section.io/engineering-education/statistical-hypothesis-testing-python-implementation/
[7] https://machinelearningmastery.com/statistical-hypothesis-tests-in-python-cheat-sheet/
[8] https://towardsdatascience.com/p-value-basics-with-python-code-ae5316197c52
[9] https://www.geeksforgeeks.org/how-to-find-a-p-value-from-a-z-score-in-python/
[10] https://www.geeksforgeeks.org/how-to-find-a-p-value-from-a-t-score-in-python/
[11] https://medium.com/@maheshhkanagavell/p-value-fdf3c6ec09ee
[12] https://rhydhamgupta.medium.com/p-value-explained-clearly-regression-pdf-discrete-45c30b5dc813
Page updated
Google Sites
Report abuse