In the realm of statistical analysis, Z critical values play a significant role in hypothesis testing and confidence interval calculations. They represent the point on the standard normal distribution curve beyond which a particular percentage of the data lies.

In this comprehensive guide, we will demonstrate how to find the Z critical value in Python, a powerful programming language popular among data analysts and researchers.

Understanding the Z-score and Z-table

Before diving into the Python implementation, it is essential to understand the Z-score and the Z-table. The Z-score, also known as the standard score, measures the number of standard deviations an observation or data point is from the mean of a normally distributed dataset. The Z-table, on the other hand, is a table that displays the probability that a score from a standard normal distribution is less than or equal to a given Z-score.

Calculating Z Critical Value in Python

To find the Z critical value in Python, you can use the scipy.stats module. Specifically, you can use the “norm” function to compute the Z critical value based on a given significance level.

Here’s an example:

				
					from scipy.stats import norm

# set the significance level (e.g., 0.05 for 95% confidence)
alpha = 0.05

# compute the Z critical value
z_critical = norm.ppf(1 - alpha/2)

print("Z critical value:", z_critical)

Output:

				
					# Z critical value: 1.959963984540054

In this code, the alpha variable represents the significance level (i.e., the probability of making a Type I error), which is typically set to 0.05 for a 95% confidence interval. The “norm.ppf” function is used to compute the Z critical value based on the specified alpha level. The “ppf” stands for “percent point function”, which is the inverse of the cumulative distribution function (CDF) of the normal distribution.

Once you run the code, you should see the Z critical value printed out to the console. Note that the Z critical value depends on the chosen alpha level and the distribution you are working with.

Left-tailed test

Suppose we have a sample of weights and we want to test the hypothesis that the true population mean weight is less than 50 kg. We can use a one-sample t-test to conduct the left-tailed test.

Here’s the Python code:

				
					import numpy as np
from scipy.stats import ttest_1samp, t

# Sample weights
weights = np.array([45, 48, 52, 49, 50, 47, 48, 46, 50, 44])

# Hypothesized population mean
hypo_mean = 50

# Set significance level
alpha = 0.05

# Conduct one-sample t-test
t_statistic, p_value = ttest_1samp(weights, hypo_mean)

# Calculate t-critical value for left-tailed test
t_critical = -1 * abs(t.ppf(alpha, df=len(weights)-1))

# Print results
print("t-statistic:", t_statistic)
print("p-value:", p_value)
print("t-critical value:", t_critical)

if t_statistic < t_critical:
    print("Reject null hypothesis, mean weight is less than 50 kg.")
else:
    print("Fail to reject null hypothesis, mean weight is not less than 50 kg.")

Output:

				
					# t-statistic: -2.6887744785908168
# p-value: 0.024846344440655466
# t-critical value: -1.8331129326536337
# Reject null hypothesis, mean weight is less than 50 kg.

In this code, we first define our sample of weights and our hypothesized population mean weight (i.e., 50 kg). We also set our significance level to 0.05.

We then conduct a one-sample t-test using the “ttest_1samp” function from scipy.stats. This returns the t-statistic and the p-value.

Next, we calculate the t-critical value for a left-tailed test using the t.ppf function from scipy.stats. We use the absolute value of the t-score and negate it to get the critical value in the left-tail of the t-distribution.

Finally, we print out the results and make our conclusion based on whether the t-statistic is less than the t-critical value (i.e., whether it falls in the left tail of the distribution).

Right-tailed test

Suppose we have a sample of test scores and we want to test the hypothesis that the true population mean test score is greater than 75. We can use a one-sample t-test to conduct the right-tailed test.

Here’s the Python code:

				
					import numpy as np
from scipy.stats import ttest_1samp, t

# Sample test scores
scores = np.array([78, 81, 76, 82, 79, 80, 77, 75, 83, 84])

# Hypothesized population mean
hypo_mean = 75

# Set significance level
alpha = 0.05

# Conduct one-sample t-test
t_statistic, p_value = ttest_1samp(scores, hypo_mean)

# Calculate t-critical value for right-tailed test
t_critical = t.ppf(1 - alpha, df=len(scores)-1)

# Print results
print("t-statistic:", t_statistic)
print("p-value:", p_value)
print("t-critical value:", t_critical)

if t_statistic > t_critical:
    print("Reject null hypothesis, mean test score is greater than 75.")
else:
    print("Fail to reject null hypothesis, mean test score is not greater than 75.")

Output:

				
					# t-statistic: 4.700096710803842
# p-value: 0.0011200178871001584
# t-critical value: 1.8331129326536333
# Reject null hypothesis, mean test score is greater than 75.

In this code, we first define our sample of test scores and our hypothesized population mean test score (i.e., 75). We also set our significance level to 0.05.

We then conduct a one-sample t-test using the “ttest_1samp” function from scipy.stats. This returns the t-statistic and the p-value.

Next, we calculate the t-critical value for a right-tailed test using the t.ppf function from scipy.stats. We use the inverse of the cumulative distribution function (CDF) of the t-distribution to find the critical value. We subtract the significance level from 1 to get the area in the right tail of the distribution.

Finally, we print out the results and make our conclusion based on whether the t-statistic is greater than the t-critical value (i.e., whether it falls in the right tail of the distribution).

Two-tailed test

Suppose we have a sample of heights and we want to test the hypothesis that the true population mean height is different from 170 cm. We can use a one-sample t-test to conduct the two-tailed test.

Here’s the Python code:

				
					import numpy as np
from scipy.stats import ttest_1samp, t

# Sample heights
heights = np.array([173, 167, 171, 175, 169, 168, 172, 174, 170, 176])

# Hypothesized population mean
hypo_mean = 170

# Set significance level
alpha = 0.05

# Conduct one-sample t-test
t_statistic, p_value = ttest_1samp(heights, hypo_mean)

# Calculate t-critical values for two-tailed test
t_critical = abs(t.ppf(alpha/2, df=len(heights)-1))

# Print results
print("t-statistic:", t_statistic)
print("p-value:", p_value)
print("t-critical values:", -t_critical, t_critical)

if t_statistic < -t_critical or t_statistic > t_critical:
    print("Reject null hypothesis, mean height is different from 170 cm.")
else:
    print("Fail to reject null hypothesis, mean height is not different from 170 cm.")

Output: 

				
					# t-statistic: 1.5666989036012806
# p-value: 0.1516274744876827
# t-critical values: -2.262157162740992 2.262157162740992
# Fail to reject null hypothesis, mean height is not different from 170 cm.

In this code, we first define our sample of heights and our hypothesized population mean height (i.e., 170 cm). We also set our significance level to 0.05.

We then conduct a one-sample t-test using the “ttest_1samp” function from scipy.stats. This returns the t-statistic and the p-value.

Next, we calculate the t-critical values for a two-tailed test using the “t.ppf” function from scipy.stats. We use the inverse of the cumulative distribution function (CDF) of the t-distribution to find the critical values. We divide the significance level by 2 to split it between the two tails of the distribution.

Finally, we print out the results and make our conclusion based on whether the t-statistic falls outside the range of the two critical values (i.e., whether it falls in either tail of the distribution).

Wrap up

Finding the T critical value in Python is straightforward with the help of the scipy.stats library. By using the t.ppf() function, you can easily calculate the T critical value for a given set of degrees of freedom and confidence level. This value is essential in various statistical analyses, such as hypothesis testing and constructing confidence interva

To learn more about SciPy documentation  check out the:
https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.t.html


Thanks for reading. Happy coding!