Z critical value is an essential concept in statistical analysis that helps determine whether our test statistics lie in a given confidence interval or not. If the test statistic is outside the confidence interval, we can reject the null hypothesis that the two populations are the same.
The determination of Z critical values depends on the significance level of the test, which is typically denoted by alpha.
In this article, we will explore how to find the Z critical value in Python for left-tailed, right-tailed, and two-tailed tests.
We will use the scipy.stats.norm.ppf() function, which calculates the percent point function (PPF) of the normal distribution. The PPF is the inverse of the cumulative distribution function (CDF) and can be used to compute probabilities for specific values of the variable, given the distribution’s mean and standard deviation.
Left-tailed test
In a left-tailed test, the null hypothesis states that the test statistic is greater than or equal to the critical value. The alternative hypothesis claims that the test statistic is less than the critical value.
Therefore, we need to find the Z critical value that corresponds to the left tail of the normal distribution at a given significance level. To find the Z critical value in Python for a left-tailed test, we need to use the scipy.stats.norm.ppf() function and input the alpha value and the loc and scale parameters of the normal distribution.
The loc parameter represents the mean of the distribution, and the scale parameter represents the standard deviation.
Here’s an example code snippet that computes the Z critical value for a left-tailed test with a significance level of 95% (i.e., alpha = 0.05), assuming a normal distribution with mean 0 and standard deviation 1:
“`python
import scipy.stats as stats
alpha = 0.05
z_critical = stats.norm.ppf(alpha, loc=0, scale=1)
print(z_critical)
“`
The output will be -1.64485362695, which is the Z critical value that corresponds to the left tail of the normal distribution at a significance level of 95%.
Right-tailed test
In a right-tailed test, the null hypothesis states that the test statistic is less than or equal to the critical value. The alternative hypothesis claims that the test statistic is greater than the critical value.
Therefore, we need to find the Z critical value that corresponds to the right tail of the normal distribution at a given significance level. To find the Z critical value in Python for a right-tailed test, we can use the same scipy.stats.norm.ppf() function as in the left-tailed test, but with a modified alpha value.
Since the right tail corresponds to 1-alpha percent of the distribution, we need to subtract the significance level from 1 to get the correct alpha value to input. Here’s an example code snippet that computes the Z critical value for a right-tailed test with a significance level of 90% (i.e., alpha = 0.10), assuming a normal distribution with mean 0 and standard deviation 1:
“`python
import scipy.stats as stats
alpha = 0.10
z_critical = stats.norm.ppf(1-alpha, loc=0, scale=1)
print(z_critical)
“`
The output will be 1.28155156554, which is the Z critical value that corresponds to the right tail of the normal distribution at a significance level of 90%.
Two-tailed test
In a two-tailed test, the null hypothesis states that the test statistic is equal to the critical value. The alternative hypothesis claims that the test statistic is not equal to the critical value.
Therefore, we need to find the Z critical values that correspond to both tails of the normal distribution at a given significance level. To find the Z critical values in Python for a two-tailed test, we can use the same scipy.stats.norm.ppf() function as in the left and right-tailed tests, but with a modified alpha value.
Since the two tails each correspond to (1-alpha)/2 percent of the distribution, we need to divide the significance level by 2 and subtract it from 1 to get the correct alpha value to input. Here’s an example code snippet that computes the Z critical values for a two-tailed test with a significance level of 99% (i.e., alpha = 0.01), assuming a normal distribution with mean 0 and standard deviation 1:
“`python
import scipy.stats as stats
alpha = 0.01/2
z_critical_left = stats.norm.ppf(alpha, loc=0, scale=1)
z_critical_right = stats.norm.ppf(1-alpha, loc=0, scale=1)
print(z_critical_left, z_critical_right)
“`
The output will be -2.57582930355 and 2.57582930355, which are the Z critical values that correspond to the left and right tails of the normal distribution at a significance level of 99%.
Conclusion
In this article, we explored how to find the Z critical value in Python for left-tailed, right-tailed, and two-tailed tests, using the scipy.stats.norm.ppf() function. We demonstrated how to input the correct alpha value and the loc and scale parameters of the normal distribution to obtain the Z critical value that corresponds to a given significance level and tail of the distribution.
By understanding how to compute the Z critical value, we can make better decisions and draw more accurate conclusions from our statistical analysis. In summary, this article explored how to find the Z critical value in Python for left-tailed, right-tailed, and two-tailed tests using the scipy.stats.norm.ppf() function.
The Z critical value is important in statistical analysis as it helps determine whether the test statistic lies in a given confidence interval or not. Knowing how to compute the Z critical value allows us to draw accurate conclusions and make better decisions based on statistical analysis.
Therefore, it is crucial to understand how to find the Z critical value and how it relates to the significance level and tail of the distribution. Takeaways from this article include the ability to input the correct alpha value and loc and scale parameters of the normal distribution to obtain the Z critical value that corresponds to a given significance level and tail of the distribution.