Hypothesis Testing in Python: Finding the critical value of T

A hypothesis test in statistics is a measure which tells us whether there is enough statistical evidence to support a particular hypothesis. Hypothesis testing enables us to make predictions with the help of the probabilities obtained from the test results about a particular sample or population.

A statistician or a scientist uses some statistical tools to measure a particular random sample data from a population. This eliminates the bias if the data is skewed in some manner. In all hypothesis testing, a random sample of the population is used to test two different types of hypothesis.

In Python, hypothesis testing and finding the critical value of t can be achieved using the Scipy library. The t-test is a statistical method used to determine if there is a significant difference between the means of two groups. It’s a powerful tool for data analysis and interpretation.

The first hypothesis that is considered is the null hypothesis. In simple words, a null hypothesis as the name suggests returns a zero or a null. It states that there is no correlation between the men of two data sets or variables, and even if there is, it is infinitesimally small due to sampling or experimental errors.

The alternative to the null hypothesis is called the alternative hypothesis. The alternate hypothesis states that the mean is not equal to zero, that is the opposite of the null hypothesis.

There are many methods of hypothesis testing. The t-test is one of them. It is very useful to draw several conclusions about data samples and their correlation with one another. Let’s look at how t-test can be done in Python!

Delving into the T-Test

The t-test is one of the most common methods of hypothesis testing. It is used to measure the mean of two data groups, determine the difference between them, and find the degree of correlation.

T-tests can be performed when the variance is unknown, and if the data sample follows normal distribution.

For example, if you want to compare the mean scores of two sections of a class of 100 students, their means and standard deviations will differ. Based on this, we can form two hypothesis statements, one will be the null hypothesis(H0 ) and the other one would be the alternate hypothesis(HA ).

The hypotheses look like this:

• H0 : μ = m, where m is the postulated value and μ is the calculated value. This is the statement which we are going to test.
• HA : The alternate hypothesis, which will be taken into consideration if the null hypothesis is rejected.

There are some assumptions that you need to keep in mind when performing a T-test. They are:

• The selected sample must be random to minimize any kind of bias.
• The data must be normally distributed.
• The variance must be homogenous when the standard variations are equal.

The t-test requires three components to be performed, which are:

• The mean difference, which is the difference between the means of the given datasets.
• The standard deviations of the data sets.
• The total number of data values in each set.

The main objective of a t-test is to determine the correlation between the two datasets.

The t-test score thus produced is the ratio of the difference between the two means of the datasets to the difference in their variance.

If the t-test score or the t-test critical value is large, then the data sets are highly different from one another. But if the t-test critical value is small, then the datasets are very similar to each other.

The t-test also produces another result, that is, the degrees of freedom. It refers to the values that are allowed to vary in a scientific hypothesis.

The formula for t-test is : T = (X̄ – μ) / S/√n, where, T is the t-test critical value, is the sample mean, μ is the hypothesized population mean from the null hypothesis, S is the standard deviation of the sample dataset and n is the number of observed values in the dataset.

Similar: T-Test Hypothesis in Python.

One-Tailed vs Two-Tailed T-Tests

There are two types of test that we can apply, they are:

One – tailed T test: In this type of test, if the value of the t score is more or less than the value stated in the null hypothesis. If you want to test if your output has a greater value than the one stated in the null hypothesis(μ > m), then you need to conduct a right tailed t -test. If you want to test whether your output has a less value than the one given in the null hypothesis(μ < m), you are going to conduct a left-tailed t test.

Two-tailed T test: This is conducted when you want to test if your t score has a completely different value than the one stated in the null hypothesis.

In these tests another very important parameter required is the critical region value which tells us the ceiling and floor values of the confidence interval. It is used to determine the region where if the T test value lies, will be used to reject the null hypothesis. In case of one tailed test, there is only one critical region value whereas there are two critical region values for the two tailed test.

Leveraging Scipy to Compute the T-Test Critical Value in Python

Scipy is a free and open source Python library that is used for complex calculations and scientific computations. It is extremely easy to use and can be used to find the critical value of t in t-tests.

To calculate the T-test critical value, we will need the critical region value or the confidence interval and the degrees of freedom.

The syntax of performing a t-test is as follows:

```scipy.stats.t.ppf(q=critical level, df=degrees of freedom)
```

Executing a One-Tailed T-Test

Let’s see how we can conduct a one tailed T test where we are going to observe if our calculated value is lesser or greater than the one postulated. Here, we’ll assume the critical level(q) is 0.05 and degrees of freedom(df) is 40.

First code block is for calculating right tailed T-test followed by it’s output and then we have the one for left tailed T-test followed by it’s own result..

```#import modules
import scipy as sp
#incase of right tailed t test
sp.stats.t.ppf(q=1-.05,df=40)
```
```#output
1.68385101138658
```
```#import required modules
import scipy as sp
#incase of left tailed t test
sp.stats.t.ppf(q=.05,df=40)
```
```#output
-1.6838510113865808
```

Implementing a Two-Tailed T-Test

The implementation of the two tailed test is very similar to the one tailed one in the previous section. You just need to divide the critical value by 2.

```# Import module
import scipy as sp
# Incase of two tailed T test
sp.stats.t.ppf(q=1-.10/2,df=34)
```

The output is :

```#output
1.6909242507706543
```

In the last case of performing two tailed test, we have two values, 1.6909 and -1.6909, which means if the test statistic has a value that is greater than 1.6909, or lesser than -1.6909, then the results are statistically significant.

Do check out: Python SciPy Tutorial.

Wrapping Up: The Power of T-Tests in Python for Hypothesis Testing

We’ve now explored the fascinating world of hypothesis testing in Python, specifically focusing on the t-test. With the power of Scipy, we’ve seen how straightforward it can be to calculate critical values and conduct meaningful statistical analysis. Remember, the key to a successful t-test lies in understanding your degrees of freedom and setting the right confidence interval. So, how will you apply these techniques in your next data analysis project?