Hypothesis Testing

What is Hypothesis Testing?

A systematic procedure is used by the researchers to predict whether the results obtained from a study supports a particular theory that is related to the population is known as hypothesis testing. It uses the sample data in order to evaluate the hypothesis of the population. It is the statistical inference method that is used to test the significance of the proposed hypothesized relation between the population statistics or the parameters and their corresponding estimators of the sample. This test consists of two hypotheses, the null hypothesis, and the alternative hypothesis. These two tests should be mutually exclusive. In most of the applications, these two methods are complementary in nature; that is one hypothesis will be the negation of the other. 


The working of this test includes the comparison of p-value to a certain level of significance. If the obtained p-value is equal to or less than that of the significance level, then the null hypothesis can be rejected. Samples of a certain size are manageable while analyzing the data as efficient computations. There is an advantage of following the method of test hypothesis as it allows what distribution or the parameters in which the data is followed. The hypothesized means a theory or a guess of the occurring event.


Null and Alternative Hypothesis

Null hypothesis was first found by Ronald Fisher in a context where he found the alternative hypothesis as less important and didn’t pay attention to the power of the test. The null hypothesis is used in statistics that mentions that there are no differences found between certain characteristics of the population. It is denoted as (H\[_{0}\]). The alternative hypothesis is used for the probability distribution of the data only informally or explicitly. It is denoted by (H\[_{1}\] or H\[_{a}\]). The comparison of these two hypotheses was done with the help of statistical significance, as according to the threshold probability of the significance level if the null hypothesis is true, then the data is unlikely to occur. The hypothesis test helps to determine which outcome can reject the null hypothesis. 


By considering the two conceptual errors, the difference between the null hypothesis and the alternative hypothesis can be found. When the null hypothesis is rejected wrongly, then the first error occurs. And when the null hypothesis is not rejected wrongly, then the second type of error occurs. These two types of errors are known as type-1 and type-2. Type-1 error is denoted by (α), it is also known as the significance level. The type-2 error is denoted by (β). (1 - β) is known as the power test. 


Difference Between the Null Hypothesis and Alternative Hypothesis:


Null Hypothesis

Alternative Hypothesis

It is the statement that has no relationship between two variables.

A statement that provides statistical significance in between the measure phenomena.

No changes observed in the opinions or actions.

Changes are observed in the case of opinions and actions.

The testing is done indirectly and the result is implicit.

Testing is done directly and the result is explicit.

It is the result of chance.

It is the result of the real effects.

Denoted as (H0).

Denoted as (H1 or Ha).


Testing Process

The statistical hypothesis plays an important role in the statistics literature. Mathematically, there are two processes that can be used to test the hypothesis.

  1. The initial hypothesis contains the truth that is unknown.

  2. The first step in this process is to find the relevant null hypothesis and alternative hypothesis. This is an important step as misstating the meaning can spoil the process. 

  3. The statistical assumptions are made about the samples that are involved in the test, these assumptions are considered.

  4. Decide which test is appropriate to conduct and mention the relevant test statistic that is denoted by T.

  5. By using the null hypothesis, the distribution of the test statistic from the assumptions is derived. This result will be well known in standard cases. 

  6. A significance level (α) of a probability threshold has to be selected, where once the value goes below it, the null hypothesis gets rejected. 

  7. The test statistic present under the null hypothesis is distributed for all the partitions of possible values of T that caused the rejection of the null hypothesis. These possible values are known as critical regions. The probability of this critical region is α. 

  8. Compute the test statistic T from the observed value t\[_{obs}\] of the observations.

  9. Depending on the T, decide whether to reject the null hypothesis or not. If the observed value t\[_{obs}\] is in the range of the critical region, then the null hypothesis has to be rejected and if it does not lie, then it can be accepted or fail to reject. 

The alternative formulation of this process is as follows:

  1. Compute the test statistic T from the observed value t\[_{obs}\] of the observations.

  2. P-value is the probability that is present under the null hypothesis, compute this p-value. The test statistic should be at least as extreme as the value that was observed. 

  3. In favour of the alternative hypothesis, the null hypothesis can be rejected if the p-value is equal to or less than the threshold of the significance level. 

The earlier mentioned process was advantageous since the test statistic at the common probability thresholds were available. The decision was taken even before calculating the probability. That process was sufficient for the operational use and the classwork but for reporting the results it was deficient. But the latter process relied on computational support and extensive tables that are not available always. To get the detailed information, the two processes are applied on the Radioactive suitcase example:

  • Geiger-counter reading is 10, and the limit is 9, thus check the suitcase.

  • Geiger-counter reading is high, where 97% of the safe suitcases have this lower reading thus the limit is 95%, and check the suitcase.

From these two examples, we can observe that the former report was adequate but the latter provides a detailed explanation of the data, which is why the suitcase is being checked. The difference between accepting or rejecting the null hypothesis is important. If the non-significant results do not provide a way to determine which of the two hypotheses are true, then the terminology fails to reject. 


Interpretation of the Statistical Hypothesis

The p-value provides the probability of the given result and it occurs under the null hypothesis. If the p-value is less than that of the selected significance threshold value, then the null hypothesis will be rejected at the selected significance level. Here if observed equivalently, the statistical significance is present in the critical region. It is similar to that of the guilty verdict. Thus by proving it wrong we can accept the alternative hypothesis. 


If the p-value is not less than that of the selected significance threshold value, then the evidence is not sufficient to provide the conclusion. Here if observed equivalently, the statistical significance is present outside the critical region. It is similar to that of the not guilty verdict. And the researcher gives an extra effort in the cases where the p-value is nearer to the significance value.


Examples:

  • Lady Tasting Tea: It is a famous example of hypothesis testing statistics. Here a female colleague of Ronald Fisher named Dr. Muriel Bristol was able to tell if the milk or tea is added to the cup at first. Where Fisher provided the eight cups in which four have variety in random order, one of the people can ask what is the probability of the number she got correct by chance. According to the null hypothesis meaning, the lady has no such ability. Here the test statistic is a simple count of the selected cups and the count obtained is four. According to the conventional probability criterion that is < 5%, among the seventy possible combinations, a pattern corresponds to one out of all the possible combinations. Thus the lady identified every cup that was considered statistically correct. Hence the alternative hypothesis was not required to conclude the result.

  • Courtroom Trial: The test procedure can be compared to the criminal trial, that is a defendant is not considered as a criminal until the guilt is not proven. And the guilt can be proven only when the prosecutor has enough evidence about the defendant’s crimes. 

Here, there are two hypotheses, H0 : The defendant is not guilty, it represents the null. Hypothesis. H1 : The defendant is guilty, it represents the alternative hypothesis. 

When the error is unlikely to occur, the hypothesis of innocence can be rejected. Because an innocent defendant cannot be considered a convict. Thus, this type of error is called an error of the first kind. Thus the occurrence of this error is rare. And the error of the second kind is more common. 



H0 is True

Not Guilty

H1 is True

Truly Guilty

Accept the null hypothesis

The decision taken is correct.

The decision taken is wrong and a type II error occurs.

Accept alternative hypothesis

The decision taken is wrong and a type I error occurs.

The decision taken is correct.


The trial of the criminal can be decided by two processes, evidence vs threshold or guilty vs not guilty. In one view, the performance of the prosecution is being judged and in the other view, the guilt of the criminal is judged. Thus the hypothesis test can be considered as the judgment of evidence or judgment of hypothesis.


Conclusion:

Hypothesized means to make the hypothesis. The statistical hypothesis can be testable depending on the data collected as a result of the collection of random variables. In some set of possible joint distributions, a set of data can be considered as a realized value of the joint probability distribution. The hypothesis test can be done in two ways. In most of the applications, these two methods are complementary in nature; that is one hypothesis will be the negation of the other.

FAQs (Frequently Asked Questions)

1. Mention the Types of a Hypothesis Test.

Ans: There are two types of a hypothesis test:

  • Null Hypothesis: It is denoted as H₀.

  • Alternative Hypothesis: IT is denoted as H₁ or Hₐ.

2. What are the Two Errors That Can Be Found While Performing the Null Hypothesis Test?

Ans: While performing the null hypothesis test there is a possibility of occurring two types of errors,

  • Type-1: The type-1 error is denoted by (α), it is also known as the significance level. It is the rejection of the true null hypothesis. It is the error of commission.

  • Type-2: The type-2 error is denoted by (β). (1 - β) is known as the power test. The false null hypothesis is not rejected. It is the error of the omission.