Tests of hypotheses

Hypothesis testing is a strategy for drawing statistical conclusions from population data. It is a tool for testing assumptions and determining how likely something is within a particular level of accuracy. Hypothesis testing allows you to see if the outcomes of an experiment are correct.

How is Hypothesis Testing related to Statistics?

Hypothesis testing makes use of population sample data to draw helpful inferences about the population probability distribution. It employs several hypothesis-testing approaches to test an assumption made about the data. The null hypothesis is either rejected or not rejected as a result of the hypothesis testing.

Definition

Hypothesis testing is a statistical tool used to determine whether or not the results of an experiment are relevant. It entails developing a null hypothesis and an alternate hypothesis. Both of these hypotheses are always mutually incompatible. That is, if the null hypothesis is true, the alternative hypothesis is false, and likewise. 

Null Hypothesis

The null hypothesis is a short mathematical declaration that indicates there is no difference between the two options. In other terms, there is no distinction between some data qualities. This hypothesis holds that the results of a study are solely determined by chance. It is symbolized by the symbol H0. Hypothesis testing is performed to determine whether or not the null hypothesis can be rejected. 

Alternative Hypothesis

The alternative hypothesis is a hypothesis that differs from the null hypothesis. It is used to demonstrate that the results of an experiment are the result of a real impact. It denotes a statistically significant difference between the two possible results and can be represented as H1 or Ha. 

P-Value

The p-value is used in hypothesis testing to show whether or not the findings of a test are statistically significant. It also represents the likelihood of falsely rejecting or not failing to reject the null. This value is always a positive integer between 0 and 1. The p-value is compared to an alpha level, often known as the level of significance. The alpha level is described as the tolerable risk of rejecting the null hypothesis mistakenly. The alpha level is chosen among 1% and 5%.

Formula

Different forms of hypothesis testing are used to assess if the null hypothesis can be rejected or not, depending on the type of data provided and its magnitude. The following are the hypothesis testing formulas for various major test statistics:

• z=x-μn.x : sample mean, μ=population mean, σ= population standard deviation, n= size of the sample.
• t=x-μn.s : sample standard deviation.
• X2=(Oi–Ei)2Ei.Oi is the observed value and Ei is the expected value.

Types of Hypothesis Testing

• Z-Test

A z-test is a hypothesis testing method that is employed with a high sample size (n 30). When the population standard deviation is known, it is used to evaluate if there is a discrepancy between the population mean and the sample mean. It is also useful for comparing the means of two samples. It is employed in the computation of the z-test statistic.

• z=x-μn.x
• z=(x1–x2)-(1–2)12n1+22n2
• T-test

The t-test is another hypothesis testing procedure that is employed with small sample size. It can also be used to compare the sample mean to the population mean. The population standard deviation, on the other hand, is unknown. The sample standard deviation, on the other hand, is known. The t-test can also be used to compare two means sets.

• z=x-μn.x
• z=(x1–x2)-(1–2)12n1+22n2
• Chi-Square

The Chi-square test is a hypothesis testing procedure used to determine whether or not the variables in a population are independent. When the test statistic is chi-squared distributed, it is utilized.

One-Tailed Hypothesis Testing

When the refusal zone is just in one direction, one-tailed hypothesis testing is used. Because the results can only be assessed in one way, it is also known as directional hypothesis testing. This form of testing is further subdivided into right and left-tailed tests.

Right-Tailed 

 The higher tail test is another name for the right-tail test. This test determines if the population parameter is higher than a certain value. The following are the null and alternative hypotheses for this test:

• H0: population parameter value
• H1: population parameter> value

Left-Tailed

The lower tail test is another name for the left tail test. It is used to see if the population parameter is less than a certain value. The following are the hypotheses for this hypothesis testing:

• H0: population parameter value
• H1: population parameter< value

Two-Tailed 

The critical zone in this hypothesis testing approach is located on both sides of the sample distribution. It’s also referred to as a non-directional hypothesis testing strategy. The two-tailed test is used to examine whether a population parameter is expected to be different from some value. The hypotheses can be organized as follows:

• H0: population parameter= value
• H1: population parameter value