Testing of Hypothesis

Introduction

The objective of testing hypotheses (Testing of Statistical Hypothesis) is to determine if an assumption about some characteristic (parameter) of a population is supported by the information obtained from the sample.

The terms hypothesis testing or testing of the hypothesis are used interchangeably. A statistical hypothesis (different from a simple hypothesis) is a statement about a characteristic of one or more populations such as the population mean. This statement may or may not be true. The validity of the statement is checked based on information obtained by sampling from the population.
Testing of Hypothesis refers to the formal procedures used by statisticians to accept or reject statistical hypotheses that include:

i) Formulation of Null and Alternative Hypothesis

Null hypothesis

A hypothesis formulated for the sole purpose of rejecting or nullifying it is called the null hypothesis, usually denoted by H₀. There is usually a “not” or a “no” term in the null hypothesis, meaning that there is “no change”.

For Example, The null hypothesis is that the mean age of M.Sc. students is 20 years. Statistically, it can be written as $H_0:\mu = 20$. Generally speaking, the null hypothesis is developed for testing.
We should emphasize that if the null hypothesis is not rejected based on the sample data we cannot say that the null hypothesis is true. In another way, failing to reject the null hypothesis does not prove that the $H_0$ is true, it means that we have failed to disprove $H_0$.

For the null hypothesis, we usually state that “there is no significant difference between “A” and “B”. For example, “the mean tensile strength of copper wire is not significantly different from some standard”.

Alternative Hypothesis

Any hypothesis different from the null hypothesis is called an alternative hypothesis denoted by $H_1$. Or we can say that a statement is accepted if the sample data provide sufficient evidence that the null hypothesis is false. The alternative hypothesis is also referred to as the research hypothesis.

It is important to remember that no matter how the problem is stated, the null hypothesis will always contain the equal sign, and the equal sign will never appear in the alternate hypothesis. It is because the null hypothesis is the statement being tested and we need a specific value to include in our calculations. The alternative hypothesis for the example given in the null hypothesis is $H_1:\mu \ne 20$.

Simple and Composite Hypothesis

If a statistical hypothesis completely specifies the form of the distribution as well as the value of all parameters, then it is called a simple hypothesis. For example, suppose the age distribution of the first-year college student follows $N(16, 25)$, and the null hypothesis is $H_0: \mu =16$ then this null hypothesis is called a simple hypothesis, and if a statistical hypothesis does not completely specify the form of the distribution, then it is called a composite hypothesis. For example, $H_1:\mu < 16$ or $H_1:\mu > 16$.

ii) Level of Significance

The level of significance (significance level) is denoted by the Greek letter alpha ($\alpha$). It is also called the level of risk (as there is the risk you take of rejecting the null hypothesis when it is true). The level of significance is defined as the probability of making a type-I error. It is the maximum probability with which we would be willing to risk a type-I error. It is usually specified before any sample is drawn so that the results obtained will not influence our choice.

In practice 10% (0.10) 5% (0.05) and 1% (0.01) levels of significance are used in testing a given hypothesis. A 5% level of significance means that there are about 5 chances out of 100 that we would reject the true hypothesis i.e. we are 95% confident that we have made the right decision. The hypothesis that has been rejected at a 0.05 level of significance means that we could be wrong with a probability of 0.05.

Selection of Level of Significance

In Testing of Hypothesis, the selection of the level of significance depends on the field of study. Traditionally 0.05 level is selected for business science-related problems, 0.01 for quality assurance, and 0.10 for political polling and social sciences.

Type-I and Type-II Errors

Whenever we accept or reject a statistical hypothesis based on sample data, there are always some chances of making incorrect decisions. Accepting a true null hypothesis or rejecting a false null hypothesis leads to a correct decision, and accepting a false hypothesis or rejecting a true hypothesis leads to an incorrect decision. These two types of errors are called type-I errors and type-II errors.
type-I error: Rejecting the null hypothesis when it is ($H_0$) true.
type-II error: Accepting the null hypothesis when $H_1$ is true.

iii) Test Statistics

The third step of Testing the Hypothesis is a procedures that enable us to decide whether to accept or reject the hypothesis or to determine whether observed samples differ significantly from expected results. These are called tests of hypothesis, tests of significance, or rules of decision. We can also say that test statistics is a value calculated from sample information, used to determine whether to reject the null hypothesis.

The test statistics for mean $\mu$ when $\sigma$ is known is $Z= \frac{\overline{X}-\mu}{\frac{\sigma}{\sqrt{n} } }$, where Z-value is based on the sampling distribution of $\overline{X}$, which follows the normal distribution with mean $\mu_{\overline{X}}$ equal to $\mu$ and standard deviation $\sigma_{\overline{X}}$ which is equal to $\frac{\sigma}{\sqrt{n}}$. Thus we determine whether the difference between $\overline{X}$ and $\mu$ is statistically significant by finding the number of standard deviations $\overline{X}$ from $\mu$ using the Z statistics. Other test statistics are also available such as $t$, $F$, and $\chi^2$, etc.

iv) Critical Region (Formulating Decision Rule)

It must be decided before the sample is drawn under what conditions (circumstances) the null hypothesis will be rejected. A dividing line must be drawn defining “Probable” and “Improbable” sample values given that the null hypothesis is a true statement. Simply a decision rule must be formulated having specific conditions under which the null hypothesis should be rejected or should not be rejected. This dividing line defines the region or area of rejection of those values that are large or small that the probability of their occurrence under a null hypothesis is rather remote i.e. Dividing line defines the set of possible values of the sample statistic that leads to rejecting the null hypothesis called the critical region.

One-tailed and two-tailed tests of significance

In testing of hypothesis if the rejection region is on the left or right tail of the curve then it is called a one-tailed hypothesis. It happens when the null hypothesis is tested against an alternative hypothesis having a “greater than” or a “less than” type.

and if the rejection region is on the left and right tail (both sides) of the curve then it is called a two-tailed hypothesis. It happens when the null hypothesis is tested against an alternative hypothesis having a “not equal to sign” type.

v) Making a Decision

In this last step of testing hypotheses, the computed value of the test statistic is compared with the critical value. If the sample statistic falls within the rejection region, the null hypothesis will be rejected or otherwise accepted. Note that only one of two decisions is possible in hypothesis testing, either accept or reject the null hypothesis. Instead of “accepting” the null hypothesis ($H_0$), some researchers prefer to phrase the decision as “Do not reject $H_0$” “We fail to reject $H_0$” or “The sample results do not allow us to reject $H_0$”.

Data Analysis in R Language

Hypothesis Testing Frequently Asked Questions

What is a statistical hypothesis?
What is a null hypothesis?
What is an alternative hypothesis?
How null and alternative hypotheses are mathematically represented?
What is the level of significance (level of risk)?
What are type-I errors and type-II errors?
What is the test statistics for one sample?
What is the test statistics for the two samples?
What is the critical region?
How decision is made in hypothesis testing?
What is a simple and composite hypothesis?
What is the calculated test value?

The P-value is a probability, with a value ranging from zero to one. It is a measure of how much evidence we have against the null hypothesis. P-value is a way to express the likelihood that $H_0$ is not true. The smaller the p-value, the more evidence we have against $H_0$. Here we will discuss about P-value and Its Interpretation.

P-value Definition

The largest significance level at which we would accept the null hypothesis. It enables us to test the hypothesis without first specifying a value for $\alpha$. OR

The probability of observing a sample value as extreme as, or more extreme than, the value observed, given that the null hypothesis is true.

P-value Interpretation

In general, the P-value interpretation is “If the P-value is smaller than the chosen significance level then $H_0$ (null hypothesis) is rejected even when it is true. If the P-value is larger than the significance level $H_0$ is not rejected”.

If the P-value is less than

0.10, we have some evidence that $H_0$ is not true
0.05, strong evidence that $H_0$ is not true
0.01, Very strong evidence that $H_0$ is not true
0.001, extremely strong evidence that $H_0$ is not true

Misinterpretation of a P-value

Many people misunderstand P-values. For example, if the P-value is 0.03 then it means that there is a 3% chance of observing a difference as large as you observed even if the two population means are the same (i.e. the null hypothesis is true). It is tempting to conclude, therefore, that there is a 97% chance that the difference you observed reflects a real difference between populations and a 3% chance that the difference is due to chance. However, this would be an incorrect conclusion. What you can say is that random sampling from identical populations would lead to a difference smaller than you observed in 97% of experiments and larger than you observed in 3% of experiments.

Note that p-values are a valuable tool in hypothesis testing, but they should be used thoughtfully and in conjunction with other analyses.