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Steps in solving hypothesis (Hypotheses and testing statistics)

Steps in solving hypothesis (Hypotheses and testing statistics)

06/03/2021
Steps in solving hypothesis (Hypotheses and testing statistics)
Edubomb Edubomb

Steps in testing hypothesis 

testing of the hypothesis consist of numbers of steps which are discussed below:

Null-Hypothesis:

in inferential statistics, the null hypothesis is a general statement that uses a test hypothesis. It is a hypothesis of no difference. It means the null hypothesis shows no difference between the actual value and our theoretical expectation. For example, if you want to test whether the average height of the girls is equal to 6 feet or not then the null hypothesis could be average height is equal to 6 feet. Again, if you want to test the average height of the girls and boys in the classroom then the null hypothesis could be if the height of boys and girls are equal. It is denoted by the H0 and written as

According to the above first example, we can write

Null hypothesis H0:µ=6 feet where µo can be regarded as  population parameter and μ is a sample parameter.

Similarly, in the case of the second example, we can write

 

Hence the null hypothesis shows a hypothesis of no difference.

 

Alternative Hypothesis:

it is a hypothesis complementary to the null hypothesis. It means it is the hypothesis of difference. If the null hypothesis is rejected then another hypothesis is set up called the alternative hypothesis. It is used as an alternative to the null hypothesis. hence in the interest of the researcher. For example, if you want to test whether the average height of the girls is equal to 6 feet or not then the alternative hypothesis could be average height is not equal (greater than or less than) to 6 feet. Again, if you want to test the average height of the girls and boys in the classroom then an alternative hypothesis could be if the height of boys and girls are not equal (or greater or less). It is denoted by the H1 and written as

According to the above first example, we can write

Alternative hypothesis

Where μ0 = 6ft which theoretical expectation over which we are testing

Test statistics:

Test statistics are tools that are used to test the given hypothesis. There are different types of test statistics that are different for different test models. Hence choosing appropriate tools only gives the correct result. It means we need to adopt a rational decision in choosing the appropriate test statistics. In simple, test statistics is given by

 

Types of error in testing hypothesis:

There are two types of errors we can commit during the testing of the hypothesis. These errors are due to our decision. When we fail to make rational decisions then such errors are seen. Sometimes errors are obvious. It is because a small sample cannot represent the true value of the entire population.  Sometimes sample statistics cannot measure the population parameter correctly due to which two types of error are observed namely I) type one error and ii) type two error.

Type I error:  when we reject the truth hypothesis then a type I error has occurred. It means when we reject those hypotheses which are correct and truth then type I error is committed. It is also called the α error.

Type II error: when we accept the false hypothesis then a type II error has occurred. It means when we accept those hypotheses which are incorrect i.e. false then type II error is committed. It is also called β error.

We can represent these errors along with hypothesis testing as follows

State of nature

Decision

Accept H0

Reject H0

Hypothesis H0 true

No error

Type I error

Hypothesis H0 false

Type II error

No error

It is obvious to commit an error when there Is lacks incomplete information of the sample parameter. When there is no complete information of the sample parameter then we cannot obtain the exact result and hypothesis testing end with an error. from the above table, it can be concluded that

Type I error: rejecting the correct null hypothesis

Type II error: accepting the false null hypothesis

To understand such an error let us take one example that there 100 eggs in a basket. Among of these 10 eggs are damaged. If we draw the sample out of 100 eggs and found exactly 10 damaged eggs then we must buy a basket of all eggs. If we make the wrong decision and reject to buy the baskets of eggs then such error is regarded as the type I error. Because we are rejecting the truth hypothesis.

On the other hand, suppose that there are 100 eggs in a basket and among which there are only 10 eggs good. If we draw samples and found all 10 goods eggs. Then we do not need to buy the eggs because all remaining eggs are damaged. If we buy such a basket of eggs then we can commit a type II error. Because we are accepting the false hypothesis.

To reduce such errors researcher, need to make the appropriate decisions based on available information. We can reduce the impact and degree of error by making an appropriate decision. Such errors are more often coincide with the level of significance. reducing a level of significance help to eliminate the type I error. It should be noted that it is not possible to reduce both types of I errors as well as type II errors simultaneously.

 

Level of significance (α):

The significance level, also denoted as alpha or α, is the probability of rejecting the null hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference. Hence in simple, the level of significance is the percentage value of type I error committed by the researcher during calculation. It means the level of significance I.e. α is the probability of error that occurred during testing of hypothesis when we reject the truth null hypothesis.  For example, a 5% level of significance means we are ready to accept a 5% error out of 100% when we reject the truth null hypothesis. It means we are ready to accept a 5% error during the rejection of the truth null hypothesis. In another word, 0.05 probability of the error is acceptable. higher the value of the α higher will be the value of type I error. hence to reduce the value of type I error, we need to reduce the value of the level of significance α. From the above explanation, we can write

α = probability of occurrence of type I error

   = probability of rejecting the null hypothesis when it is true.

Confidence level (1-α):

We define (1-α) as a confidence interval. It is the probability of accepting H0 when H0 is true. It is just opposite to the level of significance. Higher the value of confidence level lower will be the value of type I error. If there is a 5% level of significance then there will be a 95% confidence level.

Power of test (β):

it is the probability of committing the type II error. It means the power of test the probability of accepting the null hypothesis when value of the null hypothesis is false. thus, β can be defined as the

β = probability of occurrence of type II error

   = probability of accepting the null hypothesis when it is false.

Both errors are harmful to the researcher. But comparatively, type II error is more harmful than type I error. Because accepting false things than rejecting true things is more dangerous. To reduce the impact of error we can increase the sample size. Higher the sample size higher will be the level of confidence and lower will be the value of both type I error as well as the value of the type II error.

Region of acceptance and region of rejection:

The region for acceptance and region for rejection can be studied with the help of the bell-shaped curve. The region taken by the type I error is called the region of rejection. It is also called the critical region. It means it is region where there is an α level of significance. In this region, we commit errors. Hence it is a region where we reject the null hypothesis when it is true.


The region other than the rejection region is called the acceptance region. This region is covered by the level of significance I.e. (1-α). Hence it is the region where we accept the null hypothesis when it is true. Following figures are examples for the acceptance region and rejection region along with the one-tail test, and two-tail tests.

 

 

 

Two-tailed and one-tailed tests:

There are two terms to describe the tests. these terms are used to distinguish the exact relation between the sample parameter and population parameter. It means they may be equal, may greater than or less than, and may be not equal.

Two-tailed tests:

When the value of the sample parameter and hypothesized value of the population parameter are not equal then we use a two-tailed test. It means when we have to test the difference in parameter (not equal) then we use a two-tail test. In this test, critical regions are located on both sides of the probability distribution curve or bell-shaped curve.

Right tail and left tail test:

When the value of sample parameter and hypothesized value of population parameter are either greater than or less than we use right tail and left tail test. In this case the critical region is located on the right side for the right-tailed test and the left for left tailed tests.

Similarly left tailed test is given by the

 





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