Hypothesis Testing (Z-test and t-test for Single Mean)

Hypothesis Testing (Z-test and t-test for Single Mean)

INTRODUCTION
Hypothesis testing refers to the procedure that enables us to decide on the basis of sample results whether the difference between the observed sample statistic (X) and the hypothetical parameter value (u) is significant or not. We use the Z-test for large samples and t-test for small samples for studying this differences. The test procedure is based on some statistic T. This statistic T is called the test statistic.


BASIC CONCEPTS OF HYPOTHESIS TESTING

Following basic concepts are used in the study of tests of hypothesis:

(1) Large and Small Sample: Generally, a sample consisting of more than 30 items is called a large sample where as a sample consisting of less than orequal to 30 items is called a small sample.
(2) Hypothesis: A hypothesis (or statistical hypothesis) is defined as a definite statement concerning the population parameter. Thus the statement that the mean of the population is u (pronounced as Mu) is an example of statistical hypothesis.
(3) Null Hypothesis: The null hypothesis is the hypothsis which is tested for possible acceptance/rejection under the assumption that it is true. It is generally represented by Ho. For example, if we want to find out whether a particular drug is effective in curing malaria, we will take the null hypothesis that the drug is not effective in curing malaria. The rejection of a null hypothesis indicates that the difference have statistical significance and the acceptance of the null hypothesis indicates that the difference is due to chance.
(4) Alternative Hypothesis (H₁): The hypothesis opposite to null hypothesis is known as alternative hypothesis and is denoted by H₁. For example, if Ho: H₁=μ then, H₁: μμ orμ<μ or μ> μο
(5) Type I and Type II Errors: Testing of a null hypothesis involves the following two types of errors: Type I Errors and Type II Errors.
(i) Type I Errors: Type I Errors are made when we reject the null hypothesis though it is true. That is, it amounts to rejecting a lot when it is good.
(ii) Type II Errors: These are made when we accept the null hypothesis though it is actually wrong. That is, it amounts to accepting a lot when it is bad.


 

  

  
 
  
  

  
Ho (Accept)
  
  

  
Ho (Reject)
  
 
 

  

  
Ho (True)
  
  

  
Correct Decision
  
  

  
Type I Errors
  
 
 

  

  
Ho (False)
  
  

  
Type II Errors
  
  

  
Correct Decision
  
 

(6) Level of Significance: In testing a given hypothesis the maximum probability with which we would be willing to risk a type I Error is called the level of significance. It is generally pre-determined. Frequent use is made of 5% and 1% levels of significance. Others levels of significance may also be used. A 5% level of significance means that there are 5 chances in 100 that the difference could arise by chance and the null hypothesis is wrongly rejected. We are accurate in 95% of the cases. A 1% level of significance means that there is only 1 chance out in 100 that the null hypothesis is wrongly rejected. We are accurate in 99% of the cases. 25015

(7) Critical Region or Rejection Region: The critical region or rejection region is the region of the standard normal curve corresponding to a pre-determined level of significance. The region under the normal curve which is not covered by the rejection region is known as Acceptance Region. Thus, the statistic which leads to the rejection of null hypothesis Ho gives us a region known as Rejection Region or Critical Region. While those which lead to acceptance of Ho give us a region called as Acceptance Region.

(8) One Tailed Test and Two Tailed Test: A test of any statistical hypothesis where the alternative hypothesis is expressed by the symbol (<) or the symbol (>) is called a one tailed test since the entire critical region lies in one tail of the distribution of the test statistic. The critical region for all alternative hypothesis containing the symbol (>) lies entirely on the right tail of the distribution while the critical region for an alternative hypothesis containing a less than (<) symbol lies entirely in the left tail. The symbol indicates the direction where the critical region lies. A test of any statistical hypothesis where the alternative is written with a symbol '#' is called a two-tailed test, since the critical region is split in to two equal parts, one in each tail of the distribution of the statistic.
(9) Critical Value: The critical values of the standard normal variate (Z) for both the two-tailed and one tailed tests at different level of significance are very often required in hypothesis testing. 

Procedure of Testing of Hypothesis

Testing of a hypothesis passes through the following steps:

(1) Set up a null hypothesis: It is denoted Ho. Null hypothesis assumes that difference between any two values to be compared is not significant.

(2) Set up a suitable level of significance: A suitable level of significance is determined to test the null hypothesis. In practice, 5% significant level is used.

(3) Set up a suitable test of statistic: A number of test statistics like Z, t, x², F, etc., may be applied to test the null hypothesis. It is decided only the basis of available information.
(4) Doing necessary calculation: After selecting appropriate statistic, computations relating to the test statistics are made and values are worked out.
(5) Making Decision: In the process of hypothesis testing, results are interpreted at the final stage. For this purpose, we compare the computed value of a test statistic with the table value at a pre-determined level of significance. If computed value is greater that the table value at 5% or 1% level of significance, then null hypothesis is rejected. In such a situation, the sample does not represent population.
 Important Sampling Tests for Hypothesis Testing

The important sampling tests for hypothesis testing are:
(1) Z-test
(2) t-test

    

Both these tests are based on the assumption that the data is considered to be normally distributed. (1) Z-test: Z-test is a statistical hypothesis test that follows a normal distribution. Z-test is applied when we are handling large samples (n ≥ 30). This test is suitably applied if the standard deviation of the population is known.

Uses (or Applications) of Z-test

Z-test can be used for many purpose. The various applications of Z-test are as follows: 
(i) Z-test for single mean
(ii) Z-test for difference of two means
(iii) Z-test for single proportion
(iv) Z-test for difference of two proportions


https://examstudie.blogspot.com/