Stating Hypothesis in Testing of Hypothesis
Stating Hypothesis in Testing of Hypothesis
Stating Hypothesis, Relation between Interested Conjecture or alternative hypothesis and a Null conjecture or null hypothesis.
Statistics is the science of observations and therefore, one becomes interested to test his build up conjectures about the universe, based on the observations one has collected from the universe through some sampling design. Generally such conjectures are translated into determination of values of some or certain characteristics, also called indicators or variables that are measured on the sampling units. With the help of these indicators, a test statistic is built up to test if the conjecture is supported by the sample (or data). How is it done? We will discuss.
The researcher builds a hypothesis that he wants to get tested. Like suppose he wants to say Regimen A is better than Regimen B for a particular disease. Then he will conduct an experiment. He will take a set of people on which Regimen A will be applied. He will take another set of people almost similar to the group he had taken for Regimen A. Similarity of both groups will be on account of other health and disease parameters. Then he will apply Regimen B on this group. He will observe mortality in both the groups.He will calculate the proportion of people who died in both the groups during the same time period. Let these proportions are p1 for the group on which Regimen A is administered and p2 for the group on which Regimen B is administered. His conjecture that Regimen A is better than Regimen B, now can be written as hypothesis p1 < p2. This hypothesis of researchers is known as alternative hypothesis. It is written as H1 or Ha. So we write Ha : p1 < p2. Let us not puzzle with its name for the time being. Though this is the hypothesis, researcher is interested in, but even then it is called an alternative hypothesis.
As such, it is not easy to test the researcher's hypothesis, because we don't know any test-statistics having a known distribution under Ha even applying some assumptions. Moreover this hypothesis Ha stipulates that the difference between p1 and p2 exists and is greater than zero. So it might contain many distributions for a test statistic, if we could design such a test-statistic. What is done under this condition is to create a competing null hypothesis denoted by H0 as p1 = p2. Thus if this competing hypothesis H0 : p1 = p2 gets defeated by the observations in hand, it will then be predicted that Ha is supported. We don't say that Ha is proved, but we say as H0 is rejected, H1 is supported. Under H0 : p1 = p2, we know that there is a Z statistic, [ assuming sample size is large, meaning greater than 30 and both samples are independent,] given by
Z = [ (p1-p2) - E(p1-p2) ] / σ ~ N (0,1);
Under H0, we have
E(p1-p2) = 0 ;
σ^2 = [ (p1q1/n1)+(p2q2/n2) ]
= [p^. q^ (1/n1 + 1/n2) ]
Where :
p^ = [ (n1p1+n2p2) /(n1+n2); q^ = 1-p^; q1= p1; q2= p2;
n1 is the sample size of the group on which Regimen A is administered.;
n2 is the sample size of the group on which Regimen B is administered.
In such a situation, we can always test Z for its observed value Z0 whether it is more likely to come from N(0,1) or it is less likely keeping Ha p1 < p2 in mind.
As Ha is lesser inequality and test statistic Z has a numerator as p1-p2 , the test would be one tail test in the left direction. The area of rejection will be on the left hand side. Observed low value Zo of Z than the critical value Zα at a particular level of significance α in the left direction will support Ha.
In the left direction here means critical value would be negative. Observed Z viz Zo more negative than critical value Zα will reject null hypothesis H0. As a result Ha will be supported.
Standard Normal Tables are generally given for critical regions in the right hand side. So we can translate the above problem as Ha : p2 > p1 and Z statistics as Z = [ (p2-p1) - E(p2-p1) ] / σ ~ N (0,1); This Z can be tested for right hand side one tail test.
Table Standard Normal Distribution of Z. Pr ( Z ≤ Zα ) = 1- α or Pr ( Z ≥ Zα ) = α
Similar to the above, we can have another example that a researcher wants to test if the mean height of an adult in a city is μ0, he does some sample collection. Here if heights are at large variation from μ0 on either side, then that should be tested by him. Hence his concern will be to test alternative hypothesis Ha: μ ≠ μ0. His null hypothesis would be H0: μ = μ0. We know that if sample size is large enough then sample mean follows normal distribution with mean as population mean, here μ0 under H0, and variance can be estimated by S^2. The S^2 is a sum of squares of deviations of sample observations from sample mean divided by n minus one, the n is the size of the sample.
The Z statistics would be Z = (Xbar - μ0) / S ~ N(0,1) under H0. Xbar is sample mean.
The test procedure will be two tail test. Once the critical value is decided, say α , the half area in right and half area in left, both equal to α/2, will form the critical regions. An observed Zo of Z statistics falling in either of the regions will qualify for rejecting null hypothesis H0 and supporting alternative hypothesis Ha.
