A hypothesis is a statement about the value of a population parameter. The population of interest is so large that for various reasons it would not be feasible to study all the items, or persons, in the population. Analternative to measuring or interviewing the entire population is to take a sample from the population of interest. We can, therefore, test a statement to determine whether the empirical evidence does or does not support the statement. Hypothesis testing starts with a statement, or assumption, about a population parameter – such as the population mean. As noted, this statement is referred to as a hypothesis.
A hypothesis might be that the mean monthly commission of salespeople in retail computer stores is $2,000. We cannot contact all these salespeople to ascertain that the mean is in fact $2,000. The cost of locating and interviewing every computer salesperson in the whole country would be exorbitant. To test the validity of the assumption (population mean = $2,000), we must select a sample from the population consisting of all computer salespeople, calculate sample statistics, and based on certain decision rules accept or reject the hypothesis.
A sample mean of $1,000 for the computer salespeople would certainly cause rejection of the hypothesis. However, suppose the sample mean is $1,995. Is that close enough to $2,000 for us to accept the assumption that the population mean is $2,000? Can we attribute the difference of $5 between the two means to sampling (chance), or is that difference statistically significant? Hypothesis testing is a procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement and should not be rejected, or is unreasonable and should be rejected.
The null hypothesis and the alternative hypothesis
The null hypothesis is a tentative assumption made about the value of a population parameter. The alternative hypothesis is a statement that will be accepted if our sample data provide us with ample evidence that the null hypothesis is false.
Five-step procedure for testing a hypothesis
There is a five-step procedure that systematizes hypothesis testing. The
steps are: Step 1. State null and alternative hypotheses.
Step 2. Select a level of significance.
Step 3. Identify the test statistic.
Step 4. Formulate a decision rule.
Step 5. Take a sample, arrive at decision (accept H 0 or reject H 0 and accept H 1 ).
The first step is to state the hypothesis to be tested. It is called the null hypothesis, designated H 0 , and read “ H sub-zero”. The capital letter H stands for hypothesis, and the subscript zero implies “no difference”. The null hypothesis is set up for the purpose of either accepting or rejecting it. To put it another way, the null hypothesis is a statement that will be accepted if our sample data fail to provide us with convincing evidence that it is false.
It should be emphasized at this point that if the null hypothesis is accepted based on sample data, in effect we are saying that the evidence does not allow us to reject it. We cannot state, however, that the null hypothesis is true. That means, accepting the null hypothesis does not prove that H 0 is true – to prove without any doubt that the null hypothesis is true, the population parameter would have to be known. To actually determine it, we would have to test, survey, or count every item in the population and this is usually not feasible.
It should also be noted that we often begin the null hypothesis by stating: “there is no significant difference between…”. When we select a sample from a population, the sample statistic is usually different from the hypothesized population parameter. We must make a judgment about the difference: is it a significant difference, or is the difference between the sample statistic and the hypothesized population parameter due to chance (sampling)?
To answer this question, we conduct a test of significance. The alternative hypothesis describes what you will believe if you reject the null hypothesis. It is often called the research hypothesis, designated H 1 , and read “ H sub-one”, so the alternative hypothesis will be accepted if the sample data provide us with evidence that the null hypothesis is false. The level of significance
The next step, after setting up the null hypothesis and alternative hypothesis, is to state the level of significance. It is the risk we assume of rejecting the null hypothesis when it is actually true. The level of significance is designated , the Greek letter alpha.
There is no one level of significance that is applied to all studies involving sampling. A decision must be made to use the 0.05 level (often stated as the 5 percent level), the 0.01 level, the 0.10 level, or any other level between 0 and 1. Traditionally, the 0.05 level is selected for customer research projects, 0.01 for quality assurance, and 0.10 for political polling – and the chosen level is the probability of rejecting the null hypothesis when it is actually true.
The test statistic
The test statistic is a value, determined from sample information, used to accept or reject the null hypothesis. There are many test statistics: z , t , and others. The decision rule; acceptance and rejection regions
A decision rule is simply a statement of the conditions under which the null hypothesis is accepted or rejected. To accomplish this, the sampling distribution is divided into two regions, aptly called the region of acceptance and the region of rejection. The region or area of rejection defines the location of all those values that are so large or so small that the probability of their occurrence under a true null hypothesis is rather remote.
Chart 4.1 portrays the regions of acceptance and rejection for a test of significance (a one-tailed test is being applied and the 0.05 level of significance was chosen). Note in Chart 4.1:
The value 1.645 separates the regions of acceptance and rejection (the value 1.645 is called the critical value).
The area of acceptance includes the area to the left of 1.645. The area of rejection is to the right of 1.645.
Thus, the critical value is a number that is the dividing point between the region of acceptance and the region of rejection.
Chart 4.1. Sampling distribution for the statistic z ; regions of acceptance and rejection for a right-tailed test; 0.05 level of significance