Five Steps of Hypothesis Testing 

The basic logic of hypothesis testing is to prove or disprove the research question.  By only allowing an error of 5% or 1% and making correct decisions based on statistical principles, the researcher can conclude that the result must be real if chance alone could produce the same result only 5% of the time or less. These five steps consists of all the decisions a researcher needs to make in order to answer any research question using an inferntial statistical test.


The first step is to state the research problem in terms of a question that identifies the population(s) of interest to the researcher, the parameter(s) of the variable under investigation, and the hypothesized value of the parameter(s). This step makes the researcher not only define what is to be tested but what variable will be used in sample data collection.  The type of variable (or combination of variables as in relationship type research questions) whether categorical, discrete or continuous further defines the statistical test which can be performed on the collected data set.

For example:
Is the mean first salary of a newly graduated student equal to $30,000?

The population of interest is all students who have just graduated.  The parameter of interest is the mean and the variable salary is continuous.  The hypothesized value of the parameter, the mean, is $30,000. Since the parameter is a population mean of a continuous variable variable, this suggests a one sample test of a mean. 


The second step is to state the research question in terms of a null hypothesis (H0) and a alternative hypothesis (HA). The null hypothesis is the population parameter, µ = $30,000 (H0: µ = $30,000). The alternative hypothesis is the population parameter does not equal $30,000 ( HA: µ NE $30,000). This HA suggests a two-tailed test as NE $30,000 can be less than $30,000 or more than $30,000.  Sometimes the alternative hypothesis is stated in terms of a direction such as less than or greater than a value such at $30,000. A directional HA calls for a one-tailed test, in the direction stated in the HA.

The next part of step 2 is to select a significance level (Type I error) typically alpha is used at the .05 or the .01 level. A good researcher will also not neglect Type II error. In this step we are not only setting up our research question in terms of statistical hypotheses, but we must evaluate whether all the assumptions appropriate for the statistical test have been met. 

H0: µ =  $30,000 
HA: µ NE $30,000  alpha=.05

Test assumptions are 1) the population is normally distributed or sample size is approximately >=30 and 2) the sample we have used to collect the data was drawn randomly from the population. If these test assumptions have not been meet, then data collection should be reevaluated or continued under caution. 


The third step is to calculate a statistic analogous to the parameter specified by the null hypothesis. If the null hypothesis is defined by the parameter µ, then the statistics computed on our data set would be the mean (xbar) and the standard deviation (s).  A histogram of our sample data set gives us our best approximation of what we expect our population distribution to look like.

Since the best estimate of µ is xbar, our sample mean, the test statistic is based on a distribution of sample means, the sampling distribution of the mean, xbar, with n, sample size, equal to the number of data values used to compute xbar.  We have hypothesized from the research question the mean of this distribution and want to see if our sample mean is close to this value.  To determine where our sample mean fits on this sampling distribution, we convert our sample mean, xbar, to a z-score.  Thus the test statistic would be :

          z = xbar-µ (hypothesized)
                standard error of xbar

The standard error of xbar (point estimate) is s, the sample standard deviation, divided by square root of n, the sample size since the population standard deviation is unknown. 

Suppose we randomly sampled 100 high school seniors and determined their salary of their first job. The sample mean salary, xbar, was $29,000 with a standard deviation of $6,000.  Since sample size is >30, we don't have to worry about whether the population is normally distributed (Central Limit Theorem).  The test statistic would be:
            z = $29,000 - $30,000 = -$1,000 = -1.667
                  $6,000/sqrt(100)       $600 


The fourth step is to calculate the probability value (often called the p-value) which is the probability of the  test statistic for both tails since this this two-tailed test.  The probability value computed in this step is compared with the significance level selected in step 2. If the probability is less than or equal to the significance level, then the null hypothesis is rejected.  If the probability is greater than the significance level then the null hypothesis is not rejected. When the null hypothesis is rejected, the outcome is said to be "statistically significant"; when the null hypothesis is not rejected then the outcome is said be "not statistically significant."  If the outcome is statistically significant, then the null hypothesis is rejected in favor of the alternative hypothesis. 

P(z> 1.667) =.048 + P(z< -1.667)=.048, the p-value is .048+.048=.096

Since this value is greater than alpha=.05 selected when we set up out hypotheses, we accept the null hypothesis, H0: µ =  $30,000.

If we wish to use a rejection region of alpha=.05 (.025 in each tail) to determine if we accept or reject the null hypothesis, the cut-off z-score would be -1.96 and 1.96.  If our test statistic is >=1.96 or <= -1.96, then we would reject the null hypothesis at alpha=.05. We can say that our test statistic (transformed into a z-score) is in the rejection region.  In this example, our test statistic, z=-1.667 for our test statistic does not fall in the rejection region (sometimes called the acceptance region), so we must accept the null hypothesis. 


The fifth and final step is to describe the results and state correct statistical conclusions in an understandable way. The conclusions consists of two statments-one describing the results of the null hypothesis and the other describing the results of the alternative hypothesis. The first statement should state as to whether we accepted or rejected the null hypothesis and for what value of alpha or p-value for our test statistic. The second statement should answer the research question proposed in step 1 stating the sample statistic collected which estimated the parameter we hypothesized.


Accept the null hypothesis at alpha=.05 or p-value of .096.  Based on a sample mean of $25,000, the mean salary of a newly graduated student does not equal $30,000.