9.1 Null and Alternative Hypotheses
In a hypothesis test, sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we:
- Evaluate the null hypothesis, typically denoted with H0. The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ≤ or ≥)
- Always write the alternative hypothesis, typically denoted with Ha or H1, using not equal, less than or greater than symbols, i.e., (≠, <, or > ).
- If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis.
- Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.
9.2 Outcomes and the Type I and Type II Errors
In every hypothesis test, the outcomes are dependent on a correct interpretation of the data. Incorrect calculations or misunderstood summary statistics can yield errors that affect the results. A Type I error occurs when a true null hypothesis is rejected. A Type II error occurs when a false null hypothesis is not rejected.
The probabilities of these errors are denoted by the Greek letters α and β, for a Type I and a Type II error respectively. The power of the test, 1 – β, quantifies the likelihood that a test will yield the correct result of a true alternative hypothesis being accepted. A high power is desirable.
9.3 Distribution Needed for Hypothesis Testing
In order for a hypothesis test’s results to be generalized to a population, certain requirements must be satisfied.
When testing for a single population mean:
- A Student's t-test should be used if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with an unknown standard deviation.
- The normal test will work if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large.
When testing a single population proportion use a normal test for a single population proportion if the data comes from a simple, random sample, fill the requirements for a binomial distribution, and the mean number of success and the mean number of failures satisfy the conditions: np > 5 and nq > 5 where n is the sample size, p is the probability of a success, and q is the probability of a failure.
9.4 Full Hypothesis Test Examples
The hypothesis test itself has an established process. This can be summarized as follows:
- Determine H0 and Ha. Remember, they are contradictory.
- Determine the random variable.
- Determine the distribution for the test.
- Draw a graph and calculate the test statistic.
- Compare the calculated test statistic with the Z critical value determined by the level of significance required by the test and make a decision (cannot reject H0 or cannot accept H0), and write a clear conclusion using English sentences.