AN INTRODUCTION TO HYPOTHESIS TESTING

Inferential statistics deals with parameter estimation and hypothesis testing. Parameter estimation addresses such questions as 1) What is the average weight of the packages of grass seed distributed by XYZ Inc.? 2) Among all the units produced by a machine, what is the proportion of defective units? 3) How strong is the relationship/correlation between the TOEFL scores of the students at XYZ University and their grades in English? In essence, parameter estimation “asks for” a value. On the other hand, hypothesis testing answers “yes-or-no” questions such as: 1) Is the average weight of the packages of grass seed distributed by XYZ Inc. greater than 460 grams? 2) Is it reasonable to conclude that less than 3% of the units produced by the machine are defective? 3) Is there a relationship/correlation between TOEFL scores of the students and their grades in English?

Hypothesis testing is a powerful tool in quantitative research. It is used to verify whether a statement concerning the population under study is supported by sufficient evidence. Hypothesis itself means an idea or explanation of something that is based on a few known facts but that has not yet been proved to be true or correct (Oxford Advanced Learner’s Dictionary, 9^th Edition, 2015). Another definition of hypothesis is a statement about a population parameter subject to verification (Lind, 2012). Hypothesis testing is a procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement (Lind, 2012). In addition, a statistical hypothesis is an assertion or conjecture concerning one or more populations (Walpole, 1993).

Below is the 5-step procedure to test hypotheses.
1. State null and alternate hypotheses
2. Select a level of significance
3. Identify the test statistic
4. Formulate a decision rule
5. (After collecting sample data), decide whether or not the null hypothesis is rejected.

Null vs Alternate Hypothesis
The first step in hypothesis testing is to state a pair of hypotheses, i.e. null hypothesis and alternate hypothesis. Null hypothesis is a statement about the value of a population parameter developed for the purpose of testing numerical evidence. Alternate hypothesis (also called research hypothesis) is a statement that is accepted if the sample data provide sufficient evidence that the null hypothesis has to be rejected. Null hypothesis always contains equality sign (=). On the contrary, alternate hypothesis cannot contain the sign. Three possible signs for alternate hypothesis are >, <, or ≠. For example, to test whether the mean weight of the packages of grass seed distributed by XYZ Inc. greater than 460 grams, we state the following pair of hypotheses.

H₀: μ = 460 grams

H₁: μ > 460 grams

To test whether the proportion of defective units is less than 3%, the hypotheses are as follows.

H₀: π = 0.03

H₁: π < 0.03

Level of Significance
The hypothesis testing results in one of the two possible outcomes, that is, rejecting the null hypothesis or accepting it. Either we reject or accept it, we are exposed to the risk of making a false decision. It is due to the chancy nature in the sampling process. If we reject the null hypothesis, we have the risk of making false decision while the null hypothesis is true. In statistics, if we reject the null hypothesis while it is true, a type I error is committed. The probability that we make such error is called the level of significance, denoted by α. On the other hand, it is possible that we accept the null hypothesis while it is false. In this case, a type II error occurs. The probability that this type of error occurs is usually denoted by β.

Test Statistic
Test statistic is the value used to determine whether the null hypothesis must be rejected. It is determined from sample information. Different problems may call for different test statistics. To discuss this further, let’s go back to the previous examples: 1) Is the average weight of the packages of grass seed distributed by XYZ Inc. greater than 460 grams? 2) Is it reasonable to conclude that less than 3% of the units produced by the machine are defective? 3) Is there a relationship/correlation between TOEFL scores of the students and their grades in English? To address the first question, there are three available test statistics to select, i.e. , , or . To learn more about these test statistics, please refer to the this link. The test statistic suitable to the second question is binomial variable X with p = p₀ or . The other question requires another test statistic than previously mentioned.

Decision Rule
A decision rule is a statement of the specific conditions under which the null hypothesis is rejected and the condition under which it is not rejected. There are two approaches to formulate this, i.e. by using p-value or critical value. When the p-value approach is applied, the null hypothesis is rejected if the computed p-value is less than or equal to the desired significance level (α). In using the critical value, the null hypothesis is rejected if the value of the test statistic falls in the critical region (= rejection area). The critical region is determined by the critical value, which is the dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.

Possible Decisions
Every hypothesis testing may result in a rejection or an acceptance of the statement in the null hypothesis. If the test recommends the rejection of H₀, the test is said to be significant. Otherwise, the test is not significant. Regarding the significance of a statistical test, we should note the following: 1) If the test is significant, we are confident that the statement in the alternate hypothesis is correct. Why are we confident? Because the probability of making such error (= incorrectly rejecting the null hypothesis) is only as small as α. 2) If the test is not significant, it does not mean that the statement in the null hypothesis is correct. It only means that there is no sufficient evidence to reject the null hypothesis. In addition, we don’t know the probability of incorrectly accepting the null hypothesis as we do when we reject the H₀.