Hypotheses Testing

A Hypothesis
A hypothesis is a tentative explanation that accounts for a set of facts and can be tested by further investigation. It may be defined as follows:

• A hypothesis may be precisely defined as a tentative proposition suggested as a solution to a problem or as an explanation of some phenomenon (Ary, Jacobs, and Razavieh, 1984).
• A hypothesis is a conjectural statement of the relation between two or more variables (Kerlinger, 1956).
  A hypothesis is a formal statement that presents the expected relationship between an independent and dependent variable. (Creswell, 1994).
• Hypothesis relates theory to observation and observation to theory. (Ary, Jacobs, and Razavieh, 1984).
• Hypotheses are relational propositions. (Kerlinger, 1956).
• A hypothesis is a tentative explanation that accounts for a set of facts and can be tested by further investigation.
• A hypothesis is a logical supposition, a reasonable guess, and an educated conjecture. It provides a tentative explanation for a phenomenon under investigation (Leedy and Ormrod, 2001).

Importance of hypothesis
A hypothesis is important because it guides the research. An investigator may refer to the hypothesis to direct his or her thought process toward the solution of the research problem or sub-problems. The hypothesis helps an investigator to collect the right kinds of data needed for the investigation. Hypotheses are also important because they help an investigator locate information needed to resolve the research problem or sub-problems (Leedy and Ormrod, 2001)

Types of hypothesis
A hypothesis (plural hypotheses) is a precise, testable statement of what the researchers predict will be the outcome of the study. This usually involves proposing a possible relationship between two variables: the independent variable (what the researcher changes) and the dependent variable (what the research measures). In research, there is a convention that the hypothesis is written in two forms, the null hypothesis, and the alternative hypothesis (called the experimental hypothesis when the method of investigation is an experiment). The hypotheses can be expressed in the following ways:

Null Hypothesis
The null hypothesis states that there is no relationship between the two variables being studied (one variable does not affect the other). It states results are due to chance and are not significant in terms of supporting the idea being investigated. A null hypothesis also bears the features:

• Ho states the opposite of what the experimenter would expect or predict.
• The final conclusion of the investigator will either retain a null hypothesis or reject a null hypothesis in favor of an alternative hypothesis.
• Not rejecting Ho does not really mean that Ho is true. There might not be enough evidence against Ho.
• States the assumption (numerical) to be tested
• Begin with the assumption that the null hypothesis is TRUE
• Always contains the ‘=’ sign

Alternative Hypothesis
The alternative hypothesis states that there is a relationship between the two variables being studied (one variable has an effect on the other). It states that the results are not due to chance and that they are significant in terms of supporting the theory being investigated. An alternative hypothesis is a statement that suggests a potential outcome that the researcher may expect (H1 or HA). An alternative hypothesis encompasses the followings:

• Is it the opposite of the null hypothesis?
• Challenges to the status quo
• Never contains just the ‘=’ sign
• Is generally the hypothesis that is believed to be true by the researcher?
• Comes from prior literature or studies.
• It is established only when a null hypothesis is rejected.
• Often an alternative Hypothesis is the desired conclusion of the investigator.

The two types of alternative hypotheses are directional hypotheses and non-directional Hypotheses. A non-directional alternative hypothesis states that the null hypothesis is wrong. A non-directional alternative hypothesis does not predict whether the parameter of interest is larger or smaller than the reference value specified in the null hypothesis. Whereas a directional alternative hypothesis states that the null hypothesis is wrong, and also specifies whether the true value of the parameter is greater than or less than the reference value specified in the null hypothesis. The advantage of using a directional hypothesis is increased power to detect the specific effect you are interested in. The disadvantage is that there is no power to detect an effect in the opposite direction.

Derivation of Hypothesis
Inductive:
The researcher notes the observations of behavior, thinks about the problem, turns to literature for clues, makes additional observations, derives probable relationships, and hypothesizes an explanation. A hypothesis is then tested. It may be limited in scope and can lead to unconnected findings, which could explain little about the research.

Deductive:
The researcher begins by selecting a theory and derives a hypothesis leading to deductions derived through symbolic logic or mathematics. These deductions are then presented in the form of statements accompanied by an argument or a rationale for the particular proposition.

Hypothesis testing
The following 5 steps are followed when testing hypotheses.

1. Specify H0 and HA – the null and alternative hypotheses
(a)      H0:    E(X) = 10       (b)   H0:    E(X) = 10       (c)      H0:    E(X) = 10
HA:    E(X) <> 10             HA:    E(X) < 10                  HA:    E(X) > 10

Note that, in example (a), the alternative values for E(X) can be either above or below the value specified in H0. Hence, a two-tailed test is called for – that is, values for HA lie in both the upper and lower halves of the normal distribution. In example (b), the alternative values are below those specified in H0, while in example (c) the alternative values are above those specified in H0. Hence, for (b) and (c), a one-tailed test is called for.

2. Determine the appropriate test statistic
A statistic to test the hypothesis. A ‘test statistic’ should follow a probability distribution (Z, t, χ2 or F) or there should be a threshold value based on which H0 is ejected or accepted.

3. Determine the critical region
The region in which if the ‘test statistic’ falls, H0 is rejected. The critical region should be in the tail area. P-value- the tail probability; usually, the probability that the ‘test statistic’ is greater (less) than its calculated value. If the p-value is less than a given ‘level of significance’ (say 0.05), H0 is rejected.

• When p value > .10 → the observed difference is “not significant”
• When p value ≤ .10 → the observed difference is “marginally significant”
• When p value ≤ .05 → the observed difference is “significant”
• When p value ≤ .01 → the observed difference is “highly significant”

4. Compute the value of the test statistic
Say, a value of z calculated on the basis of a sample result is called a computed z-value, and is denoted by the symbols zc or simply z.

5. Make a decision
If the calculated value of the test statistic falls in the critical region, then H0is rejected. When the calculated value lies in the acceptance region, H0is not rejected.

Decision problem
How do we choose between H0and HA? The standard procedure is to assume H0is true – just as we presume innocent until proven guilty. Using probability theory, we try to determine whether there is sufficient evidence to declare H0false. We reject only when the chance is small that H0is true. Since our decisions are based on probability rather than certainty, we can make errors.

Type I Error:
A type I error occurs when the null hypothesis (H0) is wrongly rejected. For example, A type I error would occur if we concluded that the two drugs produced different effects when in fact there was no difference between them.

Type II Error:
A type II error occurs when the null hypothesis H0, is not rejected when it is in fact false. For example, A type II error would occur if it were concluded that the two drugs produced the same effect, that is, there is no difference between the two drugs on average, when in fact they produced different ones.

In conclusion, we need the null hypothesis to determine if there is a difference between the groups being tested or not. Without it, we would be swamped with possibilities making it almost impossible to test.

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