Hypothesis Testing
Although we normally base our decisions on knowledge about the problems that we are trying to solve, we always have to accept some risk of making an error. Our knowledge is never perfect and complete. Depending on the consequences of the decision, we require different degrees of confidence that the decision is correct. Non-critical questions such as whether to take an umbrella with us when we leave for work need only low levels of confidence. While critical decisions like weighing the evidence in a murder case in court, or diagnosing a certain disease in a hospital, require a high degree of confidence. The uncertainty in the process of making a decision is related to the fact that the probability of making an error is not zero.
Hypothesis testing gives us the guidelines for choosing between alternatives by either controlling or minimizing the error associated with the decision. The simplest case for a decision is the 'yes-or-no' question. In court, for example, the jurors have to decide "guilty or not guilty". These statements are two hypotheses. The normal assumption is "not guilty", in statistics this is called the null hypothesis. It is what we normally assume. Then there is an alternative hypothesis, in our example "guilty". We will accept this alternative hypothesis only when there is convincing evidence.
Hypothesis testing can be summarized in the questions: is it reasonable to assume that the value of a population parameter is equal to / larger than / less than x? This question can be applied in various situations. The population parameter can be either the mean or the variance. The value of x is either specified on the basis of prior knowledge, or an estimated parameter from another population.[1]
Hypothesis testing is always a five-step procedure:
- Formulation of the null and the alternative hypotheses
- Specification of the level of significance
- Calculation of the test statistic
- Definition of the region of rejection
- Selection of the appropriate hypothesis
Hypotheses are tentative explanations of a principle operating in nature. Hypothesis is a claim against a population parameter (µ, σ, etc.)
In statistics a hypothesis is a claim or statement about a property of a population.[2]
A tentative assumption made in order to draw out and test its logical or empirical consequences.[3]
Hypothesis testing is the use of statistics to determine the probability that a given hypothesis is true. The usual process of hypothesis testing consists of four steps.
1. Formulate the null hypothesis Ho (commonly, that the observations are the result of pure chance) and the alternative hypothesis Ha (commonly, that the observations show a real effect combined with a component of chance variation).
2. Identify a test statistic that can be used to assess the truth of the null hypothesis.
3. Compute the P-value, which is the probability that a test statistic at least as significant as the one observed would be obtained assuming that the null hypothesis were true. Note: The smaller the P-value, the stronger the evidence against the null hypothesis.
4. Compare the p-value to an acceptable significance value α (sometimes called an alpha value). If p ≤ α, that the observed effect is statistically significant, the null hypothesis is ruled out, and the alternative hypothesis is valid.[4]
