A statistical hypothesis is an assertion or conjecture about the distribution of one or more random variables.
If a statistical hypothesis completely specifies the distribution, it is referred to as a simple hypothesis; if not, it is referred to as a composite hypothesis.
General Steps in Hypothesis Testing
The testing of a statistical hypothesis is the application of an explicit set of rules for deciding whether to accept the hypothesis or to reject it.
The method of conducting any statistical hypothesis testing can be outlined in six steps:
Decide on the null hypothesis H0
The null hypothesis generally expresses the idea of no difference.
The symbol we use to denote a null hypothesis is H0•
Decide on the alternative hypothesis H1
The alternative hypothesis, which we denote by H1, expresses the idea of some difference. Alternative hypotheses may be one-sided or two-sided.
Usually the setting of the problem determines the alternative even before the data has been collected.
Calculate the appropriate test statistic
This is a value that we will calculate from the sample data.
Decide on the significance level or the critical P-value
All hypothesis testing is liable to errors. There are two basic kinds of error:
Type I error: Reject H0 when it is, in fact, true; the probability of committing a type I error is denoted by α.
Type II error: Reject H1 when it is, in fact, true; the probability of committing a type II error is denoted by β.
The objective in all hypothesis testing is to set the Type I error level, also known as the significance level, at a low enough value, and then to use a test statistic which minimizes the Type II error level for a given sample size.
As we fix the Type I error level, it is best to devise the test in such a way that the Type I error is most serious, in terms of cost.
A critical P -value is the probability that is set by the person doing the test; it is the threshold for the P-value that the tester will use to decide whether the sample is unusual enough, compared to the hypothesized population, to indicate that the null hypothesis should be rejected in favor of the alternative.
The P-value or critical region of size α
The calculated test statistic is compared to the sampling distribution that the statistic would have if the null hypothesis were true. The comparison is summarized into a probability called a P-value: this is the probability, if the null hypothesis is true, that the statistic would be at least as far from the expected value as it was observed to be in the sample.
The P-value ranges from 0.0 to 1.0. As it approaches 0.0, it indicates that the sample is a rare outcome if the population is as hypothesized.
The closer the P-value is to zero, the stronger the evidence against the null hypothesis.
When we are testing the null hypothesis H0: θ = θ0 against the two-sided alternative hypothesis H1: θ =/= θ0, the critical region consists of both tails of the sampling distribution of the test statistic. Such a test is a two-tailed test.
On the other hand, if we are testing the null hypothesis Ho: θ = θo against one-sided alternative H1: θ θ0, the critical regions are the left tail or right tail of the sampling distribution of the test statistic respectively.
Statement of conclusion
A decision is made based on the size of the P-value. When the P-value is small (i.e. less than the critical P-value), we reject the null hypothesis. When it is not small (greater than the critical P-value), we accept the null hypothesis.
In the same way, if the value of the test statistic falls in the critical region, we reject the null hypothesis.
The conclusion should, as far as possible, be devoid of statistical terminology.
However the significance level should be stated. The assumption of this test is that the variable is approximately normally distributed. This assumption is less critical the larger the sample size.
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