Hypothesis testing is realized based on one or more samples drawn from population or populations, so our decision may be right or wrong. There are 2 types of error as classification in Table 1.

Table 1 Types of error in hypothesis testing

		Hypothesis
		Right	Wrong
Decision	Accept	Correct decision	Error type 2
	Reject	Error type 1	Correct decision

In hypothesis testing, we can commit two types of error:

• type 1 : reject a true hypothesis; probability of this type of error is `alpha`.
• type 2 : accept a false hypothesis; probability of this type of error is `beta`.

Example : After a microbiological control of 5000 cans, 18 cans are contaminated. Company C announces that 99,6 % of their products are safe. Department of Quality Control D analyses 10 cans of this company and there is one can contaminated. So D decides that the announcement of C is wrong. In this case D commits type 1 error.

Example : 9 % of pupils of school A are ranked as grade A, and this school announces that there are more than 10 % of its pupils are ranked as grade A. The Department of Education D controls 20 pupils of this school, there are 3 pupils that are ranked as grade A, and D decides that the announcement of school A is right. In this case D commits type 2 error.

 

To reject a statement, we can use the rare event rule:

“if, under given assumptions, the probability of an event is very small, we consider that this event does not happen”.

Therefore, to accept an hypothesis H we can:

• prove that H is true by proving that probability `P("H")` is large (greater than 0,95 for example), then accept H. In this case we can commit type 2 error.
• or we construct an alternative hypothesis Ha, then calculate `P("Ha")`. If this probability is very small, we reject Ha, it means that we accept H. In this case we can commit type 1 error.

The second method is more frequently used.

 

In general, hypothesis testing consists of the following steps:

• From the context, identifying the parameter (or parameters) of interest.
• Stating the null hypothesis Ho and appropriate alternative hypothesis Ha.
• Determining test statistic `t` and distribution of `t`.
• Based on problem context such as confidence level, sample size, ... we can determine critical value (or values) `t`*. This one will separate the values of `t` to two regions, one corresponds to accepting Ho, other corresponds to reject Ho.
• From the data obtained, determining `t_o`.
• Comparing `t_o` and `t`*, determining the region which `t_o` belongs.
• If `t_o` belongs to rejection region (also known as critical region), we reject Ho and accept Ha. On the contrary, we accept Ho.

Note : When we construct Ho and Ha, we must be careful, because there are cases when Ho is false, Ha is not true.

Example

• Ho : `M=123`   and Ha : `M!=123`. If Ho is false then Ha is true.
• Ho : `M=123`   and Ha : `M>123`. If Ho is false ; Ha can be false too (`M=111` for example).

 

As stated above, `t`* divides all the value of `t` to 2 regions: acceptance region and rejection region (also known as critical region). Depending on Ha, there are three main cases.

Case 1

Rejection region is in the right of `t`* (Fig. 1) : to reject Ho, `t_o>t`* with `t`* is the percentage point corresponding to `alpha`.

Fig. 1 Rejection region is in the right of `t`*

This cases is usually applied when Ha has the form of `X>a` in which `X` is tested parameter, `a` is a value.

Case 2

Rejection region is in the left of `t`* (Fig. 2) : to reject Ho, `t_o< t`* with `t`* is the percentage point corresponding to `1-alpha`.

Fig. 2 Rejection region is in the left of `t`*

This cases is usually applied when Ha has the form of `Y< b` in which `Y` is tested parameter, `b` is a value.

Case 3

Rejection region consists of two zones in two sides of `t_1`* and `t_2`* (Fig. 3): to reject Ho,
`t_o< t_1`* or `t_o>t_2`* with `t_1`* is the percentage point corresponding to (`1 – alpha//2)` and `t_2`* is the percentage point corresponding `alpha//2`.

Fig. 3 Two-sided rejection region

This cases is usually applied when Ha has the form of `Z!=c` in which `Z` is tested parameter, `c` is a value.

Cases 1 and 2 are also known as one-sided (or one-tailed) hypothesis testing, case 3 is also known as two-sided (or two-tailed) hypothesis testing.

Note : When probability density function of `t` is even (Fig. 4):

`t_(1-alpha)=-t_alpha`(1)

Fig. 4 Percentage point when probability density function is even

In case of two-sided hypothesis testing, the condition to reject Ho is: `|t_o|>t_alpha`*.

 

As discussed above, `t_o` is determined from data of sample, and it is used to characterized sample. Consider case 1, when rejection region is in the right of `t`*. `p` value is defined by formula:

`p=int_(t_o)^oo f(t)dt`(2)

in which `f(t)` is probability density function of test statistic.

`p` value is also illustrated in Fig. 5.

Fig. 5 `p` value illustration

So, `p` is the highest probability to obtain the result of sample in case that Ho is right. The smaller `p` is, the higher ability Ho is rejected. When `t_o` in the rejection region, `p< alpha`.

Therefore, we have another way to reject or accept Ho: comparing `p` with `alpha`. If `p` is smaller than `alpha`, we reject Ho, if `p` is greater than `alpha`, we accept Ho.

Note that this conclusion apply for all three cases, both one-sided and two-sided hypothesis testing.