The region of rejection of h0 is called

A rejection region (a.k.a. critical region) in a Null Hypothesis Statistical Test is a part of the parameter space such that observing a result that falls under it will lead to the rejection of a the null hypothesis. In hypothesis testing usually a significance test is performed and the rejection region is given in the form of a statistic such as a t score or a z score. It can just as easily be given in terms of the actual (non-standardized) parameter value.

A rejection region has a one-to-one correspondence with the significance threshold as it is simply a more technical way to express it. Since a standardized score is expressed in terms of standard deviations (e.g. 1.644 SD or .1.96 SD) it is directly specifying the area under the parameter distribution which will lead to rejection which can easily be turned into alpha (α) during the planning stage or a p-value after the test is completed by computing the cumulative distribution function.

After the test is completed, the observed p-value is compared to the critical Z score which is at the boundary of the rejection region and if it falls with the region the null hypothesis is rejected.

Notably, if a value is not within the rejection region this does not automatically mean that the null hypothesis can be accepted: it merely states that there is not enough evidence to warrant rejecting it.

Like this glossary entry? For an in-depth and comprehensive reading on A/B testing stats, check out the book "Statistical Methods in Online A/B Testing" by the author of this glossary, Georgi Georgiev.

Hypotheses Testing is the testing of claims or statements or assumptions which we encounter very frequently in our daily lives. Although the truth or falsity of a claim or statement is never known unless we examine the entire population but practically it is almost impossible to do so hence, we take a random sample from the population under study and use information contained in the sample for decision making (whether a claim is true or false).

Some of the below examples are indicative but the list is exhaustive.

(i) The air-conditioning of certain brand saves up to 20% on electricity bill or

(ii) The motorbike of certain brand gives 60 km/liter mileage or

(iii) A soap of certain brand kills the bacteria and viruses to 99.9% or,

(iv) 95 out of 100 customers recommend brand A than brand B etc.

And in most of these cases, claims be verified statistically.

One more point to note here is that the decision maker is only interested in making inference about the population parameters, not in estimating the value of parameters.

Definition — Hypothesis

A hypothesis is a statement or a claim or an assumption about the value of a population parameter (e.g., mean, median, variance, proportion, etc.).

In hypothesis testing problems first and foremost is:

1. identifying the hypothesis to be tested and writing it in the words.

2. the identified hypothesis to be represented in symbolical form.

3. writing the complement or opposite of the claim or statement in symbolical form

For e.g. –

Motorbike Consumer may write the claim or postulate the hypothesis as

“The Motorbike of this brand gives the average mileage of 60 km/liter.”

Let μ represent the average mileage then our hypothesis in symbolical form becomes

“Motorbike of this brand gives µ = 60 km /liter.”

Complement of hypothesis be becomes

“Motorbike of this brand gives µ ≠ 60 km /liter.”

Simple and Composite Hypotheses

Simple Hypotheses — Hypothesis which specifies only one value or exact value of the

population parameter. e.g. — µ = 60 km/liter as referenced in the above motorbike example.

Also, µ1 = µ2 or µ1 — µ2 = 0, σ1 = σ2 or σ1- σ2 = 0 etc.

Composite Hypotheses — Hypothesis specifies not just one value but a range of values that the population parameter may assume. e.g. — µ1 ≠ µ2 or µ1 > µ2 or µ1 — µ2 > 0, σ1 ≠ σ2 or

σ1- σ2 ≠ 0 etc.

Null and Alternative Hypotheses

For hypothesis testing the claim and its complement are formed in such a way that they cover all possibility of the value of population parameter (i.e., entire sample space).

The rule of thumb for formulating Null and Alternative hypothesis is:

The statement containing equality is the null hypothesis.

That is, the hypothesis which contains symbols = or ≤ or ≥ is taken as null hypothesis

and the hypothesis which does not contain equality i.e., contains

≠ or < or > is taken as alternative hypothesis.

The null hypothesis in general is denoted by H0 and alternative hypothesis is denoted by H1.

In our example of motorbike above, the claim is μ = 60 km/liter and its complement

is μ ≠ 60 km/liter.

H0: μ = 60 km/liter and H1: μ ≠ 60 km/liter

As defined by Prof. R.A. Fisher,

“A null hypothesis is a hypothesis which is tested for possible rejection under the assumption that it is true.”

The hypothesis which complements to the null hypothesis is called alternative hypothesis.

Considering the sample under study, we assume that the null hypothesis is true until there is sufficient evidence to prove that it is false.

A very important point to highlight here is :

To say that null hypothesis is true it’s important to consider all observations of the sample under study. But for rejecting null hypothesis, only one observation indicating violation is enough.

For e.g. — if someone wants to test that humans have two hands then to prove that this is true, we must check all the humans whereas to prove that it is false, only one human who has one hand or no hand is sufficient.

The alternative hypothesis again has two types:

(i) Two-sided (tailed) alternative hypothesis

(ii) One-sided (tailed) alternative hypothesis

Let us not dive into one tailed and Two tailed tests as of now. I plan to write a separate tutorial to cover these with examples.

Critical Region

Let us consider, X1, X2,…, Xn as a random sample drawn from a population having unknown population parameter , p. The collection of all possible values of X1, X2,…, Xn is a set called sample space(S) and a particular value of X1, X2,…, Xn represents a point in that space.

In order to test a hypothesis, the entire sample space is partitioned into two disjoint sub-spaces, say, x and S — x = x̅. If calculated value of the test statistic lies in x , then we reject the null hypothesis and if it lies in x̅, then we do not reject the null hypothesis. The region x is called a “rejection region or critical region” and the region x̅ is called a “non-rejection region”.

The concept of critical region can be better understood from the below example:

Suppose 100 assembly items in the lot is subjected to tests in the batches of 10 for categorizing as qualified assembly item or non — qualified assembly item in a factory and each sample containing 10 units is rated on a scale of 1 to 10. Hence average ratings for each sample can be 10 (maximum)

If average of rating is 8 or more, the item within sample is qualified else non-qualified.

Let the sample be denoted by X1, X2, …, X10 .

Suppose we select one item randomly out of each sample and want to test if the selected item has a rating of 8 or more. So we can take the null and alternative hypotheses as

H0: Selected assembly item is qualified

H1: Selected assembly item is non-qualified

For taking the decision about the item, we define as the average of the ratings of all the 10 items of the sample.

n — No. of items in the sample

The range of T10 is 0 ≤ T10 ≤ 10. Now, we divide the whole space (0–10) into

two regions as non-qualified (less than 8) and qualified (greater than or equal to 8) Here, 8 is the critical value which separates the qualified and non-qualified regions.

non-qualified (less than 8) — is called as Rejection region or Critical Region or Non-acceptance region.

qualified (greater than or equal to 8) — is called as Non-Rejection region or Acceptance region.

Now basic structure of the procedure of testing of hypothesis needs two

regions :

(i) Region of rejection of null hypothesis H0

(ii) Region of non-rejection of null hypothesis H0

The region of rejection is called critical region which is denoted by α (alpha) corresponding to a cut-off value in a probability distribution of test statistic.

Here I am considering this to be two-tailed tests. However based on your hypothesis framing you can convert it to right or left tailed tests. The process is same. Only instead of considering α (alpha)/2 catering to each tail, take α (alpha) for right or left tailed tests.

The below graph gives a clear idea.

Level of Significance

The probability of type-I error is known as level of significance of a test. It is also called the size of the test or size of critical region, denoted by α. As we have discussed above that if calculated value of the test statistic lies in rejection(critical) region, then we reject the null hypothesis and if it lies in non-rejection region, then we do not reject the null hypothesis. Also, we note that when H0 is rejected then automatically the alternative hypothesis H1 is accepted.

Critical value(s) or cut-off value(s) for a known test statistic by expressing it in the form of some well-known distributions like Z, t, F etc.

p-value

Use of p-value is very popular because of the following two reasons:

· Most of the statistical software provides p-value rather than critical value.

· p-value provides more information compared to critical value as far as

rejection or non- rejection of H0 is concerned.

To explain clearly, we will consider our above example of motorbike and reiterate the hypotheses statement.

H0: μ = 60 km/liter and H1: μ ≠ 60 km/liter

Specified value of population mean = μ0= 60 km/liter,

Population standard deviation = σ = 5 km/liter,

Sample size = n = 50,

Sample mean = x̅ = 62 km/liter

Level of significance αis set as α = 0.01 (= 1 % level).

Test statistic =

Since sample size is large (n = 50 > 30) so by central limit theorem the sampling distribution of test statistic approximately follows standard normal distribution

i.e. Test statistic ~ N(0,1)

For two tailed test, T = 2.83/2 = 1.415

The critical value or cut-off value for standard normal distribution from Z-table for Two tailed test at α= 0.01 is Zα = 2.58.

To calculate Zα (Two tailed test)

1 — α = 1–0.01 = 0.99

α/2 = 0.495

After searching 0.495 in the below Z-table extracts, yields 2.58.

Now, to take the decision about the null hypothesis, we compare the calculated value of test statistic with the critical value.

Since calculated value of test statistic (= 1.415) is lesser than critical value (= 2.58), that means calculated value of test statistic lies in non-rejection region at 1% level of significance. So

we accept null hypothesis.

Thus, we conclude that sample does provide us sufficient evidence in favor of the claim so we may say that the average mileage of motorbike is 60 km/liter.

Hence here the Null Hypotheses is accepted at level of significance (α = 0.01). But there will be a level of significance(α) wherein the null hypothesis can be rejected. This smallest level of significance (α) is known as “p-value”.

Hence the p-value is the smallest value of level of significance(α) at which a null

hypothesis can be rejected using the obtained value of the test statistic.

To take the decision about the null hypothesis based on p-value, the p-value is compared with level of significance (α) and if p-value is equal or less than α

then we reject the null hypothesis and if the p-value is greater than α , we do not reject the null hypothesis.

Relation between Confidence Interval and Testing of Hypothesis

Confidence Interval = (1 — Level of Significance (α)) * 100 %

Hence for Hypotheses testing, if the value of the parameter specified by the null hypothesis lies in this confidence interval then we do not reject the null hypothesis and if this specified value does not lie in this confidence interval, then we reject the null hypothesis.

Thus, we can use three approaches (critical value, p-value and confidence interval) for taking decision about null hypothesis.

What does it mean to reject the H0?

What is the region of acceptance of H0?

What is it called when you reject the null hypothesis?

Which test is the region of rejection?