HYPOTHESIS TESTING

Presented by

Mohammad Adil Khan

DIFFINITION

HYPOTHESIS* Such a statement or assumption

which may or may not be true.* A hypothesis is a tentative explanation

that accounts for a set of facts and can be tested by further investigation.

* In statistics, a hypothesis is a claim or statement about a property of a population.

Components Of Hypothesis Test Null Hypothesis

• It is Generally denoted by H0

• H0 is any hypothesis which is to be tested for possible rejection under the assumption that it is true.

• H0 must contain condition of equality i.e. = , ≤ , or ≥ .

Alternative Hypothesis

• It is denoted by H1 or HA

• The statement that must be true if the null hypothesis is false/rejected.

• HA must contain condition of inequality i.e.

≠ , > , or < .

• Opposite to Null Hypothesis.

Test Statistics

• Its provides a basis for testing a null hypothesis.• A value computed from the sample data that is used

in Making the decision about whether to reject or accept the null hypothesis.

• Test statistic is denoted by Z. and is given by

x

x

x μz

σ

Z =Standard deviation of random variable

Value of random variable – mean of random variable

Critical Region• Set of values of the test statistic that

would cause a rejection of the null hypothesis.

• Values consistent with H0 is called acceptance region.

• Values not consistent with H0 is called rejection region.

Critical Region

Critical ValueValue or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis

Critical value(Z-score)

Reject H0

Fail to Reject H0

Type 1 Error

• The mistake of rejecting the null hypothesis when it is true.

(alpha) represent the probability of type 1 error i.e. is the probability of rejecting H0 when it is true.

=P (type 1 error)=(reject H0 / H0 is true)

Type ІІ Error

• Failing to reject Null Hypothesis when it is false.

represent the probability of type ІІ error i.e. probability of accepting H0 when H0 is false.

=P(type ІІ error)=P(accept H0 / H0 is false)

TABLE

True situationDECISION

H0 is true

H0 is false

ACCEPT H0

Reject H0

0r accept H1

Correct Decision(No error)

Wrong Decision(or type-1 error

Wrong DecisionType-2 error

Correct Decision(No error)

Significance Level

• denoted by • the probability that the test statistic will fall

in the critical region when the null hypothesis is actually true.

• common choices are 0.05, 0.01, and 0.10. =5% means there are about 5 chances

in 100 of rejecting a true null hypothesis. • In other words we say that we are 95%

confident in making the correct decision.

Two-tailed, Right-tailed,Left-tailed Tests

• The tails in a distribution are the extreme regions bounded by critical values.

• A hypothesis for which the entire rejection is located in only one of the two tails, either in the left tail or right tail of the probability distribution of the test statistic, is called one tail test or one sided test.

• If the rejection region is divided equally between the two tails of the probability distribution of the test statistics, this referred to as a two tailed test or two sided test.

• It is important to note that one tailed test and two tailed test differ only in location of the critical region, not in the size.

Right-tailed Test

H0: µ 100

H1: µ > 100

Values that differ From 100

100

Reject H0

Fail to reject H0

Left-tailed Test

H0: µ 100

H1: µ < 100

Values that differFrom 100 100

Reject H0

Fail to Reject H0

Two-tailed Test

H0: µ = 100

H1: µ 100

Means > or <

is divided equally between the two tails of the critical Region

Values that differ from 100

Reject H0 Reject H0Fail to reject H0

Controlling Type I and Type II Errors

, , and n are interrelated. If one is kept constant, then an increase in one of the remaining two will cause a decrease in the other.

• For any fixed , an increase in the sample size n will cause a decrease in

• For any fixed sample size n , a decrease in will cause an increase in .

• Conversely, an increase in will cause a decrease in .

• To decrease both and , increase the sample size n.

General Rules For Testing Hypothesis

• State your problem and formulate a null hypothesis with alternative hypothesis .

• Decide upon a significance level, of the test which is the probability of rejecting the null hypothesis if it is true.

• Choose an appropriate test statistic, determine and sketch the probability distribution of the test statistic, assuming H0 is true.

• Determine the critical region in such a way that the probability of rejecting the null hypothesis, if it is true, is equal to the significant level, .

continue• Compute the value of the test statistic in

order to decide whether to accept or reject the null hypothesis H0.

• Formulate the decision rule as below

a) Reject the null hypothesis H0, if the computed value of test statistic falls in the rejection region and conclude that H1 is true.

b) Accept the null hypothesis H0, otherwise