Presentation on

Testing of Hypothesis

By:Ankit Dubey

Chintan H.Trivedi

• Basics of Hypothesis• Procedure of Hypothesis Testing• Testing Of Hypothesis for Comparing two

related samples• Hypothesis testing of Proportion• Limitations of Test of Hypothesis

HYPOTHESIS

• For a researcher hypothesis is a formal question

that he intends to resolve.

• Hypothesis may be defined as a proposition or a

set of proposition set forth as an explanation for

the occurrence of some specified group of

phenomena.

• Example: vehicle A performs better than vehicle B

NULL HYPOTHESIS AND ALTERNATIVE HYPOTHESIS

• Null hypothesis: when population mean is equal to hypothesis mean

• Alternative hypothesis : the hypothesis which is accepted when null hypothesis fails

• A set of alternatives to null hypothesis is called alternative hypothesis

• Null hypothesis always include = to sign• Alternative hypothesis may include any of the

inequality signs

ALTERNATIVE HYPOTHESIS

ALTERNATIVE HYPOTHESIS

MEANING

H0: μ ≠ μ0 Population mean is not equal to hypothesis mean

H1 :μ>μ0 Population mean is greater than hypothesis mean

H1 :μ<μ0 Population mean is greater than hypothesis mean

TYPE I AND TYPE II ERROR POSSIBLE HYPOTHESIS TEST OUTCOMES

ACTUAL SITUATION

DECISION H0 True H0 False

ACCEPT H0 No error Type II error

Probability =1-α Probability =β

REJECT H0 Type I Error No Error

Probability =α Probability = 1-β

CRITICAL VALUE

It divides area under probability curve into two regions critical and acceptance regions In two tailed test we wish to test H0 : μ =μ0

against H1 : μ≠ μ0 we have two critical values which divides probability curve into two regions

Level of significance

• Denoted by α is the probability of Type 1 error.• It is usually 5% which should be chosen with

great care thought and reason.• If level of significance is 5% then researcher is

willing to take as much as 5% percent risk of rejecting null hypothesis if it is true.

• Maximum Value of probability of rejecting H0 when it is true.

• Determined usually in advance prior to testing of hypothesis.

Two Tailed And One Tailed Test

• Testing three types of Hypothesis given by:• H0 :μ = μ0 Against H1 :μ ≠ μ0 (two tailed test)

• H0 :μ ≤ μ0 Against H1 :μ > μ0 (Right tailed test)

• H0 :μ ≥ μ0 Against H1 :μ < μ0 (Left tailed test)

PROCEDURE FOR HYPOTHESIS TESTING

• It is used to test validity of hypothesis• Setting up the hypothesis: Formal statement

of null and alternative hypothesis• Selecting significance level:pre determined

level of significance based on sample size, variability of measurements with in sample

• Test statistic: to obtain value of mean ,variance etc on basis of distribution selected

PROCEDURE FOR HYPOTHESIS TESTING• Critical value: depends on type of distribution,

level of significance, type of test• Decision: compare the value of test statistic

and critical value • we reject null hypothesis when• Test statistic value< lower than critical

value>upper critical value• Value of test statistic > critical value• Value of test statistic < critical value

HYPOTHESIS TESTING FOR COMPARING TWO RELATED SAMPLES

• Paired t-test is a way to test for comparing two related samples,

• For a paired t-test, observations in the two samples collected in the form matched pairs

• “each observation in the one sample must be paired with an observation in the other sample in such a manner that these observations are somehow “matched” or related, in an attempt to eliminate extraneous factors which are not of interest in test.”

• Such a test is generally considered appropriate in a before-and-after-treatment study.

• Testing a group of certain students before and after training in order to know whether the training is effective, in which situation we may use paired t-test.

• To apply this test, : a. Work out the difference score for each matched

pair b. Find out the average of such differences, D c.

Sample variance of the difference score.

• If the values from the two matched samples are denoted as Xi and Yi and the differences by Di (Di = Xi – Yi), then the mean of the differences i.e.,

• And Variance of the differences:

• To work out the test statistic (t):

Where

Example

• Formulation of Null And Alternate Hypothesis

• Work out Test Statistic

• Standard deviation of differences:

HYPOTHESIS TESTING OF PROPORTION• Population is divided into exclusive and

exhaustive tests based on certain attribute• One class having attribute and other not• Important parameter is proportion of

population having that attribute H0 :π≤ π0, H1 : π> π0 (Right or Upper tail test)H0 : π≥ π0 , H1 π< π0 (left or lower tail test)H0 : π= π0 , H1 : π≠ π0 (two upper tail test)

• Test statistic for this test

Z = p - π0 √ π0 (1- π0 )/n p=sample proportion

HYPOTHESIS TESTING OF TWO PROPORTION

• Two population proportions are compared• Two population proportion are obtained from two

different samples• Hence following hypothesis is tested H0 : π1 = π2 against H1 : π1 ≠ π2

H0 : π1 = π2 against H1 : π1 > π2 or H0 π1 ≤ π2

Against H1 : π1 > π2

H0 : π1 = π2 Against H1 π1 < π2 or π1 ≥ π2 Against H1 π1 < π2

EXAMPLE

Q) A sample survey indicates that out of 3232 births, 1705 were boys and the rest were girls. Do these

figures confirm the hypothesis that the sex ratio is 50 : 50? Test at 5 per cent level of significance.

Solution: Starting from the null hypothesis that the sex ratio is 50 : 50 we may write:

Hypothesis testing for Variance

1. H0 :σ2 = σ0 2 , H1 : σ2 ≠ σ0

2 (two tailed test)2. H0 :σ2 = σ0

2 , H1 : σ2 > σ0 2 (right tailed test)

3. H0 :σ2 = σ0 2 , H1 : σ2 < σ0

2 (left tailed test)

Limitations of Test of Hypothesis

• Important limitations are as follows:• Testing is not decision-making itself; the tests are only useful

aids for decision-making.• “proper interpretation of statistical evidence is important to

intelligent decisions.”• Test do not explain the reasons as to why does the

difference exist, viz. between the means of the two samples. • They simply indicate whether the difference is due to

fluctuations of sampling or because of other reasons• The tests do not tell us the reasons causing the difference.

• Results of significance tests are based on probabilities and as such cannot be expressed with full certainty.

• Statistical inferences based on the significance tests cannot be said to be entirely correct evidences concerning the truth of the hypotheses.

• This is specially so in case of small samples where the probability of drawing erring inferences happens to be generally higher.

• For greater reliability, the size of samples be sufficiently enlarged.

• All these limitations suggest that in problems of statistical significance, the inference techniques (or the tests) must be combined with adequate knowledge of the subject-matter along with the ability of good judgement.

Thank You!