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Virtual COMSATSInferential Statistics

Lecture-17

Ossam ChohanAssistant Professor

CIIT Abbottabad

Recap of previous lectures

• We started hypothesis testing:– Why Hypothesis?.– Importance of Hypothesis.– Null and Alternative Hypothesis.– Types of Errors.

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Objective of lecture-17

• Introduction to Hypothesis testing.– One tail and two tail hypothesis.– Directional hypothesis.– Level of significance.– Type-I and type-II error.– Test Statistic(s).– Critical Region.– Conclusion.

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Errors in Hypothesis Testing

Actual StateActual State

of Affairsof Affairs

BeliefBelief ConclusionConclusion HH00 is True is True HH00 is False is False

HH00 is False is False Reject HReject H00 Type I ErrorType I Error

False PositiveFalse Positive

Correct Correct RejectionRejection

1 - 1 - PowerPower

HH00 is True is True Fail to Reject Fail to Reject HH00

Correct Correct Failure to Failure to RejectReject

1 - 1 -

Type II ErrorType II Error

False False NegativeNegative

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Example of errors: Jury’s Decision

Did Not Commit CrimeDid Not Commit Crime Committed CrimeCommitted Crime

GuiltyGuilty Type I ErrorType I ErrorConvict Innocent PersonConvict Innocent Person

Correct VerdictCorrect VerdictConvict Guilty PersonConvict Guilty PersonNo ErrorNo Error

Not GuiltyNot Guilty Correct AcquittalCorrect AcquittalFail to Convict Innocent Fail to Convict Innocent PersonPersonNo ErrorNo Error

Type II ErrorType II ErrorFail to Convict Guilty Fail to Convict Guilty PersonPerson

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Practical Problem for understanding

• We will use following example throughout the understanding phase of introduction. And we will add some more problems to make it more clear.– Example:– Does an average box of cereal contain more than 368 grams of

cereal? A random sample of 25 boxes showed = 372.5. The company has specified to be 15 grams. Test at the 0.05 level.

– The two hypotheses about a population mean: • H0: The null hypothesis ≤ 368 • H1: The alternative hypothesis > 368

x

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Important Tips

• How to represent hypothesis?

• Role of equality in hypothesis?

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Discussion

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Discussion continued

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One tail/Two tail hypothesis

• Is that example one tail or two tail? Justify

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Directional Hypothesis

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Level of Significance

Alpha: probability of committing a Type I error1. Reject H0 although it is true2. Symbolized by

• What about probability of type-II error?• What is the role of type-II error in HT?

• Probability that the test will correctly reject a false null hypothesis-Power of the Test

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• The significance level of a statistical hypothesis test is a fixed probability of wrongly rejecting the null hypothesis H0, if it is in fact true. It is the probability of a type-I error and is set by the investigator in relation to the consequences of such an error. That is, we want to make the significance level as small as possible in order to protect the null hypothesis and to prevent, as far as possible, the investigator from inadvertently making false claims.

• The significance level is usually denoted by significance level = P (type I error) = α

• Usually, the significance level is chosen to be 0.05 (or equivalently, 5%).

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Identifying ‘δ’ and observing ‘n’

• The role of population variation and sample size is highly significant in drawing inferences-statistical, as we discussed in estimation section.

• All the rules are same as we discussed in estimation.

• Selection of test statistics depends upon population variability and sample size. (obviously CLT plays role here)

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Test Statistic

• A test statistic is a quantity calculated from our sample of data. Its value is used to decide whether or not the null hypothesis should be rejected in our hypothesis test.

• There are different test statistics for different situations, same rules followed as we selected test statistics in estimation.

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Some Test Statistics

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Calculation of Test Statistics

• Calculate the standard error of the sample statistic. Use the standard error to convert the observed value of the sample statistic to a standardize value.

• Test Statistic will provide you consolidated sample evidence, and this value will be after considering variation in data.

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Critical region and Conclusions

• The critical region CR, or rejection region RR, is a set of values of the test statistic for which the null hypothesis is rejected in a hypothesis test.

• The sample space for the test statistic is partitioned into two regions;

• The critical region will lead us to reject the null hypothesis H0, the other will not.

• If the observed value of the test statistic is a member of the critical region, we conclude "Reject H0"; if it is not a member of the critical region then we conclude "Do not reject H0".

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Six steps of hypothesis testing

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