INTRODUCTION TO TESTING OF HYPOTHESIS INTRODUCTION TO TESTING OF HYPOTHESIS SHWETA MOGRE

Hypothesis Testing Definition: Hypothesis testing is a procedure for making inferences about a population. Definition: Hypothesis testing is a procedure for making inferences about a population. Parameter Population Sample Statistic Inference

Population Population means the entire spectrum of a system of interest. Population means the entire spectrum of a system of interest.Sample Sample is that part of Population which we select for the purpose of investigation. Sample is that part of Population which we select for the purpose of investigation.

What Is Hypothesis ?

A hypothesis is an assumption about the population parameter. A hypothesis is an assumption about the population parameter. A parameter is a characteristic of the population, like its mean or variance. A parameter is a characteristic of the population, like its mean or variance. The parameter must be identified before analysis. The parameter must be identified before analysis. What is a Hypothesis?

Hypothesis is a Statement. © T/Maker Co.

What is Statement ?

Statement An statement is a declarative sentence which is true or false, but not both,or in other words an statement is a declarative sentence which has a definite truth values. An statement is a declarative sentence which is true or false, but not both,or in other words an statement is a declarative sentence which has a definite truth values. A hypothesis (plural "hypotheses") is a statement which may or may not be true. A hypothesis (plural "hypotheses") is a statement which may or may not be true.

Characteristics of Hypothesis Hypothesis should be clear and precise. Hypothesis should be clear and precise. Hypothesis should be stated as far as possible in most simple terms so that the same is easily understandable by all concerned. Hypothesis should be stated as far as possible in most simple terms so that the same is easily understandable by all concerned. Hypothesis should state relationship between variables, if it happens to be a relational hypothesis. Hypothesis should state relationship between variables, if it happens to be a relational hypothesis. Hypothesis must actually explain what it claims to explain. Hypothesis must actually explain what it claims to explain.

Types of Hypothesis Null hypothesis Null hypothesis Alternative hypothesis Alternative hypothesis

Null Hypothesis It is a hypothesis which we want to test. It is a hypothesis which we want to test. It is denoted by H 0 It is denoted by H 0 H 0 : Write statement here. H 0 : Write statement here.

Null Hypothesis Null hypothesis is the hypothesis of no difference, which shows the null or neutral attitude of the statistician. Null hypothesis is the hypothesis of no difference, which shows the null or neutral attitude of the statistician.

Null Hypothesis For example, in a clinical trial of a new drug, the null hypothesis might be that the new drug is no better, on average, than the current drug. We would write For example, in a clinical trial of a new drug, the null hypothesis might be that the new drug is no better, on average, than the current drug. We would write H 0 : There is no difference between the two drugs on average. H 0 : There is no difference between the two drugs on average.

Alternative Hypothesis The hypothesis which differs from given null hypothesis. The hypothesis which differs from given null hypothesis. It is denoted by H 1 It is denoted by H 1 H 1 : Write statement here. H 1 : Write statement here.

Alternative Hypothesis The alternative hypothesis, H 1, is a statement of what a statistical hypothesis test is set up to establish. The alternative hypothesis, H 1, is a statement of what a statistical hypothesis test is set up to establish. For example, in a clinical trial of a new drug, the alternative hypothesis might be that the new drug has a different effect, on average, compared to that of the current drug. For example, in a clinical trial of a new drug, the alternative hypothesis might be that the new drug has a different effect, on average, compared to that of the current drug.

Alternative Hypothesis If we want to test the hypothesis that the population has a specified mean say μ 0 then the null hypothesis would be If we want to test the hypothesis that the population has a specified mean say μ 0 then the null hypothesis would be H 0 : μ = μ 0 H 0 : μ = μ 0 and the alternative hypothesis could be and the alternative hypothesis could be 1. H 1 : μ ≠ μ 0 2. H 1 : μ < μ 0 3. H 1 : μ > μ 0

Alternative Hypothesis The alternative hypothesis in 1 is known as a two tailed alternative in 2 & 3 are known as left tailed & right tailed alternatives. The alternative hypothesis in 1 is known as a two tailed alternative in 2 & 3 are known as left tailed & right tailed alternatives.

Critical Region The testing of hypothesis is done on the basis of the division of sample space into two mutually exclusive regions. The testing of hypothesis is done on the basis of the division of sample space into two mutually exclusive regions. One is region for acceptance of H 0 and another is region for rejection of H 0. One is region for acceptance of H 0 and another is region for rejection of H 0.

Critical Region Acceptance Region ) Rejection Region (Critical Region)

There are two decisions that we make; reject or fail to reject. Each could possibly be a wrong decision; therefore, there are two types of errors. Error

There are two types of Errors Type of I ErrorType of I Error Type of II Error Type of II Error

Types of Error Decision based On sample H o is True H o is False H o is False Reject H 0 Wrong Decision (Type I error) Right Decision (No error) Accept H 0 Right Decision (No error) Wrong Decision (Type II error)

Type I error When you reject the null hypothesis that is really true is know as Type I error. When you reject the null hypothesis that is really true is know as Type I error. The probability of committing Type I error is called the size of type I error and it is denoted by α. The probability of committing Type I error is called the size of type I error and it is denoted by α.

Type II Error The decision of accepting the null hypothesis H0 when it is false is called Type II error. The decision of accepting the null hypothesis H0 when it is false is called Type II error. The probability of committing type II error is called the size of the type II error and it is denoted by β. The probability of committing type II error is called the size of the type II error and it is denoted by β.

  Reduce probability of one error and the other one goes up.  &  Have an Inverse Relationship

Continue…. Both α & β can not be reduced at once so we try to reduce β for a prefixed α. Both α & β can not be reduced at once so we try to reduce β for a prefixed α. The seriousness of the error types is determined by the specific situations. The seriousness of the error types is determined by the specific situations.

Consider a murder trial: What are the hypotheses? Type I error – Consequence : Type II error – Consequence : Decide the defendant is guilty when really innocent An innocent person goes to prison Decide defendant is not guilty when really guilty A guilty person goes free H 0 : defendant is innocent H a : defendant is guilty

Continue…. It may be noted that in legal system the commission of the error designated as Type I error has been considered to be far more serious than a type II error, thus we say that it is a more grievous mistake to convict an innocent man than to let a guilty man go free. It may be noted that in legal system the commission of the error designated as Type I error has been considered to be far more serious than a type II error, thus we say that it is a more grievous mistake to convict an innocent man than to let a guilty man go free.

Lay’s Chip Company decides to accept a truckload of potatoes based upon results from a sample of potatoes from the truckload. What are the hypotheses? Type I error? Type II error? From the supplier’s viewpoint, which is more serious? A type I error From the chip company’s viewpoint, which is more serious? Decide the potatoes are bad when they really are good Decide the potatoes are good when they really are bad H 0 : potatoes good H a : potatoes bad A type II error Sometimes, the seriousness depends upon the person’s point-of- view

The Level of Significance The maximum value of the probability of type I error which we would be willing to risk is called level of significance. The maximum value of the probability of type I error which we would be willing to risk is called level of significance.

Continue…. Most widely used value of α are 0.01 (1%) & 0.05 (5%). Most widely used value of α are 0.01 (1%) & 0.05 (5%). 5% level of significance means that out of 100 on an average there are 5 chances of rejecting a correct H 0. 5% level of significance means that out of 100 on an average there are 5 chances of rejecting a correct H 0.

Power of The Test Probability of rejection of a wrong null hypothesis is called power of test. Probability of rejection of a wrong null hypothesis is called power of test. Symbolically it is denoted by Symbolically it is denoted by 1- β.

Normal Distribution

Level of Significance  and the Rejection Region    /2 Rejection Regions Left Tail Hypothesis Right Tail Hypothesis Two Tail Hypothesis

Statistical Tests A statistical test is a two action decision problem, when a decision of rejection or acceptance of null hypothesis is to be taken on the basis of the information supplied by the sample observation. A statistical test is a two action decision problem, when a decision of rejection or acceptance of null hypothesis is to be taken on the basis of the information supplied by the sample observation. These test may be of two types These test may be of two types 1. One tailed test 2. Two tailed test

Procedure for Testing of Significance Formulation of null hypothesis & alternative hypothesis. Formulation of null hypothesis & alternative hypothesis. Selection of test statistic. Selection of test statistic. Selection of the appropriate level of significance. Selection of the appropriate level of significance. Formulation of critical region. Formulation of critical region. Conducting study Conducting study Calculate test statistic value. Calculate test statistic value. Making decision. Making decision.

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