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- 2. A hypothesis is an assumption about the population parameter. A parameter is characteristic of the population, like its mean or variance. © 1984-1994 T/Maker Co. WHAT IS A HYPOTHESIS?
- 3. T ESTING OF HYPOTHESIS For testing of hypothesis we collect sample data , then we calculate sample statistics (say sample mean) and then use this information to judge/decide whether hypothesized value of population parameter is correct or not. Then we judge whether the difference is significant or not(difference between hypothesized value and sample value) The smaller the difference ,the greater the chance that our hypothesized value for the mean is correct.
- 4. THE NULL HYPOTHESIS, H0 The null hypothesis H0 represents a theory that has been put forward either because it is believedto be true or because it is used as a basis for an argument and has not been proven. Forexample,in a clinicaltrial ofa newdrug, thenull hypothesis might be thatthenewdrug is nobetter, on average, than the currentdrug. We would write H0: thereis no difference between the two drugs on an average.
- 5. ALTERNATIVEHYPOTHESIS new drug has a differenteffect, on average, compared to thatof the currentdrug. We would write HA: the two drugs have different effects, onaverage. or HA: the new drug is better than the currentdrug, on average. The resultof a hypothesis test: ‘Reject H0 in favourof HA’ OR ‘Do not reject H0’ The alternative hypothesis,HA, is a statement of whata statistical hypothesis test is set up to establish.For example, in the clinical trial of a new drug, the alternative hypothesis that the
- 6. SELECTINGAND INTERPRETINGSIGNIFICANCE LEVEL 1. Decidingon a criterion for accepting or rejecting the null hypothesis. 2. Significance level refers to the percentage of the means that is outside certain prescribed limits. E.g testing a hypothesis at 5% level of significancemeans ▪ thatwe reject the null hypothesis if it falls in the two regions of area 0.025. ▪ Do not reject the null hypothesis if it falls within the region of area 0.95. 3. The higher the level of significance, the higher is the probability of rejecting the null hypothesis whenit is true. (acceptance region narrows)
- 7. Critical value Critical value If our sample statistic(calculated value) fall in the non- shaded region( acceptance region), then it simply means that there is no evidence to reject the null hypothesis. (Confidence interval)
- 8. A type I error, also known as an error of the first kind, occurs when the null hypothesis (H0) is true, but is rejected. A type I error may be compared with a so called false positive. Denoted by the Greek letter α (alpha). It is usually equals to the significance level of a test. If type I error is fixed at 5 %, it means that there are about 5 chances in 100 that we will reject H0 when H0 is true.
- 9. Type II error, also known as an errorof the second kind, occurs when the null hypothesisis false, but incorrectlyfails to be rejected. Type II error means accepting the hypothesis which shouldhave been rejected. A type II error may be compared with a so-called FalseNegative. A type II error occurs when one rejects the alternative hypothesis (fails to reject the null hypothesis)when the alternativehypothesisis true. The rate of the type II error is denotedby the Greek letter β (beta) and related to the power of a test (which equals 1-β ).
- 11. EXAMPLE 1-COURT ROOM TRIAL In court room, a defendant is considered not guilty as long as his guilt is not proven. The prosecutor tries to prove the guilt of the defendant. Only when there is enough charging evidence the defendant is condemned. In the start of the procedure, there are two hypotheses H0: "the defendant is not guilty", and H1: "the defendant is guilty". The first one is called null hypothesis,and the second one is called alternative (hypothesis).
- 12. Suppose the null hypothesis, H0, is: Frank's rock climbing equipment is safe. Type I error: Frank thinks that his rock climbing equipment may not be safe when, in fact, it really is safe. Type II error: Frank thinks that his rock climbing equipment may be safe when, in fact, it is not safe. α= probability that Frank thinks his rock climbing equipment may not be safe when, in fact, it really is safe. β=probability that Frank thinks his rock climbing equipment may be safe when, in fact, it is not safe. Notice that, in this case, the error with the greater consequence is the Type II error. (If Frank thinks his rock climbing equipment is safe, he will go ahead and use it.) EXAMPLE 2
- 13. Suppose the null hypothesis, H0, is: The victim of an automobile accident is alive when he arrives at the emergency room of a hospital. Type I error: The emergency crew thinks that the victim is dead when, in fact, the victim is alive. Type II error: The emergency crew does not know if the victim is alive when, in fact, the victim is dead. α=probability that the emergency crew thinks the victim is dead when, in fact, he is really alive =P(Type I error) β= probability that the emergency crew does not know if the victim is alive when, in fact, the victim is dead =P(Type II error) The error with the greater consequence is the Type I error. (If the emergency crew thinks the victim is dead, they will not treat him.) EXAMPLE 3
- 14. EXAMPLE 4 It’s a Boy Genetic Labs claim to be able to increase the likelihood that a pregnancy will result in a boy being born. Statisticians want to test the claim. Suppose that the null hypothesis, H0 , is: It’s a Boy Genetic Labs has no effect on gender outcome. Type I error: This results when a true null hypothesis is rejected. In the context of this scenario, we would state that we believe that It’s a Boy Genetic Labs influences the gender outcome, when in fact it has no effect. The probability of this error occurring is denoted by the Greek letter alpha, α Type II error: This results when we fail to reject a false null hypothesis. In context, we would state that It’s a Boy Genetic Labs does not influence the gender outcome of a pregnancy when, in fact, it does. The probability of this error occurring is denoted by the Greek letter beta, β The error of greater consequence would be the Type I error since couples would use the It’s a Boy Genetic Labs product in hopes of increasing the chances of having a boy.
- 15. REDUCING TYPE I ERRORS Prescriptive testing is used to increase the level of confidence, which in turn reduces Type I errors. The chances of making a Type I error are reduced by increasing the level of confidence.
- 16. REDUCING TYPE II ERRORS Descriptive testing is used to better describe the test condition and acceptance criteria, which in turn reduces Type II errors. This increases the number of times we reject the Null hypothesis – with a resulting increase in the number of Type I errors (rejecting H0 when it was really true and should not have been rejected). Therefore, reducing one type of error comes at the expense of increasing the other type of error! THE SAME MEANS CANNOT REDUCE BOTH TYPES OF ERRORS SIMULTANEOUSLY!
- 17. Therefore, reducing one type of error comes at the expense of increasing the other type of error! THE SAME MEANS CANNOT REDUCE BOTH TYPES OF ERRORS SIMULTANEOUSLY!
- 18. THANK YOU