chap09PRN

chap09PRN - Chapter 9.1 Hypothesis Testing Interval...

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Unformatted text preview: Chapter 9.1 Hypothesis Testing Interval estimation leads the way to hypothesis testing. A statistical hypothesis is an assumption about the behaviour of a population. A statement of the hypothesis must be formed. The null hypothesis is denoted by (H-naught – pronounced H-not). For example, H 5 :H P This is tested against the alternative hypothesis denoted by . The form of the alternative hypothesis may arise from the specific application. Three possible options for the alternative hypothesis are: 1 H two-sided alternative 5 :H 1 zP one-sided alternative ° ¿ ° ¾ ½ ?P !P 5 :H 5 :H 1 1 Econ 325 – Chapter 9 1 To test the hypothesis, a test statistic is computed from the sample data. The decision to take can then be either: x do not reject the null hypothesis, or x reject the null hypothesis in favour of the alternative. With a sample of data it is always possible that a wrong decision will be made. Two different mistakes are: x the null hypothesis is true – but the decision is to reject it. This is called a Type I error . x the null hypothesis is false – but the test does not reject it. This is called a Type II error . Econ 325 – Chapter 9 2 For a test method: D is the probability of a Type I error, and E is the probability of a Type II error. It would be desirable to use a test method that gives a small value for both D and. But typically, there is some trade-off. By setting a lower value for this leads to reluctance to reject the null hypothesis and therefore a greater risk of a Type II error and a larger value for E . E D For a given level of D , a way to lower is to increase the sample size n . How can a decision rule be set ? A decision rule can be set to give a probability of a Type I error at some fixed level . is called the significance level of the test. Common choices for are: D D D = 0.10 , 0.05 or 0.01 . Econ 325 – Chapter 9 3 Chapter 9.2 Hypothesis Tests of the Mean Suppose economic theory proposes that the population mean for a variable of interest exceeds the value a . Does the data support this theory ? Consider testing the null hypothesis: or a :H dP a :H against the alternative hypothesis: a :H 1 !P This is a one-sided test. Note the hypothesis is stated so that if the economic theory is correct the null hypothesis will be rejected. That is, there is strong evidence to support the economic theory. From a sample of data the calculated sample mean is x . If the sample mean is substantially greater than the value a then the null hypothesis can be rejected. Econ 325 – Chapter 9 4 When the null hypothesis is true, a , and a result is: )1,0(N~ n a X V Assume that the population standard deviation is known from previous research. Choose a significance level . This sets the probability of a Type I error – rejecting a true null hypothesis. A sensible choice may be = 0.05 . From the sample of data calculate the test statistic: n a x z V The decision rule is to reject the null hypothesis if: H c zz ! where is the critical value...
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chap09PRN - Chapter 9.1 Hypothesis Testing Interval...

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