1. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement9-1 Supplement 9: The impact of Switching Null and Alternative Hypothesis *The ppt is a joint effort: Ms Yuanyuan JI discussed the impact of switching the null and alternative hypotheses with Dr. Ka-fu Wong on 12 April 2007; Ka-fu explained the problem; Yuanyuan drafted the ppt; Ka-fu revised it. Use it at your own risks. Comments, if any, should be sent to kafuwong@econ.hku.hk.

2. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement9-2 An Example In the past, 15% of the mail order solicitations for a certain charity resulted in a financial contribution. A new solicitation letter that has been drafted is sent to a sample of 200 people and 30 responded with a contribution. At the.05 significance level can it be concluded that the new letter is more effective? In the past, 15% of the mail order solicitations for a certain charity resulted in a financial contribution. In past years, the letter were drafted by a staff Mr A. A new solicitation letter has been drafted by a job applicant Mr B. The letter is sent to a sample of 200 people and 30 responded with a contribution. At the.05 significance level can it be concluded that the new letter is more effective? Can we conclude that the job applicant Mr B is better than Mr A?

3. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement9-3 Give the benefit of doubt to the old letter (or Mr A) That is, unless the new letter performs much better than the old one, we will use the old one. Step 1: State the null and the alternate hypothesis. Null Hypothesis :  .15 (old letter is more effective) Alternative Hypothesis :  >.15 (new letter is more effective) The population proportion of donation response of the new letter

4. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement9-4 Given the benefit of doubt to the old letter (or Mr A) That is, unless the new letter performs much better than the old one, we will use the old one. Step 1: State the null and the alternate hypothesis. Null Hypothesis :  .15 (old letter is more effective) Alternative Hypothesis :  >.15 (new letter is more effective) Step 2: Select the level of significance. It is.05. Step 3: Find a test statistic. A standardized test statistic is distributed with standard normal distribution. z=(p-  )/std(p), where p is the sample proportion Step 4: State the decision rule. The null hypothesis is rejected if z is greater than 1.65.

5. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement9-5 Given the benefit of doubt to the old letter (or Mr A) That is, unless the new letter performs much better than the old one, we will use the old one. Step 5: Make a decision and interpret the results. The null hypothesis is not rejected. The new letter is not more effective than the old one. Prob(Z>1.65)=0.05

6. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement9-6 Testing the hypothesis Under the null and with the greatest benefit of doubt to the null – i.e., p=0.15 – the sample proportion is distributed approximately normal with mean  =0.15 and variance  (1-  )/n = 0.15(1-0.15)/200  =0.15  =0 p z Standardized to standard normal: z=(p-  )/std(p)  =0.05 1.65 Rejection region

7. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement9-7 Switch Null and Alternative Hypothesis Give the benefit of doubt to the new letter (or Mr B) That is, unless the new letter performs much worse than the old one, we will use the new one. Step 1: State the null and the alternate hypothesis. Null Hypothesis :  >.15 (new letter is more effective) Alternative Hypothesis :  .15 (old letter is more effective) Step 2: Select the level of significance. It is.05. Step 3: Find a test statistic. A standardized test statistic is distributed with standard normal distribution. z=(p-  )/std(p), where p is the sample proportion Step 4: State the decision rule. The null hypothesis is rejected if z is smaller than -1.65.

8. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement9-8 Testing the hypothesis Under the null and with the greatest benefit of doubt to the null – i.e., p=0.15 – the sample proportion is distributed approximately normal with mean  =0.15 and variance  (1-  )/n = 0.15(1-0.15)/200  =0.15  =0 p z Standardized to standard normal: z=(p-  )/std(p) -1.65  =0.05 The null is not rejected. Rejection region

9. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement9-9 Conclusion Switching null and alternative hypothesis inappropriately will yield substantially different conclusions. Which way of stating the null is more reasonable? Give the benefit of doubt to the old letter (or Mr A) That is, unless the new letter performs much better than the old one, we will use the old one. Give the benefit of doubt to the new letter (or Mr B) That is, unless the new letter performs much worse than the old one, we will use the new one. The context in the example suggest that the former is more reasonable!!

10. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement9-10 - END - Supplement 9: The impact of Switching Null and Alternative Hypothesis