1. 1 Should Antelope Coffee Inc. open a new shop at Montana?* Example7.4 of Newbold and Carlson and Thorne, 6th edition Ka-fu Wong & Nipun Sharma University of Hong Kong 29 March 2007 Ka-fu Wong & Nipun Sharma University of Hong Kong 29 March 2007 *The ppt is a joint effort: Nipun Sharma discussed the Example with Dr. Ka-fu Wong on 28th March 2007; Ka-fu explained the problem; Nipun 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. 2 The problem at hand:  Antelope Coffee Inc. is considering the possibility of opening a coffee shop in Montana. Previous research shows that a shop will be successful if the per capita annual income > 60,000. The standard deviation is known to be 5000.  From a random sample of 36, the mean income was 62,300.  Does this sample provide enough evidence to show that the shop will be successful?

3. 3 Summarize the information and rewrite the question  Population Mean: 60,000  Standard Deviation: 5000  Sample Mean: 62,300  Sample size : 36  Standard deviation of the sample mean = 5000/36 1/2 = 5000/6 = 833.33  Sample mean of 62300 > Population mean of 60000  Naturally we tend to conclude that the mean is > 60000, but we know that there is a chance we will observe a sample mean larger than or equal to 62300 even if the true population mean is 60000 or lower.  Is it rare to observe such sample mean when the true population mean is 60000 or lower?

4. 4 Is it rare to observe a sample mean that is larger than or equal to 62300 when the true population mean is 60000?  Prob(m  62300 | =60000) =Prob((m-60000)/833.33  (62300-60000)/833.33) =Prob(Z  2.76) =0.00289  Yes! It is rare to observe a sample mean that is larger than or equal to 62300 when the true population mean is 60000.  That is, it is unlikely that the population mean is 60000.

5. 5 Is it rare to observe a sample mean that is larger than or equal to 62300 when the true population mean is 59999?  Prob(m  62300 | =59999) =Prob((m-59999)/833.33  (62300-59999)/833.33) =Prob(Z  2.7612) =0.00288  Yes! It is rare to observe a sample mean that is larger than or equal to 62300 when the true population mean is 59999.  That is, it is unlikely that the population mean is 59999. More unlikely than when the population mean is 60000.

6. 6 Is it rare to observe a sample mean that is larger than or equal to 62300 when the true population mean is 59998?  Prob(m  62300 | =59998) =Prob((m-59998)/833.33  (62300-59998)/833.33) =Prob(Z  2.7624) =0.00287  Yes! It is rare to observe a sample mean that is larger than or equal to 62300 when the true population mean is 59998.  That is, it is unlikely that the population mean is 59998. More unlikely than when the population mean is 59999.

7. 7 Concluding remarks  That is, based on the sample information, it is very likely that the population mean is larger than 60000.  Opening a new coffee shop is very likely to be a success.  What we really want to get is Prob(< 60000 | m=62300) or Prob(> 60000 | m=62300) = 1- Prob(< 60000 | m=62300)  More generally, we are interested in Prob(a < < b | m=62300) Materials in Chapter 8: Confidence Intervals.