Should Antelope Coffee Inc. open a new shop at Montana?* Example7.4 of Newbold and Carlson and Thorne, 6th edition

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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

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_at_econ.hku.hk.
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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 gt
  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?

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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
  5000/361/2 5000/6 833.33
• Sample mean of 62300 gt Population mean of 60000
• Naturally we tend to conclude that the mean is gt
  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?

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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 m60000) 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.

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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 m59999) 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.

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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 m59998) 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.

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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(m lt 60000 m62300) or
• Prob(m gt 60000 m62300) 1- Prob(m lt 60000
  m62300)
• More generally, we are interested in
• Prob(a lt m lt b m62300)
• Materials in Chapter 8 Confidence Intervals.