chapter8 - CHAPTER 8 TESTS OF SIGNIFICANCE 8.1 Tests about...

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1 CHAPTER 8: TESTS OF SIGNIFICANCE 8.1 Tests about the mean when σ is known Example: Canada Packers claims that the average fat content μ of its ground beef is 10%. A consumer agency took a random sample of n=100 packages and determined the fat content of each one. It was found that x = 10.6 (percent). Is there sufficient evidence to conclude that Canada Packers claim is false? Assume that standard deviation σ =2. Population Random Sample of 100 packages x = 10.6 Two alternative claims: 1. Canada Packers (CP): μ =10 2. Consumer Agency: μ >10 μ =10
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2 Sample Evidence: x = 10.6 Is the sample evidence strong enough to reject the Canada Packers claim? How likely is obtaining a sample of n=100 packages resulting in the sample mean of 10.6 or higher if the population mean μ were 10? ( 10.6 | 10) ? Px μ ≥= = Solution: μ =10 μ >10 x = 10.6
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3 Analysis: We assumed above that μ =10 i.e. CP is right. 1 0 1 0 . 6 x 0 3 Z Answer: A sample mean of fat content as large as the observed would occur 13 times in 10,000 samples if μ were 10. Remark: Higher than 10.6 value of the sample mean provides even stronger evidence against CP’ claim. Area under the curve to the right of 10.6 is 0.0013. This is the fraction of samples of size n=100 producing x 10.6 Area under the Z curve to the right of 3 is 0.0013. This is the fraction of samples of size n=100 producing Z 3
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4 P o p u l a t i o n w i t h μ =10 All possible samples of size 100 10.6 x values •••••••••••••••• | 3 Z 99.87% .13% of samples of samples 10 10.6 Compare the two regions under the two density curves to the right of 10.6! Obtaining a sample with the sample mean of 10.6 or higher would be much more likely if μ were larger than 10.
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5 Conclusion: Either a very rare sample has been selected or in fact μ >10. Decision: Reject the Canada Packers claim. There is strong evidence that μ >10 given the sample data. The value 0.0013 measures the risk of making an incorrect decision about the Canada Packers claim given their claim is true (i.e. μ =10). Example: Refer to Canada Packers Problem. Suppose now that a sample of 100 resulted in x = 10.2. What would be your decision about CP claim in this case? Solution: Summary: The values of x Z Risk Decision x 10.6 Z 3 .0013 x 10.2 Z 1 .1587 The larger the value of x or Z (smaller α ), the stronger the evidence against Canada Packers’ claim.
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6 Five Steps in Hypothesis Testing 1. Assumptions: Specify the variable (i.e. fat content) and parameter (i.e. mean fat content). Specify the distributional assumptions (i.e. normal population) and the method in which sample was obtained (i.e. SRS).
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This note was uploaded on 04/10/2008 for the course STAT 151 taught by Professor Henrykkolacz during the Fall '07 term at University of Alberta.

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chapter8 - CHAPTER 8 TESTS OF SIGNIFICANCE 8.1 Tests about...

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