stat104_lecture23v3_1up

# stat104_lecture23v3_1up - Stat 104 Quantitative Methods for...

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Stat 104: Quantitative Methods for Economists Class 23: Hypothesis Testing- Part II 1

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The Process The Five Steps of Hypothesis Testing Step 1: State the null and alternative hypotheses. Step 2: Choose a significance level α 2 Step 3: Choose a test statistic and use the significance level to establish a decision rule. Step 4: Compute the value of the test statistic. Step 5: Apply the decision rule and make your decision .
Review the One Sided Test s Consider the following hypotheses, and then ask yourself, “when is there evidence in favor of the alternative?” s Clearly there is evidence in favor of the alternative when the sample mean is larger than the hypothesized value. But how much larger? 3 : : o o a o H H μ = >

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Review the Decision Rule s We want to construct a decision rule that says s How do we find the value c? Reject if o H x c s It is based on our Type I error rate. s The key fact is that hypothesis testing is always done under the assumption that the null hypothesis is true. 4
Recall the Atlanta Example s To measure the effectiveness of the campaign, the speed of 40 motor vehicles were measured during rush hour in the down town core (often referred to as the downtown connector). s Their average speed was 9.3 mph. Two years ago, before the planners began their campaign, the average speed of vehicles in the downtown core during rush hour was calculated to be 7.5 mph. s Assume the population standard deviation hasn’t changed and is 4.4 mph. s We want to test 5 : 7.5 : 7.5 o a H H μ = >

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What if…. s What if our decision rule said s What would be our Type I error? s Type I error = P(reject null | null true) Reject if 7.5 o H x > 6 ( 7.5| 7.5, 4.4) 7.5 7.5 7.5 ( ) 4.4 / 40 4.4 / 40 ( 0) 0.5 P X X P P Z μ σ > = = - - = > = > = Type I error is way too high!
What if…. s What if our decision rule said s What would be our Type I error? s Type I error = P(reject null | null true) Reject if 8 o H x > 7 ( 8 | 7.5, 4.4) 7.5 8 7.5 ( ) 4.4 / 40 4.4 / 40 ( .7187) 0.236 P X X P P Z μ σ > = = - - = > = > = Type I error is still to high!

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What is going on? s IF the TRUE mean is 7.5, seeing a sample mean of 8 is still very plausible. s We need a cut-off value larger than 8! hat is, we want to find the first value that is s That is, we want to find the first value that is unlikely for the sample mean to take on, IF the true mean is 7.5 8
Different Cut-Off Values s We could repeat these past calculations for any cut-off value: 9 Interp: IF the true mean is 7.5, the chance of seeing a sample mean of 8.7 or greater is 0.0422

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The General Formula s In general, the 5% cut-off value is given by the formula 1.64 o c σ μ = + s For the Atlanta example this value is 10 n 4.4 7.5 1.64 40 8.64 c = + =
The Atlanta Decision Rule s For a 5% level test, accept the one-sided alternative hypothesis H a : μ > 15 if 11 8.64 x >

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stat104_lecture23v3_1up - Stat 104 Quantitative Methods for...

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