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Inferences on p class

# When an airplane deviates from the localizer it is

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Unformatted text preview: h Carolina; Slide 37 One Last Time! Airplanes approaching a runway for a landing are required to stay within the localizer (certain distance left and right of the runway). When an airplane deviates from the localizer, it is called an exceedence. Historically a given airline has experienced 12% exceedence at a certain airport. After major renovations to the runway a random sample of 250 landings had 35 exceedences. Is there reason to think that the exceedence rate has changed? changed? G. baker, Department of Statistics G. University of South Carolina; Slide 38 H0: Ha: α = 0.05 0.025 0.025 -1.96 Z= p-value = Ha +1.96 H0 Ha G. baker, Department of Statistics G. University of South Carolina; Slide 39 Given population parameter p and test value p0: For H0: p = p0 Ha: p ≠ p0 α/2 α/2 Ha H0 Ha: p > p0 α H0 Ha: p < p0 Ha Ha α Ha H0 G. baker, Department of Statistics G. University of South Carolina; Slide 40 There are Two Mistakes We Can Make We can reject H0, finding for Ha, when H0 iis s true. true. Or We can fail to reject H0, continuing to continuing assume H0 is true, when Ha is true. assume G. baker, Department of Statistics G. University of South Carolina; Slide 41 Controlling Risk How do we control the probability of rejecting H0 when H0 is true? rejecting when This is called a type I error and the probability of it happening is alpha. probability G. baker, Department of Statistics G. University of South Carolina; Slide 42 H0: p < 0.07 Ha: p > 0.07 n = 250 30 p = 0.07 25 20 15 10 5 0 0 0.05 0.1 0.15 0.2 0.25 p-hat 0.0965 H0 Ha G. baker, Department of Statistics G. University of South Carolina; Slide 43 Suppose p = 0.14 30 p = 0.07 25 p = 0.14 20 15 10 β 5 α 0 0 0.05 0.1 0.15 0.2 0.25 p-hat 0.0965 H0 Ha G. baker, Department of Statistics G. University of South Carolina; Slide 44 Probability of Type II error = β Probability P (fail to reject H0 | Ha iis true at p = 0.14) s ˆ P ( p < 0.0965 | p = 0.14) 0.0965 0.0965 − 0.14 Z= = −1.982 0.14(0.86) 250 P (Z < -1.982) = 0.0237 = β (Z 1.982) G. baker, Department of Statistics G. University of South Carolina; Slide 45 Power of a Hypothesis Test Power = P (reject H0 | Ha is true at p = 0.14) Power = 1 – β Power Power of this test at p = 0.14: 1 – 0.0237 = 0.9763 0.0237 G. baker, Department of Statistics G. University of South Carolina; Slide 46 How can We Control β? How 30 p = 0.07 25 p = 0.14 20 15 10 β 5 α 0 0 0.05 0.1 0.15 0.2 0.25 p-hat 0.0965 H0 Ha G. baker, Department of Statistics G. University of South Carolina; Slide 47 Summary α = P (committing Type I error) α = P (reject H0 | H0 is true) Control α by setting it prior to test Control β = P (committing Type II error) β = P (fail to reject H0 | Ha is true at some value) Control β by choosing appropriate sample size. Control Power = P (reject H0 | Ha is true at some value) Power = 1 - β Power G. baker, Department of Statistics G. University of South Carolina; Slide 48 You Try It H0: p > 0.12 Ha: p < 0.12 α = 0.05 n = 250 What is the power of this test at p = 0.07? G. baker, Department of Statistics G. University of South Carolina; Slide 49...
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