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Power of a Signficance Test

Power of a Signficance Test. Everyone get a calculator, please. The power of a test (against a specific alternative value). Is the probability that the test will reject the null hypothesis when the alternative is true

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Power of a Signficance Test

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  1. Power of a Signficance Test Everyone get a calculator, please.

  2. The power of a test (against a specific alternative value) • Is the probability that the test will reject the null hypothesis when the alternative is true • In practice, we carry out the test in hope of showing that the null hypothesis is false, so high power is important

  3. Suppose H0 is false – what if we decide to reject it? Suppose H0 is false – what if we decide to fail to reject it? We correctly reject a false H0! Suppose H0 is true – what if we decide to fail to reject it? Type I Correct a Power Suppose H0 is true – what if we decide to reject it? Correct Type II b

  4. A researcher selects a random sample of size 49 from a population with standard deviation s = 35 in order to test at the 1% significance level the hypothesis: H0: m = 680 Ha: m > 680 What is the probability of committing a Type I error? where μ is the true mean <context>. a = 0.01

  5. H0: m = 680 n = 49  = 35 Ha: m > 680 For what values of the sample mean would you reject the null hypothesis? Invnorm(0.99, 680, 35/√(49) ) = 691.632 “I would reject the mean for any value greater than or equal to 691.632 <label>.”

  6. H0: m = 680 n = 49  = 35 Ha: m > 680 If H0 is rejected, suppose that ma is 700. What is the probability of committing a Type II error? What is the power of the test? Normalcdf(-10^99, 691.632, 700, 35/√(49) ) = 0.0471 Power = 1 - 0.0471 = 0.9529

  7. H0: m = 680 n = 49  = 35 Ha: m > 680 If H0 is rejected, suppose that ma is 695. What is the probability of committing a Type II error? What is the power of the test? Normalcdf(-10^99, 691.632, 695, 35/√(49) ) = 0.2503 Power = 1 - 0.2503 = 0.7497

  8. ma Fail to Reject H0 Reject H0 a m0 Power = 1 -b b

  9. What happens to a, b, & power when the sample size is increased? Fail to Reject H0 Reject H0 • As n increases – •  stays the same •  goes down, and • Power increases. a m0 b ma

  10. Facts: • The researcher is free to determine the value of a. • The experimenter cannot control b, since it is dependent on the alternate value. • The ideal situation is to have a as small as possible and power close to 1. (Power > 0.8) • Asa increases, power increases. (But also the chance of a type I error has increased!) • Best way to increase power, without increasing a, is to increase the sample size

  11. Bottles of a popular cola are suppose to contain 300 ml of cola. A consumer group believes the company is under-filling the bottles. (Assume s = 50 ml with n = 30 at a 1% significance level.) Find the power of this test against the alternative m = 296 ml. Power = 1 – 0.9705 = 0.0295

  12. ASSIGNMENT • WS Power of a Significance Test • Due Monday, 03 March 2014 (along with BW: 10.61, 10.63, 10.64). • ASSIGNMENTS DUE TOMORROW • WS Errors WS #2 • Bookwork: 10.11, 10.15, 10.18 • QUIZ: Type I and Type II Errors on Monday, 03 March 2014. • Meet in the library’s computer lab tomorrow!!!

  13. AP StatisticsFriday, 28 February 2014 • OBJECTIVE TSW explore Power. • ASSIGNMENTS DUE • WS Errors WS #2 wire basket • Bookwork: 10.11, 10.15, 10.18  black tray • QUIZ: Type I and Type II Errors on Monday, 03 March 2014. • REMINDERS • PI Day Celebration: Friday, 14 March 2014.

  14. Assignment • WS Internet Activity: Power Applet • Due on Tuesday, 04 March 2014. • WS Power & Errors • Due by the end of the period today. • QUIZ: Type I and Type II Errors on Monday, 03 February 2014.

  15. AP StatisticsMonday, 03 March 2014 • OBJECTIVE TSW work on Errors and Power. • QUIZ: Type I and Type II Errors will be tomorrow. • ASSIGNMENTS DUE • WS Power of a Significance Test wire basket • ASSIGNMENT DUE TOMORROW • WS Internet Activity: Power Applet • Bookwork: 10.61, 10.63, and 10.64 and 10.65 • 4th Period will be due on Thursday, 06 March 2014. • REMINDERS • PI Day Celebration: Friday, 14 March 2014.

  16. Facts, Definitions, etc. •  is the probability of a Type I error. • Rejecting H0 when it is true. •  is the probability of a Type II error. • Failing to reject H0 when it is false. • Poweris the probability of rejecting H0 when Ha is true. • Power = 1 -  • Significance level is the probability of rejecting the null hypothesis when it is true. • The best way to increase power: increase n

  17. What is the Power? H0: μ = 444 mi. Ha: μ < 444 mi.  = 7 mi. α = 0.01 n = 37 Calculate  when μa = 440 mi. Find and interpret . 1st, find the value at which your critical value exists. Then, find . If the mean is 440 mi, about 13% of all samples would result in x-bar values greater than 441.322 mi and the nonrejection of H0: μ = 444 mi. Now, find Power. Type I: Reject H0 when it’s true (α). Type II: Fail to reject H0 when it’s false ().

  18. AP StatisticsTuesday, 04 March 2014 • OBJECTIVE TSW (1) work on Errors and Power, and (2) quiz over Errors and Power. • QUIZ: Type I and Type II Errors. • ASSIGNMENTS DUE • WS Power of a Significance Test wire basket • ASSIGNMENT DUE TOMORROW • WS Internet Activity: Power Applet • Bookwork: 10.61, 10.63, and 10.64 and 10.65 • 4th Period will be due on Thursday, 06 March 2014. • REMINDERS • PI Day Celebration: Friday, 14 March 2014.

  19. AP StatisticsThursday, 06 March 2014 • OBJECTIVE TSW work on Errors and Power. • ASSIGNMENTS DUE • WS Mixed Review Inference on Means wire basket • QUIZ: Inference with Means and Errors will be tomorrow, Friday, 07 March 2014. • TEST: Inference with Means and Errors will be on Tuesday, 11 March 2014. • REMINDERS • PI Day Celebration: Friday, 14 March 2014.

  20. Problem 10.65 Let μ denote the true mean diameter for bearings of a certain type. A test of H0: μ = 0.5 versus Ha: μ≠ 0.5 will be based on a sample of n bearings. The diameter distribution is believed to be normal. Determine the value of  in each of the following cases: a)n = 15, α = 0.05,  = 0.02, μ = 0.52 b)n = 15, α= 0.05,  = 0.02, μ= 0.48 c)n = 15, α= 0.01,  = 0.02, μ= 0.52 d)n = 15, α= 0.05,  = 0.02, μ= 0.54 e)n= 15, α= 0.05,  = 0.04, μ= 0.54 f)n = 20, α= 0.05,  = 0.04, μ= 0.54 Complete b-f. You will turn it in later.

  21. Problem 10.65  = 0.02761 a) b) c) d) e) f)  = 0.02761  = 0.0972  = 3.5279 x 10−9  = 0.02761  = 0.005942

  22. Assignment • WS Power and the Power Plant • Problem #1: Also include the hypotheses. • Problem #4: Also compute the power. • Problem #5: Also compute the power. • Due before you leave today. • QUIZ: Inference with Means and Errors is tomorrow, Friday, 07 March 2014.

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