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Chapter 9

Chapter 9. Significance Tests Type I and II errors Power. Pizza Delivery. Pines Pizza claims that on average they deliver in 20 min or less. Design an experiment to test their claim. Remember your three principles of experimental design( control, replication, randomization.

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Chapter 9

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  1. Chapter 9 • Significance Tests • Type I and II errors • Power

  2. Pizza Delivery • Pines Pizza claims that on average they deliver in 20 min or less. • Design an experiment to test their claim. • Remember your three principles of experimental design( control, replication, randomization. • What would constitute sufficient evidence that they are wrong in their claim. • Your group has 10 minutes.

  3. Pizza Delivery • FACTS…. • Claim is 20 minutes or less. So the true mean is assumed to be u = 20 • Lets use 1.5 minutes as the true standard deviation • We need a sample mean from our experiment that we can use as evidence against the claim.

  4. Pizza Delivery • FACTS…. • Choose a sample mean that is greater than 20 minutes but not too much greater. • Also choose a sample size, how many deliveries do you think is necessary to test this claim.

  5. Pizza Delivery • FACTS…. • Claim is 20 minutes or less. Ho: u = 20 • If you were going to argue the claim you are saying it is over 20 minutes Ha: u > 20

  6. Conclusions • The whole idea is to make the correct conclusion. • There are 2 scenarios based on your p-value. • The sample mean drives your p-value and conclusion…..use your brain!!!

  7. This p-value is low enough to reject Ho at the ____ level. • This is evidence to suggest that pines pizza may have a mean delivery time of more than 20 min. • This p-value is too high to reject Ho at the ____ level. • There is not enough evidence to suggest that pines pizza may have a mean delivery time of more than 20 min.

  8. Check Your Pulse • Most peoples pulse rate ranges from 70 to 100 beats per minute. • A previous study gave a mean of 73 and standard deviation 11. • Let’s see how our pulse rates compare to these.

  9. Let’s use the following Hypothesis Ho: μ = 73 vs Ha: μ ≠ 73, where μ is the true mean pulse rate for our class today. We will also use the standard deviation from the previous study σ=11

  10. How to take your pulse • Place 2 fingers on your other wrist and count the beats for 15 seconds, then multiply by 4. • You can also count the beats for 60 seconds(takes more concentration)

  11. Ho: μ = 73 vs Ha: μ ≠ 73, where μ is the true mean pulse rate for students at Rancho. Assumptions: We have an independent random sample of # pulse rates taken during class. The population of all students at school is more than 10x our sample. Our normal condition is met by the CLT. 1-sample Z Test for means Mean = n = Z = P-value = . This p-value is __________to reject Ho at ____ level. Based on this sample, there is _______________ evidence to suggest the the pulse rates for Rancho students are different than 73.

  12. What if our sample was poor and the p-value led us to the wrong conclusion? • We may have made an error…

  13. Type 1 and 2 errors • Type 1 error — Rejecting Ho when it’s true. • Type 2 error — Failing to reject Ho when it’s false. Write these down

  14. Ho: μ = 73 vs Ha: μ ≠ 73, where μ is the true mean pulse rate for students at Rancho. Type I error: You determine that the pulse rate for our class is different than 73 when it actually was 73. • Type I error: Reject Ho when true Consequence: Students all make appointments with their doctor for no reason, waste time and money, parents get irritated at Mr. Pines. • Type II error: Fail to Reject Ho when its false Type II error: You determine that the pulse rate for our class is 73 when it was actually not 73. Consequence: Students feel comfortable and satisfied with their pulse rates and when they should be seeking medical attention.

  15. Which is worse? Is type II always worse? • Probably a type II error, better to be safe and get checked out from a doctor. Type I is more of a nuisance, but no one is in danger. • Every problem is different, you have to read it, decide, and give justification for your choice

  16. How does this affect the alpha level? • If you think a type II error is more serious, you want to avoid that error. • Make it easier to make a type I error….make it easy to reject Ho. • This can be done by using a high alpha level like 10%.

  17. CH 11 Tests

  18. Significance Tests Use the HAN SOLO acronym • H: hypotheses Ho and Ha • A: assumptions/Conditions • N: name the Test • S: stats from calc • O: obtain a p-value • L: low enough to reject Ho? • O: outcome in context

  19. What is a P-Value? • P-value assumes Ho is true. • It is a percent. • It is the probability of your sample happening by chance if Ho is true.

  20. What is a P-value? When the p-value is very low like p = .0123 it means that the data you collected could only happen 1.23% of the time if Ho is true. Since 1.23% is very rare, what does this probably suggest? It means that Ho is probably not true. So we Reject Ho

  21. What is a P-value? Now, lets say a p-value is higher. p = .3567. This means that the data we collected could happen about 35.67% of the time, if Ho is true. This percent is very likely to happen. There is no reason to doubt Ho Do not reject Ho

  22. Alpha Levels The popular alpha levels are: • α = 0.01 which is 1% • α = 0.05 which is 5% • α = 0.10 which is 10% These levels are the cutoff points for rejecting Ho

  23. Practice the Hard Parts • The following slides will help you write your alternative hypotheses(Ha) • Help with your p-values • Help write your conclusions.

  24. Jack claims he can throw a football 50 yards. He throws the ball 75 times and averages 49.5 yards. Carry out a significance test at the α = .05 level. Ho: μ = 50 vs Ha: μ < 50 Where μ is the true mean distance Jack can throw a football After crunching the #’s on the calculator a p-value gives P = .0766 What do we do with Ho? Write your conclusion. Is p < .05? No, we will not Reject Ho There is not enough evidence at the 5% level to suggest that Jack throws a football less than 50 yards.

  25. A manufacturer claims that a new brand of air-conditioning unit uses only 6.5 kilowatts of electricity per day. A consumer agency believes the true figure is higher and runs a test on a sample size of 50. Carry out a significance test at the α = .10 level. Ho: μ = 6.5 vs Ha: μ > 6.5 Where μ is the true mean kilowatts of electricity used per day After crunching the #’s on the calculator a p-value gives P = .0590 What do we do with Ho? Write your conclusion. Is p < .10? Yes, we will Reject Ho This evidence suggests that the true mean kilowatts of electricity used per day might be more than 6.5 kilowatts.

  26. Mr. Pines claims that the mean GPA for athletes at Rancho Alamitos High School is 3.25. Carry out a significance test at the α = .01 level. Ho: μ = 3.25 vs Ha: μ ≠ 3.25 Where μ is the true mean GPA of athletes at Rancho After crunching the #’s on the calculator a p-value gives P = .0083 What do we do with Ho? Write your conclusion. Is p < .01? Yes, we will Reject Ho This evidence suggests that the true mean GPA of athletes at Rancho might be different than 3.25.

  27. No alpha(α) level given Lets say our calculator gives a p-value of p = .0452 If there is no alpha level given, how do you know if you should reject the Ho? Typical alpha levels .01 .05 .10 You need to reject at the level where it fits best, this p-value of p =.0452 is low enough to reject at the 5% level but not at the 1% level.

  28. Understand this….. • It is NOT our job to find the true mean or the true proportion in a significance test. • We just have to make a decision on the claim. • Our data helps us make a decision

  29. One and Two-sided tests One sided tests: Ha: μ < 20 Ha: μ > 20 Two sided test Ha: μ ≠ 20

  30. One-sided test Ho: μ = 10 vs Ha: μ > 10 Let’s say you are not given any data, all you have is a z-score Z = 2.78 How do you get a p-value? normalcdf(2.78,1000) = .0027

  31. Two-sided test Ho: μ = 10 vs Ha: μ ≠ 10 Let’s say you are not given any data, all you have is a z-score Z = 2.78 How do you get a p-value? normalcdf(2.78,1000) = .0027 Because this is a two-sided test(≠) you need to multiply your p-value by 2 2(.0027) = .0054

  32. P-value? • Caution!!! • You only need to multiply your p-value by 2 if it is a two-sided test and if you had to use normalcdf to calculate it.

  33. P-value If you used one of the tests from the test menu on your calculator, the p-value is already taken care of. This was done from the TEST MENU, do NOT multiply your p-value by 2

  34. Type 1 and 2 errors • Type 1 error — Rejecting Ho when it’s true. • Type 2 error — Failing to reject Ho when it’s false.

  35. Here is the dilemna • It might not be safe to launch the space shuttle today. • How is this a type I, type II error situation?

  36. Type 1 and 2 errors Ho: The shuttle is safe, launch rockets Ha: The shuttle is not safe, delay the launch What is a type I error in this context? What are its consequences? What is a type II error in this context? What are its consequences? Which is worse?

  37. Type I error: Reject Ho but it was true. Assuming the shuttle is not safe, delaying the launch, but the shuttle was safe. Consequence: Not launching on time, have to reschedule. But no one is in danger Type II error: Fail to Reject Ho, when it is false. Assuming shuttle is safe, when it actually wasn’t safe. Consequence: Shuttle blows up.

  38. Which type of error is more serious? …. You have to read the problem! Decreasing the chance of a type I error increases the chance of a type II error…and vice versa

  39. Type 1 and 2 errors • What is the probability of that error? • Type 1 is the alpha level. • Type 2 is β and complex and you do not have to know.

  40. Power STUDY THIS PAGE • What is power? • Power = 1 - β • It is the probability of correctly rejecting Ho when it is false. Ways to increase power Increase sample size Increase alpha level.....ex.(use α = .10 instead of α = .05 or α = .01)

  41. QUIZ…….. yes, a quiz • In a criminal trial, the defendant is held to be innocent until shown to be guilty beyond a reasonable doubt. • Ho: defendant is innocent • Ha: defendant is guilty

  42. Criminal Trial • Give the type I and type II errors in this context. • Give the consequences for each • Which is worse? • Based on your answer to #3, is it better to use a significance test with α = .05 or α = .01? Explain your answer. • Explain what power would be in the context of THIS setting.

  43. Are these dice “FAIR”? Lets run a significance test.

  44. 1-sample Z-test for means • Ho: u = 3.5 vs Ha: u ≠ 3.5 where u is the true mean of this real die. • Assumptions: We have an independent random sample of 30 rolls of a real die. Our normal condition is met by the CLT.

  45. z = X-bar = σ = 1.70783 P-value = This p-value is __________to reject Ho at ___________ There is ___________evidence to suggest that the true mean of this die is __________ than 3.5

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