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Hypothesis Testing

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Hypothesis Testing

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Hypothesis Testing

  1. 1. Hypothesis Testing
  2. 2. The logic of statistical hypothesis testing follows the logic of judicial decision making.
  3. 3. A jury is asked to decide whether a defendant is guilty or not guilty.
  4. 4. A jury is asked to decide whether a defendant is guilty or not guilty. It is a dicho-tomous decision, guilty or not guilty.
  5. 5. A jury is asked to decide whether a defendant is guilty or not guilty. It is a dicho-tomous decision, guilty or not guilty. There is no in-between or partial decision.
  6. 6. A jury is asked to decide whether a defendant is guilty or not guilty. It is a dicho-tomous decision, guilty or not guilty. There is no in-between or partial decision. The jury does not begin its decision-making process in a neutral position.
  7. 7. A jury is asked to decide whether a defendant is guilty or not guilty. It is a dicho-tomous decision, guilty or not guilty. There is no in-between or partial decision. The jury does not begin its decision-making process in a neutral position. The default position is “not guilty.”
  8. 8. A jury is asked to decide whether a defendant is guilty or not guilty. It is a dicho-tomous decision, guilty or not guilty. There is no in-between or partial decision. The jury does not begin its decision-making process in a neutral position. The default position is “not guilty.” The prosecution must mount enough evidence to convince the jury to move from its default position of not guilty to a verdict of guilty.
  9. 9. The jury will make a decision which may or may not coincide with reality.
  10. 10. When the jury decides “not guilty” and the defendant is, in reality, not guilty, It is true because the not guilty (negative) decision aligns with the not guilty (negative) reality.
  11. 11. When the jury decides “not guilty” and the defendant is, in reality, not guilty, they have made a correct decision called a “true negative decision.” It is true because the not guilty (negative) decision aligns with the not guilty (negative) reality.
  12. 12. When the jury decides “not guilty” and the defendant is, in reality, not guilty, they have made a correct decision called a “true negative decision.” It is true because the not guilty (negative) decision aligns with the not guilty (negative) reality. not guilty and I really wasn’t guilty! true negative
  13. 13. When the jury decides “guilty” and the defendant is, in reality, guilty, It is true because the guilty (positive) decision aligns with the guilty (positive) reality.
  14. 14. When the jury decides “guilty” and the defendant is, in reality, guilty, they have made a correct decision called a “true positive” decision. It is true because the guilty (positive) decision aligns with the guilty (positive) reality.
  15. 15. When the jury decides “guilty” and the defendant is, in reality, guilty, they have made a correct decision called a “true positive” decision. It is true because the guilty (positive) decision aligns with the guilty (positive) reality. guilty and I really WAS guilty! true positive
  16. 16. When the jury decides “not guilty” and the defendant is, in reality, guilty,
  17. 17. When the jury decides “not guilty” and the defendant is, in reality, guilty, they have made an incorrect decision called a “false negative error” which is also called a Type II or beta error.
  18. 18. When the jury decides “not guilty” and the defendant is, in reality, guilty, they have made an incorrect decision called a “false negative error” which is also called a Type II or beta error. It is false because the “not guilty” (negative) decision does not align with the guilty (positive) reality. not guilty and I really WAS guilty! false negative
  19. 19. When a jury decides “guilty” and the defendant is, in reality, not guilty,
  20. 20. When a jury decides “guilty” and the defendant is, in reality, not guilty, they have made an incorrect decision called a “false positive error”
  21. 21. When a jury decides “guilty” and the defendant is, in reality, not guilty, they have made an incorrect decision called a “false positive error” which is also called a Type I or alpha error.
  22. 22. When a jury decides “guilty” and the defendant is, in reality, not guilty, they have made an incorrect decision called a “false positive error” which is also called a Type I or alpha error. It is false because the “guilty” (positive) decision is not aligned with the not guilty (negative) reality. guilty but I really WASN’T guilty! false positive
  23. 23. Although we prefer correct decisions, if we cannot be correct, we prefer the false negative error over the false positive error. In other words you’d rather render a “NOT GUILTY” verdict when there is GUILT. Than a “GUILTY” verdict where there is NO GUILT.
  24. 24. Although we prefer correct decisions, if we cannot be correct, we prefer the false negative error over the false positive error. In other words you’d rather render a “NOT GUILTY” verdict when there is GUILT. not guilty and I really WAS guilty! Than a “GUILTY” verdict where there is NO GUILT.
  25. 25. Although we prefer correct decisions, if we cannot be correct, we prefer the false negative error over the false positive error. In other words you’d rather render a “NOT GUILTY” verdict when there is GUILT. not guilty and I really WAS guilty! Than a “GUILTY” verdict where there is NO GUILT. guilty but I really WASN’T guilty!
  26. 26. In judicial decisions we would rather let a guilty defendant go free . . . than convict and imprison an innocent defendant. Our default position of “not guilty” supports this preference and protects against the least favorable condition.
  27. 27. In judicial decisions we would rather let a guilty defendant go free . . . than convict and imprison an innocent defendant. Our default position of “not guilty” supports this preference and protects against the least favorable condition.
  28. 28. In judicial decisions we would rather let a guilty defendant go free . . . than convict and imprison an innocent defendant. Our default position of “not guilty” supports this preference and protects against the least favorable condition.
  29. 29. Review the following slide and answer the questions that follow:
  30. 30. Review the following slide and answer the questions that follow: What type of decision is made when a guilty (+) verdict is rendered and the person is guilty (+)?
  31. 31. Review the following slide and answer the questions that follow: What type of decision is made when a guilty (+) verdict is rendered and the person is guilty (+)?
  32. 32. Review the following slide and answer the questions that follow: What type of decision is made when a not guilty (-) verdict is rendered and the person is not guilty (-)?
  33. 33. Review the following slide and answer the questions that follow: What type of decision is made when a not guilty (-) verdict is rendered and the person is not guilty (-)?
  34. 34. Review the following slide and answer the questions that follow: What type of decision is made when a guilty (+) verdict is rendered and the person is not guilty (-)?
  35. 35. Review the following slide and answer the questions that follow: What type of decision is made when a guilty (+) verdict is rendered and the person is not guilty (-)?
  36. 36. Review the following slide and answer the questions that follow: What type of decision is made when a not guilty (-) verdict is rendered and the person is guilty (+) ?
  37. 37. Review the following slide and answer the questions that follow: What type of decision is made when a not guilty (-) verdict is rendered and the person is guilty (+) ?
  38. 38. Each conviction protects against Type I error at a different stringency according to the gravity of the punishment to be imposed.
  39. 39. The haunting reality is that we really never know the reality of the guilt or innocence of defendants. We make our best decisions knowing that there is a probability that we have made an error.
  40. 40. The haunting reality is that we really never know the reality of the guilt or innocence of defendants. We make our best decisions knowing that there is a probability that we have made an error.
  41. 41. Statistical hypothesis testing and decision-making are directly analogous to judicial decision making. Judicial Decisions Statistical Decisions
  42. 42. Let’s consider an example: A statistician is asked to decide whether a difference exists between two groups of people in terms of some attribute (e.g., excitability). It is a dichotomous decision (meaning only two options), different or not different. There is no in-between or partial decision.
  43. 43. Let’s consider an example: A statistician is asked to decide whether a difference exists between two groups of people in terms of some attribute (e.g., excitability). It is a dichotomous decision (meaning only two options), different or not different. There is no in-between or partial decision.
  44. 44. Let’s consider an example: A statistician is asked to decide whether a difference exists between two groups of people in terms of some attribute (e.g., excitability). It is a dichotomous decision (meaning only two options), different or not different. There is no in-between or partial decision. x
  45. 45. The statistician does not begin her decision-making in a neutral position. The default position is “not different.” This is also called the “null hypothesis.”
  46. 46. The statistician does not begin her decision-making in a neutral position. The default position is “not different.” This is also called the “null hypothesis.”
  47. 47. The statistician does not begin her decision-making in a neutral position. The default position is “not different.” This is also called the “null hypothesis.”
  48. 48. The research findings must present sufficient evidence to convince the statistician to move from her default position of no difference to a conclusion that the groups are different in terms of the attribute.
  49. 49. The statistician will make a decision which may or may not coincide with reality. The apparent differences may be due to chance or may be real.
  50. 50. The statistician will make a decision which may or may not coincide with reality. The apparent differences may be due to chance or may be real. OR Something that is really happening
  51. 51. When the statistician decides “not different” (fails to reject the null hypothesis, maintains the default position) and the groups are, in reality, not different, she has made a correct decision called a “true negative decision.” true negative
  52. 52. It is true because the “no difference” (negative) decision aligns with “no difference” reality. not guilty and I really WASN’T guilty! true negative
  53. 53. When the statistician decides that there is a difference (rejects the null hypothesis, moves off of the default position) and the groups are, in reality, different, she has made a correct decision called a true positive decision. true positive
  54. 54. It is true because the “different” (positive) decision aligns with the “different” (positive) reality. guilty and I really WAS guilty! true positive
  55. 55. When the statistician decides “not different” (fails to reject the null hypothesis, maintains the default position) and the group are, in reality different, she has made a false negative error. false negative
  56. 56. It is false because the decision of no difference (negative) does not align with difference (positive) reality. not guilty false negative Ha ha! and I really WAS guilty!
  57. 57. Although we prefer correct decisions, if we cannot be correct, we then prefer false negative error over the alternative error.
  58. 58. When a statistician decides that there is a difference (positive) between the groups and rejects the null hypothesis of no difference and, in reality, there is no difference, she has made a false positive error (also called Type I error or alpha error.) false positive
  59. 59. It is false because the “difference” (positive) decision does not align with the “no difference” (negative) reality. guilty but I really WASN’T guilty! false positive
  60. 60. Our hypothesis testing conventions protect against false positive, Type I error by holding a default position of the null hypothesis. α Beware of Type I Error
  61. 61. We set a standard of evidence that is required before rejecting the default null hypothesis.
  62. 62. The standard of evidence is based on the probability density of the sampling distribution.
  63. 63. Using probability density we can estimate the probability of Type I error. If the mean of the sample is here, then we have a .0001 or .01% chance that we made a Type I error.
  64. 64. Or in other words, we have a .01% chance of rejecting the null hypothesis that the group scores come from two different populations (claiming guilty) and being wrong when both groups were really part of the same population (not guilty)
  65. 65. When the probability of Type I error is at a low enough level, we reject the default, null hypothesis. Like in our previous example.
  66. 66. The conventional level of tolerable Type I error is .05. 95% .05 or 5% chance that we selected a sample from this population and claimed it was a sample from another population = false positive
  67. 67. This means that out of 100 similar decisions based on these data … 10 11 09 10 11 12 08 09 10 11 12 13 07 08 09 10 11 12 13 14 06 07 08 09 10 11 12 13 14 15 05 06 07 08 09 10 11 12 13 14 15 16 5 10 15
  68. 68. … we will be wrong (make a Type I error) less than 5 times. 10 11 09 10 11 12 08 09 10 11 12 13 07 08 09 10 11 12 13 14 06 07 08 09 10 11 12 13 14 15 05 06 07 08 09 10 11 12 13 14 15 16 5 10 15
  69. 69. One advantage that statisticians have over juries is that we can estimate the probability of Type I error while they cannot. I can estimate the probability of being right or wrong Not sure of the probability of being right or wrong
  70. 70. (Or, at least it is easier for us to do so than for them. There is some recent research in rape cases that has estimated how frequently juries make Type I errors in such cases.)
  71. 71. Even so, we do not get to make the similar decision 100 times.
  72. 72. We tend to make the decision once. The haunting reality is that we never know in this one decision whether it is one of the probably occurring Type I errors.
  73. 73. In other words, we take a sample of 30 persons and get a score of 7. 10 11 09 10 11 12 08 09 10 11 12 13 07 08 09 10 11 12 13 14 06 07 08 09 10 11 12 13 14 15 05 06 07 08 09 10 11 12 13 14 15 16 5 10 15
  74. 74. In other words, we take a sample of 30 persons and get a score of 7. And then another sample and get a score of 12, and another with a score of 11, and so on and so on until the distribution below emerges. 10 11 09 10 11 12 08 09 10 11 12 13 07 08 09 10 11 12 13 14 06 07 08 09 10 11 12 13 14 15 05 06 07 08 09 10 11 12 13 14 15 16 5 10 15
  75. 75. But since, in real life, we usually only take one sample of 30 for our research purposes, 10 11 09 10 11 12 08 09 10 11 12 13 07 08 09 10 11 12 13 14 14 14 06 07 08 09 10 11 12 13 15 05 06 07 08 09 10 11 12 13 15 16 5 10 15
  76. 76. But since, in real life, we usually only take one sample of 30 for our research purposes, 10 10 11 09 10 11 12 08 09 10 11 12 13 07 08 09 10 11 12 13 14 14 14 06 07 08 09 10 11 12 13 15 05 06 07 08 09 10 11 12 13 15 16 5 10 15
  77. 77. But since, in real life, we usually only take one sample of 30 for our research purposes, we don’t know if the sample was selected from the far left of the distribution below 10 11 09 10 11 12 08 09 10 11 12 13 07 08 09 10 11 12 13 14 14 14 06 07 08 09 10 11 12 13 15 06 05 06 07 08 09 10 11 12 13 15 16 5 10 15
  78. 78. But since, in real life, we usually only take one sample of 30 for our research purposes, we don’t know if the sample was selected from the far left of the distribution below or the far right 10 11 09 10 11 12 08 09 10 11 12 13 07 08 09 10 11 12 13 14 14 14 06 07 08 09 10 11 12 13 15 05 06 07 08 09 10 11 12 13 15 16 5 10 15 16
  79. 79. But since, in real life, we usually only take one sample of 30 for our research purposes, we don’t know if the sample was selected from the far left of the distribution below or the far right or the middle. 10 11 11 09 10 11 12 08 09 10 11 12 13 07 08 09 10 11 12 13 14 14 14 06 07 08 09 10 11 12 13 15 05 06 07 08 09 10 11 12 13 15 16 5 10 15
  80. 80. But since, in real life, we usually only take one sample of 30 for our research purposes, we don’t know if the sample was selected from the far left of the distribution below or the far right or the middle. So, we examine the probability that the sample did or did not come from the far left or the far right. 10 11 09 10 11 12 08 09 10 11 12 13 07 08 09 10 11 12 13 14 14 14 06 07 08 09 10 11 12 13 15 05 06 07 08 09 10 11 12 13 15 16 5 10 15
  81. 81. But since, in real life, we usually only take one sample of 30 for our research purposes, we don’t know if the sample was selected from the far left of the distribution below or the far right or the middle. So, we examine the probability that the sample did or did not come from the far left or the far right. 10 11 09 10 11 12 08 09 10 11 12 13 07 08 09 10 11 12 13 14 14 14 06 07 08 09 10 11 12 13 15 05 06 07 08 09 10 11 12 13 15 16 5 10 15 Hmm. . . What are the chances the sample came from the far right or left of the Distri-bution?
  82. 82. So let’s say we want to know if the students who go to a college party are more excited to be there than little girls at a birthday party.
  83. 83. Here are the sampling distributions of the excitability of young girls at a birthday party.
  84. 84. Let’s say we don’t have the same kind of distribution for college student excitability at a party. ?
  85. 85. We want to know if there is a statistical difference between the girls at the birthday party and the excitability of college students at a Friday night party.
  86. 86. We randomly select a group of college students at a party and measure their levels of excitability.
  87. 87. Our random selection is “13”. 13
  88. 88. Our random selection is “13”. Since this number does not lie in the extreme ends we would reject the null hypothesis or render a judgment of “not guilty”. 13
  89. 89. Our random selection is “13”. Since this number does not lie in the extreme ends we would reject the null hypothesis or render a judgment of “not guilty”. College Students and little girls show no difference. 13
  90. 90. However, what if we randomly selected a college student sample with an average excitability value of “05”. 05
  91. 91. However, what if we randomly selected a college student sample with an average excitability value of “05”. Wow! This is a rare occurrence. 05
  92. 92. Because the chance of that happening is so rare we would reject the null hypothesis. 05
  93. 93. Because the chance of that happening is so rare we would reject the null hypothesis. We would say “guilty!” 05
  94. 94. Because the chance of that happening is so rare we would reject the null hypothesis. We would say “guilty!” But if in reality there is no difference, 05
  95. 95. Because the chance of that happening is so rare we would reject the null hypothesis. We would say “guilty!” But if in reality there is no difference, then we have made a type I error. 05
  96. 96. Because the chance of that happening is so rare we would reject the null hypothesis. We would say “guilty!” But if in reality there is no difference, then we have made a type I error. 05 Researchers are willing to take that chance.
  97. 97. In conclusion, hypothesis testing, is a way of determining the probability of our default position (not guilty or no difference) being correct or incorrect.
  98. 98. In conclusion, hypothesis testing, is a way of determining the probability of our default position (not guilty or no difference) being correct or incorrect. We determine the likelihood of being right or wrong based on the results.
  99. 99. In conclusion, hypothesis testing, is a way of determining the probability of our default position (not guilty or no difference) being correct or incorrect. We determine the likelihood of being right or wrong based on the results. Then we decide if we are willing to maintain our default position (no difference) or go out on a limb and change our default position (yes there is a difference).
  100. 100. What follows are exercises to help you check your understanding.
  101. 101. Go as far as you feel you need to until you have a good feel for what you know.
  102. 102. First Set of Questions
  103. 103. 1. Which expression below from the world of judicial decision-making best describes the “Null-hypothesis”?
  104. 104. 1. Which expression below from the world of judicial decision-making best describes the “Null-hypothesis”? A. “Guilty as charged” B. “Not guilty until proven innocent” C. “Pleading no contest”
  105. 105. 1. Which expression below from the world of judicial decision-making best describes the “Null-hypothesis”? A. “Guilty as charged” B. “Not guilty until proven innocent” C. “Pleading no contest”
  106. 106. 1. Which expression below from the world of judicial decision-making best describes the “Null-hypothesis”? A. “Guilty as charged” B. “Not guilty until proven innocent” C. “Pleading no contest” 2. What is another way to say “Null-hypothesis”?
  107. 107. 1. Which expression below from the world of judicial decision-making best describes the “Null-hypothesis”? A. “Guilty as charged” B. “Not guilty until proven innocent” C. “Pleading no contest” 2. What is another way to say “Null-hypothesis”? A. Not clear B. Not different C. Not important
  108. 108. 1. Which expression below from the world of judicial decision-making best describes the “Null-hypothesis”? A. “Guilty as charged” B. “Not guilty until proven innocent” C. “Pleading no contest” 2. What is another way to say “Null-hypothesis”? A. Not clear B. Not different C. Not important
  109. 109. With hypothesis testing we are attempting to set up a default position of not guilty. We stay in that position unless we have enough evidence to overturn it.
  110. 110. With hypothesis testing we are attempting to set up a default position of not guilty. We stay in that position unless we have enough evidence to overturn it. Let’s say our null-hypothesis is the following:
  111. 111. With hypothesis testing we are attempting to set up a default position of not guilty. We stay in that position unless we have enough evidence to overturn it. Let’s say our null-hypothesis is the following: There is no difference in IQ between children who are exposed to classical music between the ages of 0 and 3 and those who were not.
  112. 112. With hypothesis testing we are attempting to set up a default position of not guilty. We stay in that position unless we have enough evidence to overturn it. Let’s say our null-hypothesis is the following: There is no difference in IQ between children who are exposed to classical music between the ages of 0 and 3 and those who were not. This is our default position. We are not neutral, we are claiming at the outset that there is no difference.
  113. 113. But then along comes some evidence that over turns that position. So we reject the null hypothesis and claim there is a probable difference.
  114. 114. But then along comes some evidence that over turns that position. So we reject the null hypothesis and claim there is a probable difference. Notice how we don’t say “there is a difference”. We say there is a probable or statistical difference. This just means that with statistics we are never 100% certain. We just say that the probability that we are wrong is a certain percent. Usually that percent needs to be pretty low.
  115. 115. If we have estimated that there is a 60% chance that we are wrong, that is a risk not worth taking. If you were told that you had a 60% chance of losing a lot of money and a 40% chance of making a lot of money, would you take that chance? Probably not. But if you were told that you had only a 5% chance of losing a lot of money and a 95% of earning a lot, that might be a chance you would be willing to take. The same holds true with hypothesis testing.
  116. 116. If we have estimated that there is a 60% chance that we are wrong, that is a risk not worth taking. If you were told that you had a 60% chance of losing a lot of money and a 40% chance of making a lot of money, would you take that chance? Probably not. But if you were told that you had only a 5% chance of losing a lot of money and a 95% of earning a lot, that might be a chance you would be willing to take. The same holds true with hypothesis testing.
  117. 117. Based on that instruction, consider your answer to these questions again and explain the correct answer in your own words.
  118. 118. Based on that instruction, consider your answer to these questions again and explain the correct answer in your own words. 1. Which expression below from the world of judicial decision-making best describes the “Null-hypothesis”? A. “Guilty as charged” B. “Not guilty until proven innocent” C. “Pleading no contest” 2. What is another way to say “Null-hypothesis”? A. Not clear B. Not different C. Not important
  119. 119. Second Set of Questions – see if you can answer these questions, if not go to the instruction that follows and you’ll be given an opportunity to respond to the questions armed with the instruction.
  120. 120. 3. When the jury decides “not guilty” and the defendant really is “not guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were right to do so. B. REJECT the null hypothesis and it turns out - - - you were right to do so.
  121. 121. 3. When the jury decides “not guilty” and the defendant really is “not guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were right to do so. B. REJECT the null hypothesis and it turns out - - - you were right to do so.
  122. 122. 3. When the jury decides “not guilty” and the defendant really is “not guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were right to do so. B. REJECT the null hypothesis and it turns out - - - you were right to do so.
  123. 123. 3. When the jury decides “not guilty” and the defendant really is “not guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were right to do so. B. REJECT the null hypothesis and it turns out - - - you were right to do so.
  124. 124. 3. When the jury decides “not guilty” and the defendant really is “not guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were right to do so. B. REJECT the null hypothesis and it turns out - - - you were right to do so. 4. When the jury decides “guilty” and the defendant actually was “not guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were wrong to do so. B. REJECT the null hypothesis and it turns out - - - you were wrong to do so.
  125. 125. 3. When the jury decides “not guilty” and the defendant really is “not guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were right to do so. B. REJECT the null hypothesis and it turns out - - - you were right to do so. 4. When the jury decides “guilty” and the defendant actually was “not guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were wrong to do so. B. REJECT the null hypothesis and it turns out - - - you were wrong to do so.
  126. 126. 3. When the jury decides “not guilty” and the defendant really is “not guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were right to do so. B. REJECT the null hypothesis and it turns out - - - you were right to do so. 4. When the jury decides “guilty” and the defendant actually was “not guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were wrong to do so. B. REJECT the null hypothesis and it turns out - - - you were wrong to do so.
  127. 127. 3. When the jury decides “not guilty” and the defendant really is “not guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were right to do so. B. REJECT the null hypothesis and it turns out - - - you were right to do so. 4. When the jury decides “guilty” and the defendant actually was “not guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were wrong to do so. B. REJECT the null hypothesis and it turns out - - - you were wrong to do so.
  128. 128. 5. When the jury decides “not guilty” and the defendant actually was “guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were wrong to do so. B. REJECT the null hypothesis and it turns out - - - you were wrong to do so.
  129. 129. 5. When the jury decides “not guilty” and the defendant actually was “guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were wrong to do so. B. REJECT the null hypothesis and it turns out - - - you were wrong to do so.
  130. 130. 5. When the jury decides “not guilty” and the defendant actually was “guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were wrong to do so. B. REJECT the null hypothesis and it turns out - - - you were wrong to do so.
  131. 131. 5. When the jury decides “not guilty” and the defendant actually was “guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were wrong to do so. B. REJECT the null hypothesis and it turns out - - - you were wrong to do so.
  132. 132. 5. When the jury decides “not guilty” and the defendant actually was “guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were wrong to do so. B. REJECT the null hypothesis and it turns out - - - you were wrong to do so. 6. When the jury decides “guilty” and the defendant really is “guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were right to do so. B. REJECT the null hypothesis and it turns out - - - you were right to do so.
  133. 133. 5. When the jury decides “not guilty” and the defendant actually was “guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were wrong to do so. B. REJECT the null hypothesis and it turns out - - - you were wrong to do so. 6. When the jury decides “guilty” and the defendant really is “guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were right to do so. B. REJECT the null hypothesis and it turns out - - - you were right to do so.
  134. 134. 5. When the jury decides “not guilty” and the defendant actually was “guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were wrong to do so. B. REJECT the null hypothesis and it turns out - - - you were wrong to do so. 6. When the jury decides “guilty” and the defendant really is “guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were right to do so. B. REJECT the null hypothesis and it turns out - - - you were right to do so.
  135. 135. 5. When the jury decides “not guilty” and the defendant actually was “guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were wrong to do so. B. REJECT the null hypothesis and it turns out - - - you were wrong to do so. 6. When the jury decides “guilty” and the defendant really is “guilty”, in statistics that is the same as saying: A. ACCEPT the null hypothesis and it turns out - - - you were right to do so. B. REJECT the null hypothesis and it turns out - - - you were right to do so.
  136. 136. Accepting the null-hypothesis is essentially like saying “not guilty” or that we accept the default position of innocence or no difference. Rejecting the null-hypothesis is essentially like saying “guilty” or that we reject the default position of innocence or there is enough evidence to suggest there is a difference.
  137. 137. Here is a visual:
  138. 138. Here is a visual: Null-hypothesis ACCEPTED!
  139. 139. Here is a visual: Null-hypothesis ACCEPTED! I was found NOT GUILTY!
  140. 140. Here is a visual: Null-hypothesis ACCEPTED! I was found NOT GUILTY! Na, na, . . . nanana! There is NOT enough statistical evidence to convict or reject the null-hypothesis!
  141. 141. Here is a visual: Null-hypothesis ACCEPTED! I was found NOT GUILTY! Na, na, . . . nanana! There is NOT enough statistical evidence to convict or reject the null-hypothesis! Not Guilty = Accept the Null
  142. 142. Here is a visual: Null-hypothesis REJECTED!
  143. 143. Here is a visual: Null-hypothesis REJECTED! I was found GUILTY!
  144. 144. Here is a visual: Null-hypothesis REJECTED! I was found GUILTY! Wa, Wa! There IS enough statistical evidence to convict or reject the null- Hypothesis!
  145. 145. Here is a visual: Null-hypothesis REJECTED! I was found GUILTY! Wa, Wa! There IS enough statistical evidence to convict or reject the null- Hypothesis! Guilty = Reject the Null
  146. 146. Third Set of Questions - see if you can answer these questions, if not go to the instruction that follows and you’ll be given an opportunity to respond to the questions armed with the instruction.
  147. 147. 7. When the jury decides “guilty” (reject the null) and the defendant actually was “not guilty” (shouldn’t have rejected the null), what type of error has been committed? A. Type I error B. Type II error
  148. 148. 7. When the jury decides “guilty” (reject the null) and the defendant actually was “not guilty” (shouldn’t have rejected the null), what type of error has been committed? A. Type I error B. Type II error 8. When the jury decides “not guilty” (accept the null) and the defendant actually was “guilty” (reject the null), what type of error has been committed? A. Type I error B. Type II error
  149. 149. 9. Which type of error is preferable? A. Type I error B. Type II error
  150. 150. 9. Which type of error is preferable? A. Type I error B. Type II error 10. Question: What is the haunting reality? Answer: We actually never know for sure if we have committed a type I or II error. All we are doing is determining the probability that we . . . have committed an error. are correct in our hypothesis.
  151. 151. 9. Which type of error is preferable? A. Type I error B. Type II error 10. Question: What is the haunting reality? Answer: We actually never know for sure if we have committed a type I or II error. All we are doing is determining the probability that we . . . have committed an error. are correct in our hypothesis.
  152. 152. 9. Which type of error is preferable? A. Type I error B. Type II error 10. Question: What is the haunting reality? Answer: We actually never know for sure if we have committed a type I or II error. All we are doing is determining the probability that we . . . A. have committed an error. B. are correct in our hypothesis.
  153. 153. Let’s consider each type of error
  154. 154. 1. State your null-hypothesis; There is no significant difference between females and males in terms of their preference of certain sports-car colors 2. Collect your evidence, 3. Determine if the evidence merits accepting or rejecting the null-hypothesis, 4. You accept the null 5. In reality (and you could never know this for sure) you were wrong. In actuality there is a difference between men and women sports-car color preference and you should have rejected the null. 6. This is a type I error
  155. 155. 1. State your null-hypothesis; There is no significant difference between females and males in terms of their preference of certain sports-car colors 2. Collect your evidence 3. Determine if the evidence merits accepting or rejecting the null-hypothesis, 4. You accept the null 5. In reality (and you could never know this for sure) you were wrong. In actuality there is a difference between men and women sports-car color preference and you should have rejected the null. 6. This is a type I error
  156. 156. 1. State your null-hypothesis; There is no significant difference between females and males in terms of their preference of certain sports-car colors 2. Collect your evidence 3. Determine if the evidence merits accepting or rejecting the null-hypothesis 4. You accept the null 5. In reality (and you could never know this for sure) you were wrong. In actuality there is a difference between men and women sports-car color preference and you should have rejected the null. 6. This is a type I error
  157. 157. 1. State your null-hypothesis; There is no significant difference between females and males in terms of their preference of certain sports-car colors 2. Collect your evidence 3. Determine if the evidence merits accepting or rejecting the null-hypothesis 4. You accept the null 5. In reality (and you could never know this for sure) you were wrong. In actuality there is a difference between men and women sports-car color preference and you should have rejected the null. 6. This is a type I error
  158. 158. 1. State your null-hypothesis; There is no significant difference between females and males in terms of their preference of certain sports-car colors 2. Collect your evidence 3. Determine if the evidence merits accepting or rejecting the null-hypothesis 4. You accept the null 5. In reality (and you could never know this for sure) you were wrong. In actuality there is a difference between men and women sports-car color preference and you should have rejected the null. 6. This is a type I error
  159. 159. 1. State your null-hypothesis; There is no significant difference between females and males in terms of their preference of certain sports-car colors 2. Collect your evidence 3. Determine if the evidence merits accepting or rejecting the null-hypothesis 4. You accept the null 5. In reality (and you could never know this for sure) you were wrong. In actuality there is a difference between men and women sports-car color preference and you should have rejected the null. This is a type I error
  160. 160. 1. State your null-hypothesis; There is no significant difference between females and males in terms of their preference of certain sports-car colors 2. Collect your evidence 3. Determine if the evidence merits accepting or rejecting the null-hypothesis 4. You reject the null 5. In reality (and you could never know this for sure) you were wrong. In actuality there is NO difference between men and women sports-car color preference and you should have accepted the null This is a type II error
  161. 161. 1. State your null-hypothesis; There is no significant difference between females and males in terms of their preference of certain sports-car colors 2. Collect your evidence 3. Determine if the evidence merits accepting or rejecting the null-hypothesis 4. You reject the null 5. In reality (and you could never know this for sure) you were wrong. In actuality there is NO difference between men and women sports-car color preference and you should have accepted the null This is a type II error
  162. 162. 1. State your null-hypothesis; There is no significant difference between females and males in terms of their preference of certain sports-car colors 2. Collect your evidence 3. Determine if the evidence merits accepting or rejecting the null-hypothesis 4. You reject the null 5. In reality (and you could never know this for sure) you were wrong. In actuality there is NO difference between men and women sports-car color preference and you should have accepted the null This is a type II error
  163. 163. 1. State your null-hypothesis; There is no significant difference between females and males in terms of their preference of certain sports-car colors 2. Collect your evidence 3. Determine if the evidence merits accepting or rejecting the null-hypothesis 4. You reject the null 5. In reality (and you could never know this for sure) you were wrong. In actuality there is NO difference between men and women sports-car color preference and you should have accepted the null This is a type II error
  164. 164. 1. State your null-hypothesis; There is no significant difference between females and males in terms of their preference of certain sports-car colors 2. Collect your evidence 3. Determine if the evidence merits accepting or rejecting the null-hypothesis 4. You reject the null 5. In reality (and you could never know this for sure) you were wrong. In actuality there is NO difference between men and women sports-car color preference and you should have accepted the null This is a type II error
  165. 165. 1. State your null-hypothesis; There is no significant difference between females and males in terms of their preference of certain sports-car colors 2. Collect your evidence 3. Determine if the evidence merits accepting or rejecting the null-hypothesis 4. You reject the null 5. In reality (and you could never know this for sure) you were wrong. In actuality there is NO difference between men and women sports-car color preference and you should have accepted the null This is a type II error
  166. 166. You’ll never know if you committed a type I or II error. You can only estimate the probability that you did!
  167. 167. That’s because with statistics we deal in probability, not certainty.
  168. 168. Based on the instruction you just received, respond to these questions again. Explain your reasoning for selecting the options you did.
  169. 169. 7. When the jury decides “guilty” (reject the null) and the defendant actually was “not guilty” (shouldn’t have rejected the null), what type of error has been committed? A. Type I error B. Type II error
  170. 170. 7. When the jury decides “guilty” (reject the null) and the defendant actually was “not guilty” (shouldn’t have rejected the null), what type of error has been committed? A. Type I error B. Type II error
  171. 171. 7. When the jury decides “guilty” (reject the null) and the defendant actually was “not guilty” (shouldn’t have rejected the null), what type of error has been committed? A. Type I error B. Type II error 8. When the jury decides “not guilty” (accept the null) and the defendant actually was “guilty” (reject the null), what type of error has been committed? A. Type I error B. Type II error
  172. 172. 7. When the jury decides “guilty” (reject the null) and the defendant actually was “not guilty” (shouldn’t have rejected the null), what type of error has been committed? A. Type I error B. Type II error 8. When the jury decides “not guilty” (accept the null) and the defendant actually was “guilty” (reject the null), what type of error has been committed? A. Type I error B. Type II error
  173. 173. 9. Which type of error is preferable? A. Type I error B. Type II error
  174. 174. 9. Which type of error is preferable? A. Type I error B. Type II error
  175. 175. 9. Which type of error is preferable? A. Type I error B. Type II error 10. Question: What is the haunting reality? Answer: We actually never know for sure if we have committed a type I or II error. All we are doing is determining the probability that we . . . have committed an error. are correct in our hypothesis.
  176. 176. 9. Which type of error is preferable? A. Type I error B. Type II error 10. Question: What is the haunting reality? Answer: We actually never know for sure if we have committed a type I or II error. All we are doing is determining the probability that we . . . have committed an error. are correct in our hypothesis.
  177. 177. 9. Which type of error is preferable? A. Type I error B. Type II error 10. Question: What is the haunting reality? Answer: We actually never know for sure if we have committed a type I or II error. All we are doing is determining the probability that we . . . A. have committed an error. B. are correct in our hypothesis. Answers: 7-A, 8-B, 9-B, 10-A
  178. 178. 9. Which type of error is preferable? A. Type I error B. Type II error 10. Question: What is the haunting reality? Answer: We actually never know for sure if we have committed a type I or II error. All we are doing is determining the probability that we . . . A. have committed an error. B. are correct in our hypothesis. Answers: 7-A, 8-B, 9-B, 10-A
  179. 179. Fourth Set of Questions - see if you can answer these questions, if not go to the instruction that follows and you’ll be given an opportunity to respond to the questions armed with the instruction.
  180. 180. 11. Question: How do we decide how much evidence is required before we will reject the null hypothesis? Answer: We estimate the probability of being ______ a certain percent of the time (e.g., .05 or 5% of the time). a. right b. wrong
  181. 181. 11. Question: How do we decide how much evidence is required before we will reject the null hypothesis? Answer: We estimate the probability of being ______ a certain percent of the time (e.g., .05 or 5% of the time). a. right b. wrong
  182. 182. 11. Question: How do we decide how much evidence is required before we will reject the null hypothesis? Answer: We estimate the probability of being ______ a certain percent of the time (e.g., .05 or 5% of the time). a. right b. wrong 12. Question: What does a .05 rejection level mean? Answer: If we were to take the same small sample 100 times from a population, we would be willing to _____________________ .05 or 5% of the time a. . . . take the chance of being wrong . . . b. . . . reject the null hypothesis . . .
  183. 183. 11. Question: How do we decide how much evidence is required before we will reject the null hypothesis? Answer: We estimate the probability of being ______ a certain percent of the time (e.g., .05 or 5% of the time). a. right b. wrong 12. Question: What does a .05 rejection level mean? Answer: If we were to take the same small sample 100 times from a population, we would be willing to _____________________ .05 or 5% of the time a. . . . take the chance of being wrong . . . b. . . . reject the null hypothesis . . .
  184. 184. In statistics we generally ask ourselves, “What is the probability that we have made a type I error?”
  185. 185. In statistics we generally ask ourselves, “What is the probability that we have made a Type I Error?” Type I errors are considered a bigger issue because if we are wrong, than we might waste a lot of money or impact people negatively (e.g., spend millions of dollars on a new drug that doesn’t work).
  186. 186. In statistics we generally ask ourselves, “What is the probability that we have made a Type I Error?” Type I errors are considered a bigger issue because if we are wrong, than we might waste a lot of money or impact people negatively (e.g., spend millions of dollars on a new drug that doesn’t work). Type II errors are considered less of an issue because if we are wrong, than we may stop or continue researching.
  187. 187. We have to have determine a cut-off point as to when we will reject the null-hypothesis. No matter what cut-off point we could have chosen, the decision would always have been somewhat arbitrary.
  188. 188. We have to have determine a cut-off point as to when we will reject the null-hypothesis. No matter what cut-off point we could have chosen, the decision would always have been somewhat arbitrary. Would we be satisfied with a 75% chance of committing a type I error? Probably not. That means out of 100 experiments we would live with being wrong about our conclusions 75 times.
  189. 189. Would we be satisfied with a .01% chance of committing a type I error? Probably not. That means out of 10,000 experiments we would live with being wrong about our conclusions only once. If that were the case, then almost no null-hypothesis could ever be rejected.
  190. 190. Would we be satisfied with a .01% chance of committing a type I error? Probably not. That means out of 10,000 experiments we would live with being wrong about our conclusions only once. If that were the case, then almost no null-hypothesis could ever be rejected. In the discipline of statistics .05 or 5% of a chance of committing a type I error has been deemed an acceptable arbitrary cut-off point. This means that out of 100 experiments we will live with being wrong five times.
  191. 191. Based on the instruction you just received, respond to these questions again. Explain your reasoning for selecting the options you did.
  192. 192. 11. Question: How do we decide how much evidence is required before we will reject the null hypothesis? Answer: We estimate the probability of being ______ a certain percent of the time (e.g., .05 or 5% of the time). a. right b. wrong
  193. 193. 11. Question: How do we decide how much evidence is required before we will reject the null hypothesis? Answer: We estimate the probability of being ______ a certain percent of the time (e.g., .05 or 5% of the time). a. right b. wrong
  194. 194. 11. Question: How do we decide how much evidence is required before we will reject the null hypothesis? Answer: We estimate the probability of being ______ a certain percent of the time (e.g., .05 or 5% of the time). a. right b. wrong
  195. 195. 11. Question: How do we decide how much evidence is required before we will reject the null hypothesis? Answer: We estimate the probability of being ______ a certain percent of the time (e.g., .05 or 5% of the time). a. right b. wrong 12. Question: What does a .05 rejection level mean? Answer: If we were to take the same small sample 100 times from a population, we would be willing to _____________________ .05 or 5% of the time a. . . . take the chance of being wrong . . . b. . . . reject the null hypothesis . . .
  196. 196. 11. Question: How do we decide how much evidence is required before we will reject the null hypothesis? Answer: We estimate the probability of being ______ a certain percent of the time (e.g., .05 or 5% of the time). a. right b. wrong 12. Question: What does a .05 rejection level mean? Answer: If we were to take the same small sample 100 times from a population, we would be willing to _____________________ .05 or 5% of the time a. . . . take the chance of being wrong . . . b. . . . reject the null hypothesis . . . Answers: 11-B, 12-A
  197. 197. 11. Question: How do we decide how much evidence is required before we will reject the null hypothesis? Answer: We estimate the probability of being ______ a certain percent of the time (e.g., .05 or 5% of the time). a. right b. wrong 12. Question: What does a .05 rejection level mean? Answer: If we were to take the same small sample 100 times from a population, we would be willing to _____________________ .05 or 5% of the time a. . . . take the chance of being wrong . . . b. . . . reject the null hypothesis . . . Answers: 11-B, 12-A

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