{"id":5312,"date":"2022-08-19T16:56:57","date_gmt":"2022-08-19T16:56:57","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&p=5312"},"modified":"2022-08-19T16:56:57","modified_gmt":"2022-08-19T16:56:57","slug":"11a-preview","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/11a-preview\/","title":{"raw":"11A Preview","rendered":"11A Preview"},"content":{"raw":"Preparing for the next class\r\n\r\nIn the next in-class activity, you will need to understand the basic idea of null and alternative hypotheses, that an event with low probability is very unlikely but still may occur, and how probability can be used as statistical evidence.\r\n\r\nSuppose that you are playing a game with your friend that involves flipping a coin. Each round consists of flipping the coin 10 times. Your friend is favored if more of the flips land on heads, and you are favored if more of the flips land on tails.\r\n
\r\n

Question 1<\/h3>\r\nIn one round of play, your friend gets 8 heads out of the 10 total flips.\r\n
    \r\n \t
  1. What is the probability of obtaining 8 or more heads in 10 flips of a fair coin?\r\nHint: You can find this probability using the binomial model (what are\u00a0[latex] p [\/latex] and\u00a0[latex] n[\/latex] here?).\u00a0 Use technology.<\/li>\r\n \t
  2. Are you surprised that your friend got that many heads?\r\n
      \r\n \t
    1. a) Yes, because the probability is high.<\/li>\r\n \t
    2. b) Yes, because the probability is low.<\/li>\r\n \t
    3. c) No, because the probability is high.<\/li>\r\n \t
    4. d) No, because the probability is low.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
    5. You begin to suspect that your friend is not using a fair coin. If your friend\u2019s\u00a0 coin is weighted on the heads side, what can you say about the probability,\u00a0 [latex] p [\/latex] , of obtaining heads with your friend\u2019s coin?\r\n
        \r\n \t
      1. a)\u00a0[latex] p [\/latex] < 5<\/li>\r\n \t
      2. b) [latex] p [\/latex] = 5<\/li>\r\n \t
      3. c) [latex] p [\/latex] = 8<\/li>\r\n \t
      4. d) [latex] p [\/latex] > 5\r\nHint: Remember that the 8 heads obtained was only in one round of 10 flips.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn this situation, you had a baseline assumption when you started playing the game with your friend: the coin is fair. This assumption is called the null hypothesis. Then you suspected that your friend\u2019s coin was weighted in favor of heads. This guess is called the alternative hypothesis. If you wanted to claim that your friend was indeed cheating, you would need to produce evidence in order to prove that their coin was not fair.\r\n\r\nStatistics is a very useful tool in this scenario and can be used to test these kinds of hypotheses. In a statistical hypothesis test:\r\n
          \r\n \t
        • The null hypothesis, [latex] H_{0} [\/latex], is what we assume to be true to begin with. It is often a statement of no change from the previous value or from what is expected (e.g., we expect a coin to be fair).\r\n
            \r\n \t
          • The null hypothesis, [latex] H_{0} [\/latex], is always given in the form: [latex] parameter = value [\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
          • The alternative hypothesis, [latex] H_{A} [\/latex], is what we consider to be plausible if the null hypothesis is false. Often, it is a change from the null hypothesis that we would like to test the accuracy of. The alternative hypothesis answers the question, \u201cDo we think the actual parameter is larger than, smaller than, or just different from the null value, where the null value is the value specified in the null hypothesis?\u201d\r\n
              \r\n \t
            • The alternative hypothesis, [latex] H_{A} [\/latex], is always given as an inequality: [latex] parameter>null value, parameter<null~value[\/latex], or\u00a0[latex] parameter \\neq null~value[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
            • The evidence used is probability. The statistical evidence that we gather is always evidence in support of the alternative hypothesis and against the null hypothesis. We ask ourselves the question, \u201cDo we have enough evidence to reject the null hypothesis?\u201d<\/li>\r\n \t
            • The outcomes of the hypothesis test are either:\r\n
                \r\n \t
              • We reject the null hypothesis (we have gathered enough evidence).<\/li>\r\n \t
              • We fail to reject the null hypothesis (we have not gathered sufficient evidence, so we cannot reject the starting assumption).<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n
                \r\n

                Question 2<\/h3>\r\nIn the context of the coin-flipping game with your friend (where\u00a0[latex] p [\/latex] is the probability of\u00a0 obtaining heads), what is the null hypothesis, [latex] H_{0} [\/latex]?\r\n
                  \r\n \t
                1. a)\u00a0[latex] p = 0.8 [\/latex]<\/li>\r\n \t
                2. b)\u00a0[latex] p > 0.5 [\/latex]<\/li>\r\n \t
                3. c) [latex] p = 0.5 [\/latex]<\/li>\r\n \t
                4. d)\u00a0[latex] p < 0.5 [\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
                  \r\n

                  Question 3<\/h3>\r\nIn the context of the coin-flipping game, what is the alternative hypothesis, [latex] H_{A} [\/latex]?\r\n
                    \r\n \t
                  1. a) [latex] p = 0.8 [\/latex]<\/li>\r\n \t
                  2. b) [latex] p \\neq 0.5 [\/latex]<\/li>\r\n \t
                  3. c) [latex] p = 0.5 [\/latex]<\/li>\r\n \t
                  4. d) [latex] p > 0.5 [\/latex]<\/li>\r\n \t
                  5. e) [latex] p < 0.5 [\/latex]<\/li>\r\n<\/ol>\r\nHint: What is it that you would like to gather evidence of? What do you suspect might\u00a0 be true about your friend\u2019s coin?\r\n\r\n<\/div>\r\nYour friend claims the coin is fair, but you aren\u2019t convinced. You can\u2019t prove for certain\u00a0 that your friend\u2019s coin is weighted unfairly (without special equipment, of course), but\u00a0 you can test your hypotheses with a sample of coin flips. If the proportion of heads in\u00a0 your sample is high enough, it provides strong evidence that your friend\u2019s coin is\u00a0 weighted in their favor. In other words, a high enough proportion of heads would be\u00a0 sufficient evidence for you to reject the assumption that your friend\u2019s coin is fair. Let\u2019s\u00a0 suppose that you flip the coin 20 times for your sample.\r\n
                    \r\n

                    Question 4<\/h3>\r\nWhich of the following would be stronger evidence that the coin is weighted unfairly\u00a0 towards the heads side (i.e., stronger evidence that the null hypothesis is false)?\r\n
                      \r\n \t
                    1. a) A higher proportion of heads, because that would be less likely if the coin is fair<\/li>\r\n \t
                    2. b) A higher proportion of heads, because that would be more likely if the coin is fair<\/li>\r\n \t
                    3. c) A smaller proportion of heads, because that would be less likely if the coin is fair<\/li>\r\n \t
                    4. d) A smaller proportion of heads, because that would be more likely if the coin is fair<\/li>\r\n<\/ol>\r\n<\/div>\r\n
                      \r\n

                      Question 5<\/h3>\r\nIf your sample of 20 coin flips all landed on heads, would it prove with complete\u00a0 certainty that your friend\u2019s coin is weighted?\r\n
                        \r\n \t
                      1. a) Yes, it is impossible for a fair coin to land on heads 20 out of 20 times.<\/li>\r\n \t
                      2. b) No, even though the probability is very small, it is still possible for a fair coin to\u00a0 land on heads 20 out of 20 times.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
                        \r\n

                        Question 6<\/h3>\r\nSuppose that in your sample of 20 coin flips, you obtain 17 heads. The probability of\u00a0 obtaining 17 or more heads in 20 flips of a fair coin is 0.0013. Would you conclude\u00a0 that your friend\u2019s coin is weighted? Explain.\r\n\r\n<\/div>\r\nLooking ahead\r\n\r\nNotice that above, you assumed your friend\u2019s coin is fair, and that assumption was used\u00a0 to calculate the probability of obtaining 17 or more heads. The probability of 0.0013 is\u00a0 the probability of obtaining 17 or more heads out of 20 flips if the coin is fair, and this\u00a0 low probability is evidence that the coin is not fair. Note that no matter what probability\u00a0 you obtain, even if it is a high probability, it is not evidence that the coin is fair. This is\u00a0 because fairness was already the given assumption, and the probability was computed\u00a0 using the assumption that the coin is fair. If we assume something is true, we can\u2019t use\u00a0 that assumption to prove that it\u2019s true (that would be using circular logic). From our\u00a0 sample of coin flips, we either get enough evidence to say the coin is weighted, or we\u00a0 don\u2019t. In hypothesis-testing language, we either get enough evidence to reject the null\u00a0 hypothesis, or we don\u2019t. When we don\u2019t get enough evidence to reject the null\u00a0 hypothesis, we fail to reject the null hypothesis. We never accept the null hypothesis\u00a0 because that was already our assumption to begin with.\r\n
                        \r\n

                        Question 7<\/h3>\r\nWhich of the following are the possible outcomes of your sample of 20 coin flips?\u00a0 There may be more than one correct answer.\r\n
                          \r\n \t
                        1. a) You obtain enough heads that you conclude your friend\u2019s coin is weighted.<\/li>\r\n \t
                        2. b) You obtain enough heads that you conclude your friend\u2019s coin is fair.<\/li>\r\n \t
                        3. c) You do not obtain enough heads to conclude that your friend\u2019s coin is weighted.<\/li>\r\n \t
                        4. d) You do not obtain enough heads to conclude that your friend\u2019s coin is fair.<\/li>\r\n<\/ol>\r\n<\/div>\r\n ","rendered":"

                          Preparing for the next class<\/p>\n

                          In the next in-class activity, you will need to understand the basic idea of null and alternative hypotheses, that an event with low probability is very unlikely but still may occur, and how probability can be used as statistical evidence.<\/p>\n

                          Suppose that you are playing a game with your friend that involves flipping a coin. Each round consists of flipping the coin 10 times. Your friend is favored if more of the flips land on heads, and you are favored if more of the flips land on tails.<\/p>\n

                          \n

                          Question 1<\/h3>\n

                          In one round of play, your friend gets 8 heads out of the 10 total flips.<\/p>\n

                            \n
                          1. What is the probability of obtaining 8 or more heads in 10 flips of a fair coin?
                            \nHint: You can find this probability using the binomial model (what are\u00a0[latex]p[\/latex] and\u00a0[latex]n[\/latex] here?).\u00a0 Use technology.<\/li>\n
                          2. Are you surprised that your friend got that many heads?\n
                              \n
                            1. a) Yes, because the probability is high.<\/li>\n
                            2. b) Yes, because the probability is low.<\/li>\n
                            3. c) No, because the probability is high.<\/li>\n
                            4. d) No, because the probability is low.<\/li>\n<\/ol>\n<\/li>\n
                            5. You begin to suspect that your friend is not using a fair coin. If your friend\u2019s\u00a0 coin is weighted on the heads side, what can you say about the probability,\u00a0 [latex]p[\/latex] , of obtaining heads with your friend\u2019s coin?\n
                                \n
                              1. a)\u00a0[latex]p[\/latex] < 5<\/li>\n
                              2. b) [latex]p[\/latex] = 5<\/li>\n
                              3. c) [latex]p[\/latex] = 8<\/li>\n
                              4. d) [latex]p[\/latex] > 5
                                \nHint: Remember that the 8 heads obtained was only in one round of 10 flips.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n

                                In this situation, you had a baseline assumption when you started playing the game with your friend: the coin is fair. This assumption is called the null hypothesis. Then you suspected that your friend\u2019s coin was weighted in favor of heads. This guess is called the alternative hypothesis. If you wanted to claim that your friend was indeed cheating, you would need to produce evidence in order to prove that their coin was not fair.<\/p>\n

                                Statistics is a very useful tool in this scenario and can be used to test these kinds of hypotheses. In a statistical hypothesis test:<\/p>\n

                                  \n
                                • The null hypothesis, [latex]H_{0}[\/latex], is what we assume to be true to begin with. It is often a statement of no change from the previous value or from what is expected (e.g., we expect a coin to be fair).\n
                                    \n
                                  • The null hypothesis, [latex]H_{0}[\/latex], is always given in the form: [latex]parameter = value[\/latex].<\/li>\n<\/ul>\n<\/li>\n
                                  • The alternative hypothesis, [latex]H_{A}[\/latex], is what we consider to be plausible if the null hypothesis is false. Often, it is a change from the null hypothesis that we would like to test the accuracy of. The alternative hypothesis answers the question, \u201cDo we think the actual parameter is larger than, smaller than, or just different from the null value, where the null value is the value specified in the null hypothesis?\u201d\n
                                      \n
                                    • The alternative hypothesis, [latex]H_{A}[\/latex], is always given as an inequality: [latex]parameter>null value, parameter\n<\/ul>\n<\/li>\n
                                    • The evidence used is probability. The statistical evidence that we gather is always evidence in support of the alternative hypothesis and against the null hypothesis. We ask ourselves the question, \u201cDo we have enough evidence to reject the null hypothesis?\u201d<\/li>\n
                                    • The outcomes of the hypothesis test are either:\n
                                        \n
                                      • We reject the null hypothesis (we have gathered enough evidence).<\/li>\n
                                      • We fail to reject the null hypothesis (we have not gathered sufficient evidence, so we cannot reject the starting assumption).<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
                                        \n

                                        Question 2<\/h3>\n

                                        In the context of the coin-flipping game with your friend (where\u00a0[latex]p[\/latex] is the probability of\u00a0 obtaining heads), what is the null hypothesis, [latex]H_{0}[\/latex]?<\/p>\n

                                          \n
                                        1. a)\u00a0[latex]p = 0.8[\/latex]<\/li>\n
                                        2. b)\u00a0[latex]p > 0.5[\/latex]<\/li>\n
                                        3. c) [latex]p = 0.5[\/latex]<\/li>\n
                                        4. d)\u00a0[latex]p < 0.5[\/latex]<\/li>\n<\/ol>\n<\/div>\n
                                          \n

                                          Question 3<\/h3>\n

                                          In the context of the coin-flipping game, what is the alternative hypothesis, [latex]H_{A}[\/latex]?<\/p>\n

                                            \n
                                          1. a) [latex]p = 0.8[\/latex]<\/li>\n
                                          2. b) [latex]p \\neq 0.5[\/latex]<\/li>\n
                                          3. c) [latex]p = 0.5[\/latex]<\/li>\n
                                          4. d) [latex]p > 0.5[\/latex]<\/li>\n
                                          5. e) [latex]p < 0.5[\/latex]<\/li>\n<\/ol>\n

                                            Hint: What is it that you would like to gather evidence of? What do you suspect might\u00a0 be true about your friend\u2019s coin?<\/p>\n<\/div>\n

                                            Your friend claims the coin is fair, but you aren\u2019t convinced. You can\u2019t prove for certain\u00a0 that your friend\u2019s coin is weighted unfairly (without special equipment, of course), but\u00a0 you can test your hypotheses with a sample of coin flips. If the proportion of heads in\u00a0 your sample is high enough, it provides strong evidence that your friend\u2019s coin is\u00a0 weighted in their favor. In other words, a high enough proportion of heads would be\u00a0 sufficient evidence for you to reject the assumption that your friend\u2019s coin is fair. Let\u2019s\u00a0 suppose that you flip the coin 20 times for your sample.<\/p>\n

                                            \n

                                            Question 4<\/h3>\n

                                            Which of the following would be stronger evidence that the coin is weighted unfairly\u00a0 towards the heads side (i.e., stronger evidence that the null hypothesis is false)?<\/p>\n

                                              \n
                                            1. a) A higher proportion of heads, because that would be less likely if the coin is fair<\/li>\n
                                            2. b) A higher proportion of heads, because that would be more likely if the coin is fair<\/li>\n
                                            3. c) A smaller proportion of heads, because that would be less likely if the coin is fair<\/li>\n
                                            4. d) A smaller proportion of heads, because that would be more likely if the coin is fair<\/li>\n<\/ol>\n<\/div>\n
                                              \n

                                              Question 5<\/h3>\n

                                              If your sample of 20 coin flips all landed on heads, would it prove with complete\u00a0 certainty that your friend\u2019s coin is weighted?<\/p>\n

                                                \n
                                              1. a) Yes, it is impossible for a fair coin to land on heads 20 out of 20 times.<\/li>\n
                                              2. b) No, even though the probability is very small, it is still possible for a fair coin to\u00a0 land on heads 20 out of 20 times.<\/li>\n<\/ol>\n<\/div>\n
                                                \n

                                                Question 6<\/h3>\n

                                                Suppose that in your sample of 20 coin flips, you obtain 17 heads. The probability of\u00a0 obtaining 17 or more heads in 20 flips of a fair coin is 0.0013. Would you conclude\u00a0 that your friend\u2019s coin is weighted? Explain.<\/p>\n<\/div>\n

                                                Looking ahead<\/p>\n

                                                Notice that above, you assumed your friend\u2019s coin is fair, and that assumption was used\u00a0 to calculate the probability of obtaining 17 or more heads. The probability of 0.0013 is\u00a0 the probability of obtaining 17 or more heads out of 20 flips if the coin is fair, and this\u00a0 low probability is evidence that the coin is not fair. Note that no matter what probability\u00a0 you obtain, even if it is a high probability, it is not evidence that the coin is fair. This is\u00a0 because fairness was already the given assumption, and the probability was computed\u00a0 using the assumption that the coin is fair. If we assume something is true, we can\u2019t use\u00a0 that assumption to prove that it\u2019s true (that would be using circular logic). From our\u00a0 sample of coin flips, we either get enough evidence to say the coin is weighted, or we\u00a0 don\u2019t. In hypothesis-testing language, we either get enough evidence to reject the null\u00a0 hypothesis, or we don\u2019t. When we don\u2019t get enough evidence to reject the null\u00a0 hypothesis, we fail to reject the null hypothesis. We never accept the null hypothesis\u00a0 because that was already our assumption to begin with.<\/p>\n

                                                \n

                                                Question 7<\/h3>\n

                                                Which of the following are the possible outcomes of your sample of 20 coin flips?\u00a0 There may be more than one correct answer.<\/p>\n

                                                  \n
                                                1. a) You obtain enough heads that you conclude your friend\u2019s coin is weighted.<\/li>\n
                                                2. b) You obtain enough heads that you conclude your friend\u2019s coin is fair.<\/li>\n
                                                3. c) You do not obtain enough heads to conclude that your friend\u2019s coin is weighted.<\/li>\n
                                                4. d) You do not obtain enough heads to conclude that your friend\u2019s coin is fair.<\/li>\n<\/ol>\n<\/div>\n

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