1 00:00:00,060 --> 00:00:05,250 Now that we understand the idea of the central limit theorem, we can use it in inferential statistics 2 00:00:05,250 --> 00:00:07,230 to start testing hypotheses. 3 00:00:07,230 --> 00:00:13,680 And the focus of this lesson is going to be the hypothesis testing procedure that will follow each time. 4 00:00:13,680 --> 00:00:18,120 And more specifically, we'll look at this first step of our hypothesis testing procedure. 5 00:00:18,120 --> 00:00:23,280 So in statistics, as we've already talked about, the whole idea is that we collect a sample from the 6 00:00:23,280 --> 00:00:28,440 population, we investigate that sample and then we try to make a clear statement about whether or not 7 00:00:28,440 --> 00:00:33,300 that sample supports some original statement that we've made about our population. 8 00:00:33,300 --> 00:00:40,920 So this process is the hypothesis testing process or the process of inferential statistics, because 9 00:00:40,920 --> 00:00:45,840 we're using information about the sample to make inferences about the population. 10 00:00:45,840 --> 00:00:52,980 Now, the first step in this process is to state our hypothesis, which is a statement of expectation 11 00:00:52,980 --> 00:00:59,070 about some population parameter that we develop for the purpose of testing it, and we can hypothesize 12 00:00:59,070 --> 00:01:00,000 about anything. 13 00:01:00,000 --> 00:01:06,990 So our hypothesis might be that mean order processing time is 17 hours from the time the customer places 14 00:01:06,990 --> 00:01:08,970 the order to the time that we ship it out. 15 00:01:08,970 --> 00:01:14,550 Or we might hypothesize that employees in one department of our company work longer hours than employees 16 00:01:14,550 --> 00:01:15,660 in another department. 17 00:01:15,660 --> 00:01:18,240 Our hypothesis can be anything we want it to be. 18 00:01:18,240 --> 00:01:22,620 It's just that we hypothesize about whatever it is we want to test. 19 00:01:22,620 --> 00:01:28,200 And what we need to realize is that in this hypothesis testing process, in the process of doing inferential 20 00:01:28,200 --> 00:01:33,510 statistics, we actually always create a pair of opposing hypothesis statements. 21 00:01:33,510 --> 00:01:40,020 One of those statements is the null hypothesis and the other is the alternative hypothesis. 22 00:01:40,020 --> 00:01:43,410 We can indicate the null hypothesis as h sub zero. 23 00:01:43,410 --> 00:01:49,530 Sometimes we call it h not and the alternative hypothesis as h sub A We can remember that if we think 24 00:01:49,530 --> 00:01:51,720 about the A standing for alternative. 25 00:01:51,720 --> 00:01:56,640 Now the alternative hypothesis is the abnormality we're looking for in the data. 26 00:01:56,640 --> 00:01:59,010 It's the significance that we're hoping to find. 27 00:01:59,010 --> 00:02:04,740 Once we have our alternative hypothesis, then we always state the opposite claim and that opposite 28 00:02:04,740 --> 00:02:06,720 claim is the null hypothesis. 29 00:02:06,720 --> 00:02:12,300 So as an example, let's say we want to test the hypothesis that employees in the billing department 30 00:02:12,300 --> 00:02:14,910 of our company are working more than full time. 31 00:02:14,910 --> 00:02:16,890 They're working more than 40 hours a week. 32 00:02:16,890 --> 00:02:25,770 In that case, our alternative hypothesis would be that the mean number of hours worked is greater than 33 00:02:25,770 --> 00:02:26,580 40. 34 00:02:26,580 --> 00:02:33,120 Therefore, the null hypothesis is the opposite statement, which means that the null hypothesis is 35 00:02:33,120 --> 00:02:37,320 that the employees work less than or equal to 40 hours a week. 36 00:02:37,320 --> 00:02:43,890 Now, interestingly, when we think about hypotheses, we always tend to think in terms of the alternative 37 00:02:43,890 --> 00:02:47,220 hypothesis, because that's the thing that we're really trying to test. 38 00:02:47,220 --> 00:02:51,960 We're wondering here, do employees in the billing department work more than 40 hours a week? 39 00:02:51,960 --> 00:02:56,160 So we sort of frame up our thinking in terms of the alternative hypothesis. 40 00:02:56,160 --> 00:03:01,920 But when it comes to this hypothesis testing procedure, what we're actually going to be doing is testing 41 00:03:01,920 --> 00:03:02,970 the null hypothesis. 42 00:03:02,970 --> 00:03:08,670 So our focus going forward is going to be on the null hypothesis, even though when we think about what 43 00:03:08,670 --> 00:03:12,180 it is that we want to test, we think about the alternative hypothesis. 44 00:03:12,180 --> 00:03:13,560 That's not wrong. 45 00:03:13,560 --> 00:03:18,150 It's just that we might frame up our thinking in terms of the alternative hypothesis state. 46 00:03:18,150 --> 00:03:20,820 That state the null hypothesis is the opposite. 47 00:03:20,820 --> 00:03:26,910 And then going forward we'll focus on the null hypothesis that opposite statement and work on testing 48 00:03:26,910 --> 00:03:31,020 that hypothesis instead of really looking at the alternative hypothesis. 49 00:03:31,020 --> 00:03:34,650 As we work through this hypothesis testing process. 50 00:03:34,650 --> 00:03:40,680 Now when it comes to setting up the null and alternative hypothesis, there are only three ways that 51 00:03:40,680 --> 00:03:42,090 we can set these up. 52 00:03:42,090 --> 00:03:48,030 This second option here is the one we used in our example where we said that employees in the billing 53 00:03:48,030 --> 00:03:50,160 department worked more than 40 hours a week. 54 00:03:50,160 --> 00:03:56,430 The alternative hypothesis has the mean set greater than some value, which means that the null hypothesis 55 00:03:56,430 --> 00:03:57,390 is the opposite. 56 00:03:57,390 --> 00:04:01,320 Claim that the mean is less than or equal to that same value. 57 00:04:01,560 --> 00:04:07,920 We can have the opposite situation where we set the alternative hypothesis as the mean less than some 58 00:04:07,920 --> 00:04:13,620 value, in which case the null hypothesis is that the mean is greater than or equal to that same value 59 00:04:13,620 --> 00:04:14,400 that we chose. 60 00:04:14,400 --> 00:04:19,950 And then the third option is that we choose an alternative hypothesis where the mean is not equal to 61 00:04:19,950 --> 00:04:20,730 some value. 62 00:04:20,730 --> 00:04:25,980 So in this case, maybe we don't have a suspicion that the employees in the billing department work 63 00:04:25,980 --> 00:04:27,420 more than 40 hours a week. 64 00:04:27,420 --> 00:04:31,740 Maybe we just think that they work some number of hours other than 40. 65 00:04:31,740 --> 00:04:36,930 We're not sure if they work fewer than 40 hours or more than 40 hours, but we don't think that they 66 00:04:36,960 --> 00:04:39,030 work exactly 40 hours a week. 67 00:04:39,030 --> 00:04:44,160 Our suspicion, our hypothesis, the thing we want to test is that the employees in the billing department 68 00:04:44,160 --> 00:04:45,900 do not work 40 hours a week. 69 00:04:45,900 --> 00:04:52,500 So our alternative hypothesis would be that the mean is not equal to 40, and therefore the null hypothesis 70 00:04:52,500 --> 00:04:56,220 that goes along with that would be that the mean is equal to 40. 71 00:04:56,220 --> 00:04:59,760 So again, in this example we said that the billing department were. 72 00:04:59,850 --> 00:05:05,160 It's more than 40 hours a week, which means that we described a relationship between the mean and some 73 00:05:05,160 --> 00:05:06,180 constant value. 74 00:05:06,210 --> 00:05:10,650 We can also set up hypothesis statements that compare to groups. 75 00:05:10,650 --> 00:05:16,770 So if we think that the billing department works more hours than the payroll department, we could say 76 00:05:16,770 --> 00:05:17,640 something like this. 77 00:05:17,640 --> 00:05:25,170 So the mean for the billing department is greater than the mean for the payroll department, in which 78 00:05:25,170 --> 00:05:30,660 case the null hypothesis is the opposite statement that the mean for the billing department is less 79 00:05:30,660 --> 00:05:34,110 than or equal to the mean for the payroll department. 80 00:05:34,110 --> 00:05:41,190 And we could compare means like this with all three of these different statement types, these different 81 00:05:41,190 --> 00:05:43,170 pairs of hypothesis statements. 82 00:05:43,170 --> 00:05:50,310 But that being said, it's important for us to say that we should only ever use these second and third 83 00:05:50,310 --> 00:05:56,280 types here when we have good reason to believe that there's this kind of directionality that we're indicating 84 00:05:56,280 --> 00:05:56,730 here. 85 00:05:56,730 --> 00:06:01,680 In other words, we only want to set up these kind of directional hypothesis statements where we use 86 00:06:01,680 --> 00:06:06,660 either a greater than or less than simple in the alternative hypothesis, when we have a pretty strong 87 00:06:06,660 --> 00:06:12,930 suspicion that that is the directionality that actually exists in the population to continue on with 88 00:06:12,930 --> 00:06:18,780 this example we've been using here, maybe we have data from previous pay stubs for the employees in 89 00:06:18,780 --> 00:06:22,950 our billing department to indicate that they do tend to work more than 40 hours a week. 90 00:06:22,950 --> 00:06:26,280 And what we're doing here is just testing the current pay period. 91 00:06:26,280 --> 00:06:31,500 We already have some kind of prior knowledge, some strong suspicion that they are, in fact, working 92 00:06:31,500 --> 00:06:32,370 more than 40 hours. 93 00:06:32,370 --> 00:06:34,680 And we're looking to test that here. 94 00:06:34,680 --> 00:06:40,890 So we need to have some kind of suspicion for some good reason if we're going to use either of these 95 00:06:40,890 --> 00:06:42,720 sets of hypothesis statements. 96 00:06:42,720 --> 00:06:49,290 Otherwise, we really need to use this first one because this first pair of hypothesis statements has 97 00:06:49,290 --> 00:06:51,270 no directionality to it at all. 98 00:06:51,270 --> 00:06:56,490 And so it's sort of a more conservative approach to take what we're saying here with this alternative 99 00:06:56,490 --> 00:07:02,100 hypothesis that the mean is not equal to some value is that we don't know if the mean is less than that 100 00:07:02,100 --> 00:07:03,960 value or greater than that value. 101 00:07:03,960 --> 00:07:07,140 We don't really have any suspicion at all about the directionality. 102 00:07:07,140 --> 00:07:08,430 We're not sure which way it goes. 103 00:07:08,430 --> 00:07:12,330 We're just saying maybe the mean is not equal to that particular value. 104 00:07:12,330 --> 00:07:17,370 Whereas with these two sets of hypothesis statements, we're being more specific, we're indicating 105 00:07:17,370 --> 00:07:21,780 directionality, we're saying no, we think the mean is greater than some value, not less than the 106 00:07:21,780 --> 00:07:23,970 value, not equal to the value, but greater than the value. 107 00:07:23,970 --> 00:07:26,820 Or here the mean is less than that value. 108 00:07:26,850 --> 00:07:29,700 We're sort of presupposing that directionality. 109 00:07:29,700 --> 00:07:34,320 And so we really need to have some kind of a good reason to think that this is the case. 110 00:07:34,320 --> 00:07:38,970 If we're going to use either one of these pairs of hypothesis statements. 111 00:07:38,970 --> 00:07:40,950 Otherwise we should use this first one. 112 00:07:40,950 --> 00:07:42,870 It's just more conservative. 113 00:07:42,870 --> 00:07:49,260 It doesn't suppose to much if we don't really have any evidence for supposing a particular direction. 114 00:07:49,260 --> 00:07:56,640 So to summarize here, the first step in our hypothesis testing procedure is to restate the null and 115 00:07:56,640 --> 00:07:58,050 alternative hypotheses. 116 00:07:58,080 --> 00:08:02,880 Remember that we'll frame up our thinking around the alternative hypothesis, so we'll sort of think 117 00:08:02,880 --> 00:08:04,350 about what we want to test. 118 00:08:04,350 --> 00:08:10,920 That'll be our alternative hypothesis, and then the null hypothesis will be the opposite of that statement. 119 00:08:11,040 --> 00:08:17,280 But that being said, the alternative hypothesis always needs to be that the mean is not equal to some 120 00:08:17,280 --> 00:08:20,040 value greater than a value or less than a value. 121 00:08:20,070 --> 00:08:26,160 We can't say in the alternative hypothesis that the mean is equal to some value less than or equal to 122 00:08:26,160 --> 00:08:27,630 or greater than or equal to a value. 123 00:08:27,630 --> 00:08:33,929 We need to stick with these symbols in the alternative hypothesis, which means that the null hypothesis 124 00:08:33,929 --> 00:08:39,659 will be the opposite statement and therefore will always only see these values in the null hypothesis. 125 00:08:39,659 --> 00:08:45,660 So we'll save equal to less than or equal to and greater than or equal to for the null hypothesis statement. 126 00:08:45,660 --> 00:08:48,780 So we frame up our thinking around the alternative hypothesis. 127 00:08:48,780 --> 00:08:54,270 We state that as the mean is not equal to some value greater than some value or less than some value. 128 00:08:54,270 --> 00:08:59,730 Then we make the null hypothesis the opposite statement with the opposite sign, which means that we 129 00:08:59,730 --> 00:09:05,970 will end up with either this first pair here of hypothesis statements, this second pair, or this third 130 00:09:05,970 --> 00:09:06,450 pair. 131 00:09:06,450 --> 00:09:12,540 And once we have our pair of hypothesis statements, we can move on to step two, which is about determining 132 00:09:12,540 --> 00:09:13,830 the level of significance. 133 00:09:13,830 --> 00:09:19,800 But keep in mind that as we're going through this hypothesis testing procedure, it's important to always 134 00:09:19,800 --> 00:09:27,780 remember that the goal here of this whole process is to provide support for the claim that's being made 135 00:09:27,780 --> 00:09:30,150 by our alternative hypothesis. 136 00:09:30,150 --> 00:09:34,860 The goal is not to prove that our alternative hypothesis is true. 137 00:09:34,860 --> 00:09:41,430 This hypothesis testing procedure can never actually prove the statement being made in our alternative 138 00:09:41,430 --> 00:09:42,060 hypothesis. 139 00:09:42,060 --> 00:09:46,560 We can never prove either of these hypothesis statements to be wrong or right. 140 00:09:46,560 --> 00:09:50,550 All we can do is lend support or thought about the other way. 141 00:09:50,550 --> 00:09:53,760 Fail to lend support to the statement that we're trying to make. 142 00:09:53,760 --> 00:09:59,580 So if we go through this whole process, the best we can do is find support for or fail to. 143 00:09:59,620 --> 00:10:02,380 Find support for the alternative hypothesis. 144 00:10:02,380 --> 00:10:08,470 And the way that that's going to manifest itself in the end is with sort of this inverted language, 145 00:10:08,470 --> 00:10:13,780 which can be a little confusing, but we will either reject the null hypothesis, and you can imagine 146 00:10:13,780 --> 00:10:19,180 that if we reject the null hypothesis, that's going to give support to the alternative hypothesis since 147 00:10:19,180 --> 00:10:25,030 these are opposite statements, or if this process doesn't give us enough evidence to reject the null, 148 00:10:25,060 --> 00:10:28,870 then the conclusion that will make is that we failed to reject the null. 149 00:10:28,870 --> 00:10:31,600 We didn't have enough evidence to reject the null. 150 00:10:31,600 --> 00:10:34,120 We couldn't say that the null was wrong. 151 00:10:34,120 --> 00:10:39,640 And if we can't say that the null is wrong, that means that we don't really have any support to throw 152 00:10:39,640 --> 00:10:41,530 behind the alternative hypothesis. 153 00:10:41,530 --> 00:10:42,790 Our original statement. 154 00:10:42,790 --> 00:10:48,550 So just remember that this whole process is about support only, not proof, and that what we're going 155 00:10:48,550 --> 00:10:54,910 to do is break down each of these steps to eventually get to the conclusion where we try to reject the 156 00:10:54,910 --> 00:11:01,090 null in order to lend support to the alternative or fail to reject the null, which means that we've 157 00:11:01,090 --> 00:11:03,910 failed to support the alternative.