1 00:00:00,060 --> 00:00:04,270 So we've already covered the first two steps of the hypothesis testing procedure. 2 00:00:04,290 --> 00:00:08,940 We've learned how to state null and alternative hypotheses as opposite statements. 3 00:00:08,940 --> 00:00:14,460 And we talked in the last lecture about how to determine the level of significance or the alpha value 4 00:00:14,460 --> 00:00:19,470 based on how important it is that we don't commit a type one error. 5 00:00:19,500 --> 00:00:24,010 Now we want to talk about the third step, which is how to calculate the test statistic. 6 00:00:24,030 --> 00:00:29,070 Now, regardless of what kind of test we're running and we'll talk a little bit more about that later 7 00:00:29,070 --> 00:00:30,240 in this lecture. 8 00:00:30,270 --> 00:00:36,120 The short answer here is that our test statistic formula is always given by this formula here. 9 00:00:36,120 --> 00:00:41,910 So we're looking at the test statistic formula for a T score because we're assuming that population 10 00:00:41,910 --> 00:00:45,750 standard deviation is unknown, which is almost always the case. 11 00:00:45,750 --> 00:00:51,390 If we do happen to know population standard deviation, then we can calculate a Z score instead of a 12 00:00:51,390 --> 00:00:56,610 T score, in which case the formula is exactly the same, except that we have Z equals instead of T 13 00:00:56,610 --> 00:00:57,240 equals. 14 00:00:57,240 --> 00:01:01,860 And instead of using sample standard deviation, we would use population standard deviation. 15 00:01:01,860 --> 00:01:07,530 But again, since we almost never know, population standard deviation will assume here that we are 16 00:01:07,530 --> 00:01:09,280 calculating a T score. 17 00:01:09,300 --> 00:01:15,150 So you may recognize here that the values in this test statistic formula are the sample mean. 18 00:01:15,150 --> 00:01:20,490 So the mean that we calculate from the sample that we take from our population and then this mu sub 19 00:01:20,490 --> 00:01:25,050 zero value here or mu not is our hypothesized value. 20 00:01:25,050 --> 00:01:30,450 So in the last lecture when we were talking about determining the level of significance, we gave these 21 00:01:30,450 --> 00:01:37,830 hypothesis statements where we said that the alternative hypothesis was mu greater than 14, where we 22 00:01:37,830 --> 00:01:43,950 said that we were hypothesizing that mean order processing time was longer than 14 hours, which means 23 00:01:43,950 --> 00:01:50,040 that the alternative hypothesis is the opposite statement, which is that MU is less than or equal to 24 00:01:50,070 --> 00:01:51,090 14. 25 00:01:51,090 --> 00:01:55,410 That mean order processing time is less than or equal to 14 hours. 26 00:01:55,440 --> 00:02:04,200 This value right here in our hypothesis statements, this is mu not it is our hypothesized value. 27 00:02:04,200 --> 00:02:10,690 So we have sample mean, we have the hypothesized value and then here in the denominator you may recognize 28 00:02:10,710 --> 00:02:16,200 the formula for standard error, the standard deviation of the sampling distribution of the sample mean 29 00:02:16,200 --> 00:02:21,660 which is given by sample standard deviation divided by the square root of the sample size. 30 00:02:21,660 --> 00:02:27,900 N So let's say here again that working with this set of hypothesis statements, again, the idea here 31 00:02:27,900 --> 00:02:34,800 is that we are running a warehouse, a distribution center, we are shipping out orders and we're hypothesizing 32 00:02:34,800 --> 00:02:38,160 here that mean order processing time is longer than 14 hours. 33 00:02:38,160 --> 00:02:43,800 So from the time the customer places the order online to the time that we actually ship, the order 34 00:02:43,800 --> 00:02:47,370 out of the warehouse is greater than 14 hours. 35 00:02:47,370 --> 00:02:55,830 Let's say that we pull from our warehouse a simple random sample, and we use a sample size of 50 orders 36 00:02:55,830 --> 00:03:02,280 and we calculate a sample mean of maybe we'll say 16.5 hours. 37 00:03:02,280 --> 00:03:08,880 So for our 50 order sample, we found that the mean order processing time was 16 and a half hours. 38 00:03:08,880 --> 00:03:13,950 We already said that our hypothesized value was 14. 39 00:03:14,400 --> 00:03:20,310 And then let's say we also calculate it from that sample of 50 orders that we pulled a sample standard 40 00:03:20,310 --> 00:03:24,480 deviation equal to, let's maybe say 1.5 hours. 41 00:03:24,480 --> 00:03:28,410 So that's enough information to calculate the test statistic. 42 00:03:28,410 --> 00:03:39,000 We would say that t is equal to our sample mean 16.5 minus the hypothesized value 14 divided by sample 43 00:03:39,000 --> 00:03:44,880 standard deviation, which is 1.5 hours divided by square root of sample size. 44 00:03:44,970 --> 00:03:50,130 We said that our sample size was and equals 50, so we divide by square root 50. 45 00:03:50,130 --> 00:03:56,490 And if we use a calculator to find the value of this fraction, here we get an approximate T score of 46 00:03:56,490 --> 00:04:00,300 about 11.79 and that's it. 47 00:04:00,300 --> 00:04:07,470 That is the value of our test statistic and we are done with the third step of our hypothesis test. 48 00:04:07,470 --> 00:04:13,350 But we still want to take this opportunity to talk about one extra factor in our hypothesis testing 49 00:04:13,350 --> 00:04:19,050 procedure before we go forward, because we'll need this, which is the idea of a one tailed test versus 50 00:04:19,050 --> 00:04:20,130 a two tailed test. 51 00:04:20,130 --> 00:04:26,640 So if you remember from before, we talked about our null and alternative hypotheses and we said that 52 00:04:26,640 --> 00:04:32,910 we had three options, we could use an equal sign in the null hypothesis, in which case the alternative 53 00:04:32,910 --> 00:04:38,400 hypothesis has to have a not equal to sign or in the null we could have less than or equal to, in which 54 00:04:38,400 --> 00:04:43,500 case the alternative has a greater than sign or the null has a greater than or equal to, in which case 55 00:04:43,500 --> 00:04:45,510 the alternative has a less than sign. 56 00:04:45,750 --> 00:04:51,270 These are our only three options when we state our null and alternative hypotheses. 57 00:04:51,270 --> 00:04:55,920 Well, these three options correspond to one and two tailed tests. 58 00:04:55,920 --> 00:04:59,940 So if we use this first option here where the null is. 59 00:05:00,090 --> 00:05:03,100 An equal sign, and the alternative is a not equal to sign. 60 00:05:03,120 --> 00:05:08,820 That means we're using a two tailed test, which we also sometimes call a non directional test or a 61 00:05:08,820 --> 00:05:09,790 two sided test. 62 00:05:09,810 --> 00:05:12,420 Those are three different ways of saying the same thing. 63 00:05:12,450 --> 00:05:18,900 In contrast, if we use either the second or the third option, then we're conducting a one tailed test, 64 00:05:18,900 --> 00:05:22,470 also called a directional test, also called a one sided test. 65 00:05:22,500 --> 00:05:29,280 Now, previously when we talked about these three options, we said that this first option here, option 66 00:05:29,280 --> 00:05:32,700 number one, was our most conservative option. 67 00:05:32,730 --> 00:05:39,450 Using this option says that we don't really have an idea or a presupposition about the direction or 68 00:05:39,450 --> 00:05:41,560 the directionality of our test. 69 00:05:41,580 --> 00:05:47,330 We're not sure if we should use a greater than value or less than value in our alternative hypothesis. 70 00:05:47,340 --> 00:05:51,180 We really don't have any preconceived idea about the direction. 71 00:05:51,180 --> 00:05:58,650 But the reason in actuality that this is more conservative, we can see visually if we look at the normal 72 00:05:58,650 --> 00:05:59,600 distribution here. 73 00:05:59,610 --> 00:06:01,620 So we talked about this before too. 74 00:06:01,650 --> 00:06:08,010 If we have our probability distribution, then we have our region of acceptance in the middle here that 75 00:06:08,010 --> 00:06:12,060 is based on the confidence level percentage. 76 00:06:12,060 --> 00:06:16,950 And if we're running a two tailed test or a non directional test, then we're going to have a region 77 00:06:16,950 --> 00:06:21,720 of rejection over here in the lower tail and a region of rejection over here in the upper tail. 78 00:06:21,750 --> 00:06:28,890 The amount of area under the curve in this lower tail region of rejection is alpha over two and the 79 00:06:28,890 --> 00:06:31,650 amount of area in the upper tail is alpha over two. 80 00:06:31,680 --> 00:06:36,210 Remember that alpha is given by one minus the confidence level. 81 00:06:36,210 --> 00:06:46,230 So if we choose a confidence level of let's say 95%, then by definition the alpha value has to be 5%, 82 00:06:46,230 --> 00:06:53,640 which means then that we would have 2.5% of all the area under the curve in this lower tail and 2.5% 83 00:06:53,640 --> 00:06:55,770 of all the area under the curve in the upper tail. 84 00:06:55,770 --> 00:07:00,380 So this lower tail region of rejection would make up two and one half percent of total area. 85 00:07:00,390 --> 00:07:04,320 The upper tail region of rejection would make up two and one half percent of total area. 86 00:07:04,470 --> 00:07:09,690 And then this whole region of acceptance in the middle would be 95% of the area. 87 00:07:09,720 --> 00:07:16,080 Contrast that with a directional test in either direction where we have a one tailed test. 88 00:07:16,080 --> 00:07:20,720 So if we have this situation here, this is situation number two. 89 00:07:20,730 --> 00:07:26,310 So this is two and three, this is one and two. 90 00:07:26,310 --> 00:07:30,480 And then while we're at it here, we'll say that this is scenario three. 91 00:07:30,720 --> 00:07:36,300 So instead of two here, the alternative hypothesis states that we think the population mean is greater 92 00:07:36,300 --> 00:07:38,880 than some hypothesized value. 93 00:07:38,880 --> 00:07:44,430 And because we use this greater than value, that means we're using what we call an upper tailed test. 94 00:07:44,430 --> 00:07:48,740 And the entire region of rejection is squeezed into this upper tail. 95 00:07:48,750 --> 00:07:52,410 There is no other region of rejection down here in the lower tail. 96 00:07:52,410 --> 00:07:57,150 So we just have the region of acceptance on the left and the region of rejection on the right. 97 00:07:57,150 --> 00:08:03,660 Whereas if we state in our alternative hypothesis that we think the population mean is less than some 98 00:08:03,660 --> 00:08:10,110 hypothesized value, so less than that means we're running a lower tailed test and it pushes the entire 99 00:08:10,110 --> 00:08:12,270 region of rejection into the lower tail. 100 00:08:12,270 --> 00:08:18,410 So we just have the singular region of rejection on the left and the region of acceptance on the right. 101 00:08:18,420 --> 00:08:25,140 Now, going back here to this idea of the two tailed test being more conservative than either of the 102 00:08:25,140 --> 00:08:27,090 one tailed directional tests. 103 00:08:27,180 --> 00:08:34,620 The reason we say it's more conservative is because let's say that we have chosen a 95% confidence level, 104 00:08:34,620 --> 00:08:38,010 like we said, and therefore an alpha value of 5%. 105 00:08:38,220 --> 00:08:43,409 That means for a two tailed test, if we're running a two tailed test based on the fact that we have 106 00:08:43,409 --> 00:08:49,080 stated null and alternative hypotheses with these equal and not equal to signs, if we're running a 107 00:08:49,080 --> 00:08:55,500 two tailed test, then that means that this region of acceptance in the middle here is 95% of the area, 108 00:08:55,500 --> 00:08:59,460 because that 95% area is centered around the mean. 109 00:08:59,460 --> 00:09:03,060 It's symmetrical here in the middle of this distribution. 110 00:09:03,060 --> 00:09:11,040 That means that the boundary here between this lower tail region of rejection and the region of acceptance 111 00:09:11,040 --> 00:09:17,330 is here at 2.5% or at 0.0250. 112 00:09:17,340 --> 00:09:22,530 That would be our Z score value if we were thinking about this in terms of Z scores. 113 00:09:22,530 --> 00:09:28,500 And then the boundary here at the right edge of the region of acceptance and the lower edge of this 114 00:09:28,500 --> 00:09:37,950 upper region of rejection, this boundary right here would be at a Z score of 97.5 or 0.9750. 115 00:09:37,950 --> 00:09:42,990 We can see that two and one half percent over on the left, the two and one half percent over on the 116 00:09:42,990 --> 00:09:49,680 right above 97.5 and the 95% in the middle between 97.5 and 2.5. 117 00:09:49,680 --> 00:09:55,890 But if we run a one tailed or directional test, let's say we're running an upper tail test here and 118 00:09:55,890 --> 00:09:59,580 we look at this boundary again with the same. 119 00:10:00,000 --> 00:10:01,650 Its level of 95%. 120 00:10:01,680 --> 00:10:09,210 That means that we have the entire 95% over here, which means that this Z score has to be associated 121 00:10:09,210 --> 00:10:12,630 with 0.9500. 122 00:10:12,660 --> 00:10:18,060 It is the Z score that gives 0.9500 in the body of the Z table. 123 00:10:18,060 --> 00:10:25,860 So these are the Z scores that give 0.0250 and 0.9750 in the body of the Z table. 124 00:10:25,860 --> 00:10:29,450 So that is the boundary value we'd be looking for. 125 00:10:29,460 --> 00:10:35,160 And then similarly with this lower tail test, again, keeping our confidence level the same at 95%, 126 00:10:35,250 --> 00:10:40,290 that means the entire alpha value, 5% has to get squeezed into this lower tail. 127 00:10:40,290 --> 00:10:50,850 And so this boundary right here has to be associated with a Z score that gives 0.0500 in the body of 128 00:10:50,850 --> 00:10:51,740 the Z table. 129 00:10:51,750 --> 00:10:59,520 Now, what we see ultimately is that we are more likely to land in the region of acceptance with a two 130 00:10:59,520 --> 00:11:00,420 tailed test. 131 00:11:00,450 --> 00:11:06,930 It's going to be harder to reject the null hypothesis, harder to land within the region of rejection 132 00:11:06,960 --> 00:11:13,470 if we're using a to tail test, because we can see here that with this two tailed test on this upper 133 00:11:13,470 --> 00:11:19,890 side, we have to clear a bar of 0.975 in order to reject the null hypothesis. 134 00:11:19,890 --> 00:11:24,400 Whereas here with the upper tail test we only have to clear a bar of 0.95. 135 00:11:24,420 --> 00:11:31,170 In other words, our test statistic doesn't have to be as extreme when we run this upper tail test as 136 00:11:31,170 --> 00:11:32,910 when we run this two tailed test. 137 00:11:32,910 --> 00:11:39,570 And similarly here with the lower tail test, the test statistic does not need to be as extreme on the 138 00:11:39,570 --> 00:11:46,760 low end in order to be to the left of 0.05, as it does to be to the left of 0.0 to 5. 139 00:11:46,770 --> 00:11:53,880 So running a one tailed test makes it easier to reject the null hypothesis, which makes it easier to 140 00:11:53,880 --> 00:11:56,490 lend support to our alternative hypothesis. 141 00:11:56,490 --> 00:12:03,120 And so we've kind of lowered the bar, we've lowered the strictness of our test by running a one tail 142 00:12:03,120 --> 00:12:03,570 test. 143 00:12:03,570 --> 00:12:09,960 And that's why we said previously that it's super important that we have some idea, some evidence ahead 144 00:12:09,960 --> 00:12:15,690 of time about the directionality of the test if we're going to move forward with one of these pairs 145 00:12:15,690 --> 00:12:17,250 of hypothesis statements. 146 00:12:17,280 --> 00:12:22,170 Otherwise, if we don't really have any evidence of directionality either way or we want to be more 147 00:12:22,170 --> 00:12:27,840 conservative, we want to be safer and set the bar for ourselves as high as we can to make sure that 148 00:12:27,840 --> 00:12:34,350 our conclusion is more rock solid, Then we should run a two tailed non directional test. 149 00:12:34,350 --> 00:12:41,460 In other words, if we sort of line up these top two distributions here and we extend the boundary in 150 00:12:41,460 --> 00:12:45,720 this second distribution between the region of acceptance and the region of rejection, we sort of pull 151 00:12:45,720 --> 00:12:48,780 that up here into the first distribution. 152 00:12:48,780 --> 00:12:51,870 Maybe that boundary is right around here. 153 00:12:51,870 --> 00:12:57,420 If we extend that up, then what we're saying is that if we're running a two tailed test, we have to 154 00:12:57,420 --> 00:13:03,150 find a test statistic that clears this boundary in order to reject the null hypothesis. 155 00:13:03,150 --> 00:13:08,730 But if we were running a one tailed test only, we would need to find a test statistic that only clears 156 00:13:08,730 --> 00:13:12,840 this bar, this boundary in order to reject the null. 157 00:13:12,840 --> 00:13:18,900 And so any test statistics that we find between these two, any test statistics that fall within this 158 00:13:18,900 --> 00:13:24,990 interval, we will fail to reject the null if we're running a two tailed test but succeed at rejecting 159 00:13:24,990 --> 00:13:27,270 the null if we're running a one tailed test. 160 00:13:27,270 --> 00:13:33,180 And so that's why we see that two tailed test as being more conservative with all of this out of the 161 00:13:33,180 --> 00:13:33,540 way. 162 00:13:33,540 --> 00:13:39,090 And now that we know how to calculate a test statistic, we can move on to the last couple of steps 163 00:13:39,090 --> 00:13:45,240 of the hypothesis testing procedure, which will be determining whether or not this test statistic clears 164 00:13:45,240 --> 00:13:51,570 one of these bars, these hurdles, and lands us in one of these regions of rejection instead of the 165 00:13:51,570 --> 00:13:52,620 region of acceptance. 166 00:13:52,620 --> 00:13:58,170 In other words, is our test statistic severe enough to land us within a region of rejection? 167 00:13:58,170 --> 00:13:59,430 That's what we're going to look at next. 168 00:13:59,430 --> 00:14:04,230 How to determine whether or not the test statistic shows us enough significance. 169 00:14:04,230 --> 00:14:10,770 If it does, then we'll be landing within one of these regions of rejection, which will allow us to 170 00:14:10,770 --> 00:14:15,360 reject the null hypothesis and lend support for our alternative hypothesis. 171 00:14:15,360 --> 00:14:21,570 So now that we have this test statistic, we'll move on to determining these regions of acceptance and 172 00:14:21,570 --> 00:14:28,470 rejection and looking at whether or not it'll be acceptable for us to reject the null hypothesis.