1 00:00:00,090 --> 00:00:05,670 So we've said that the standard normal distribution is the Z distribution, and that we can use the 2 00:00:05,670 --> 00:00:10,110 Z table to look up probabilities associated with that Z distribution. 3 00:00:10,140 --> 00:00:14,190 Now we want to talk about what's called the student's T distribution. 4 00:00:14,190 --> 00:00:20,040 So it is also a probability distribution, like the standard normal distribution represented by Z. 5 00:00:20,070 --> 00:00:22,270 So we're talking about the student's T distribution. 6 00:00:22,290 --> 00:00:28,680 It is also associated with its own T table in the same way that the Z distribution is associated with 7 00:00:28,680 --> 00:00:29,760 the Z table. 8 00:00:29,760 --> 00:00:35,250 And of course the T table gives us probabilities associated with this T distribution. 9 00:00:35,280 --> 00:00:41,640 Now the T distribution is similar to the Z distribution in the sense that it is symmetrical, it's bell 10 00:00:41,640 --> 00:00:45,160 shaped and it's centered around this mean of zero. 11 00:00:45,180 --> 00:00:51,690 The mean is always zero, but the T distribution is flatter and wider than the standard normal distribution, 12 00:00:51,690 --> 00:00:58,320 which means that more of the area under the T distribution is pushed out toward the tails, whereas 13 00:00:58,320 --> 00:01:02,320 more of the area in the Z distribution is collected around the mean. 14 00:01:02,340 --> 00:01:07,920 What that means, of course, is that the standard deviation of the T distribution is larger than the 15 00:01:07,920 --> 00:01:11,940 standard deviation of the Z distribution or the standard normal distribution. 16 00:01:11,970 --> 00:01:18,750 Now, the reason that we have this T distribution is because if we're ever using a small enough sample, 17 00:01:18,780 --> 00:01:24,450 the Z distribution becomes unreliable, as we've seen before, and we can kind of intuitively think 18 00:01:24,450 --> 00:01:25,170 about this. 19 00:01:25,170 --> 00:01:30,960 The smaller our sample size, the less reliably that sample is representative of the population. 20 00:01:30,960 --> 00:01:37,200 And so because the sample is less reliable, of course, the standard deviation is going to be larger, 21 00:01:37,200 --> 00:01:42,050 since we can't be as certain that we'll find a value around the mean. 22 00:01:42,060 --> 00:01:47,100 In other words, a smaller sample means less certainty, which means a larger standard deviation, which 23 00:01:47,100 --> 00:01:51,030 means more area is pushed out toward the tails of the distribution. 24 00:01:51,060 --> 00:01:57,030 Now, it turns out that when we're taking small samples, like a sample size of ten or a sample size 25 00:01:57,030 --> 00:02:02,850 of 15, that sample size is too small to reliably use the Z distribution. 26 00:02:02,850 --> 00:02:05,220 So instead we'll use the T distribution. 27 00:02:05,220 --> 00:02:12,240 But if we keep increasing that sample size, the magic number is a sample size of about 30. 28 00:02:12,270 --> 00:02:19,950 A sample size of 30, at which point this T distribution matches almost exactly the standard normal 29 00:02:19,950 --> 00:02:20,780 distribution. 30 00:02:20,790 --> 00:02:26,070 In other words, you can almost imagine here with this T distribution, the smaller the sample is, 31 00:02:26,070 --> 00:02:31,740 as we said before, the larger the standard deviation, which means the flatter this blue curve becomes. 32 00:02:31,740 --> 00:02:38,130 So the tip of the blue curve, the tip of the T distribution moves down like this and the curve gets 33 00:02:38,130 --> 00:02:41,310 flatter as our sample size gets smaller. 34 00:02:41,310 --> 00:02:48,510 But as the sample size gets larger, the peak of this blue curve increases, it moves up, which pulls 35 00:02:48,510 --> 00:02:54,270 in the area from the left and right side pulls in the area closer in toward the mean, pulling an area 36 00:02:54,270 --> 00:02:57,900 away from the tails and in toward the center of the distribution. 37 00:02:57,900 --> 00:03:06,480 And so the top of the distribution starts to rise and it's at this sample size and equals 30 where the 38 00:03:06,480 --> 00:03:13,380 T distribution has risen enough to match almost exactly the standard normal distribution, the Z distribution. 39 00:03:13,380 --> 00:03:20,130 And so what we can say is that for sample sizes of 30 or greater, in other words, and greater than 40 00:03:20,130 --> 00:03:26,730 or equal to 30, the values we would get from the T distribution and Z distribution are virtually identical. 41 00:03:26,730 --> 00:03:33,360 And so at that point we can revert to our Z distribution and use Z scores looking up values in the Z 42 00:03:33,360 --> 00:03:36,540 table, but for samples that are smaller than 30. 43 00:03:36,540 --> 00:03:42,960 So when NW is less than 30, that's when the T distribution is going to give us more accurate values. 44 00:03:42,960 --> 00:03:46,860 And so we want to find T scores and look them up in the T table. 45 00:03:46,860 --> 00:03:53,970 So we'll use values of T when N is less than 30, we'll use values of Z when N is greater than or equal 46 00:03:53,970 --> 00:03:54,940 to 30. 47 00:03:54,960 --> 00:04:01,200 Now, to go back to this point of the T distribution, we said that the peak of this distribution, 48 00:04:01,200 --> 00:04:08,190 the height of it increases or moves up as sample size increases and moves down as sample size decreases. 49 00:04:08,220 --> 00:04:13,710 Effectively, what we're saying there is that the shape of the T distribution changes based on the sample 50 00:04:13,710 --> 00:04:20,910 size and more specifically, we say that it changes based on what we call the degrees of freedom of 51 00:04:20,910 --> 00:04:28,440 the sample and the degrees of freedom is equal to n minus one where N is the sample size. 52 00:04:28,440 --> 00:04:32,880 So if our sample size is 20, then we'll have 19 degrees of freedom. 53 00:04:32,880 --> 00:04:36,630 If our sample size is ten, we'll have nine degrees of freedom. 54 00:04:36,630 --> 00:04:42,150 Keep in mind that sample size isn't the only condition that determines when we should use a T score 55 00:04:42,150 --> 00:04:43,290 versus a Z score. 56 00:04:43,290 --> 00:04:49,380 For instance, whenever population standard deviation is unknown, we want to use a T score instead 57 00:04:49,380 --> 00:04:50,250 of a Z score. 58 00:04:50,250 --> 00:04:56,940 So we'll be using the T score and the T table very often, which is why it's critical that we look at 59 00:04:56,940 --> 00:04:59,910 it here along with the Z table before we move. 60 00:04:59,980 --> 00:05:00,490 Forward. 61 00:05:00,520 --> 00:05:06,180 So when it comes to looking up values in the tea table, here is our tea table. 62 00:05:06,190 --> 00:05:13,600 Notice that along this left hand side here, down this first column, we show degrees of freedom, starting 63 00:05:13,600 --> 00:05:15,750 at one all the way up to 30. 64 00:05:15,760 --> 00:05:21,940 And the reason that we stop there, you can find tea tables on line for larger degrees of freedom. 65 00:05:21,940 --> 00:05:27,820 But the reason that the standard tea table stops here is because, remember, this is our magic number, 66 00:05:27,820 --> 00:05:34,210 at which point the Z table can take over because the values from the T and Z tables are almost identical 67 00:05:34,210 --> 00:05:38,710 for larger sample sizes, specifically sample sizes greater than or equal to 30. 68 00:05:38,710 --> 00:05:43,510 So in our standard tea table, we just list degrees of freedom from 1 to 30. 69 00:05:43,510 --> 00:05:49,810 So looking up values in the tea table requires us to know the degrees of freedom, which again is given 70 00:05:49,810 --> 00:05:56,470 by n minus one where n is the sample size and to know either upper tail probability which we see here 71 00:05:56,470 --> 00:06:02,390 across the top of the tea table or confidence level, which we see here along the bottom of the table. 72 00:06:02,410 --> 00:06:06,700 Now, we'll talk more later about upper tail probability and confidence level. 73 00:06:06,730 --> 00:06:11,410 All we want to focus on right now is just how to locate values in the tea table. 74 00:06:11,410 --> 00:06:13,510 So we understand that at minimum. 75 00:06:13,510 --> 00:06:18,490 And then later on, we'll dive into these two concepts further so that we can understand them, pair 76 00:06:18,490 --> 00:06:23,350 them with degrees of freedom, and find the correct value within the body of the tea table. 77 00:06:23,350 --> 00:06:29,860 So when it comes to just locating a value in the tea table, we look for the intersection in the body 78 00:06:29,860 --> 00:06:34,660 of the table between degrees of freedom and either upper tail probability or confidence level. 79 00:06:34,660 --> 00:06:40,000 So let's say we know that our upper tail probability is 0.15 or 15%. 80 00:06:40,060 --> 00:06:43,090 That puts us in this third column here. 81 00:06:43,090 --> 00:06:45,940 And let's say we know that we have six degrees of freedom. 82 00:06:46,120 --> 00:06:51,460 Well, of course we would come down here to six degrees of freedom over to this third column and we 83 00:06:51,460 --> 00:06:57,970 would locate this value right here, 1.134 as the intersection of six degrees of freedom, meaning a 84 00:06:57,970 --> 00:07:06,370 sample size of seven and 0.15 upper tail probability or 15% of the area under the tea distribution in 85 00:07:06,370 --> 00:07:07,390 the upper tail. 86 00:07:07,390 --> 00:07:12,850 Now, we said that we could use upper tail probability or confidence level, and the reason is because 87 00:07:12,880 --> 00:07:17,680 upper tail probability and confidence level are kind of two different ways of saying the same thing 88 00:07:17,680 --> 00:07:19,810 or describing the same scenario. 89 00:07:19,810 --> 00:07:25,420 So an upper tail probability of 0.15, if we come all the way down to the bottom of the column here, 90 00:07:25,420 --> 00:07:33,520 we see this 70% figure for confidence level, a confidence level of 70% and an upper tail probability 91 00:07:33,850 --> 00:07:36,790 of 0.15 mean the same thing. 92 00:07:36,790 --> 00:07:42,250 What you can notice if you compare all of the confidence levels with all of the upper tail probabilities, 93 00:07:42,370 --> 00:07:46,240 is that if we look here at confidence level, let's say we have 70% here. 94 00:07:46,240 --> 00:07:50,740 If we subtract 70% from 100%, we get 30%. 95 00:07:50,740 --> 00:07:54,490 If we divide that value by two, we get 15%. 96 00:07:54,490 --> 00:08:00,670 And we see that that is the value here for upper tail probability 0.15 and that holds across the board. 97 00:08:00,670 --> 00:08:07,930 So if we go over here to a 99% confidence level, if we subtract that from 100%, we get 1%. 98 00:08:07,960 --> 00:08:11,560 If we divide 1% by two, we get half a percent. 99 00:08:11,560 --> 00:08:16,900 And if we come up to the top of the column here, we can see that that upper tail probability is half 100 00:08:16,900 --> 00:08:17,800 of 1%. 101 00:08:17,800 --> 00:08:24,700 So really, we're looking up the same thing, whether we look up 0.15 upper tail probability or 70% 102 00:08:24,700 --> 00:08:26,050 for the confidence level. 103 00:08:26,050 --> 00:08:30,430 But sometimes we'll have confidence level and sometimes we'll have upper tail probability. 104 00:08:30,430 --> 00:08:35,650 And so we include both of those in the tea table so that we can quickly find whatever value it is that 105 00:08:35,650 --> 00:08:40,419 we're working with and then look up the corresponding value from the body of the tea table. 106 00:08:40,419 --> 00:08:45,130 Now, the only other thing we really want to say at this point is that, of course all of these values 107 00:08:45,130 --> 00:08:53,740 are important and useful, but the values that we will use by far the most often are given by 90 or 108 00:08:53,740 --> 00:08:59,350 95% confidence or 99% confidence. 109 00:08:59,350 --> 00:09:08,350 And of course, those are associated with upper tail probabilities of 5%, 2.5% and half of a percent. 110 00:09:09,140 --> 00:09:10,160 Respectively. 111 00:09:10,190 --> 00:09:15,020 Again, like we said earlier, we'll talk about these things later, but it is very common for us to 112 00:09:15,020 --> 00:09:19,870 choose a confidence level of 90%, 95% or 99%. 113 00:09:19,880 --> 00:09:25,580 Those are common confidence levels to work with, sort of think about them as industry standards. 114 00:09:25,580 --> 00:09:31,430 And so these are the values we'll see the most, which means that we will very often find ourselves 115 00:09:31,580 --> 00:09:39,890 within these columns of the tea table, not that we won't use the other columns ever, but these will 116 00:09:39,890 --> 00:09:43,600 probably be the columns we'll use most often. 117 00:09:43,610 --> 00:09:47,120 And so we just really want to be aware of these things. 118 00:09:47,120 --> 00:09:51,740 In particular, 0.05 upper table probability is equal to a 90% confidence level. 119 00:09:51,830 --> 00:09:59,570 0.0 to 5 upper tail probability is the same thing as 95% confidence level and 0.005 upper tail probability 120 00:09:59,570 --> 00:10:02,300 is equivalent to a 99% confidence level. 121 00:10:02,300 --> 00:10:08,060 If we can be very comfortable remembering these relationships, then that'll help us a lot moving forward. 122 00:10:08,060 --> 00:10:10,250 So that's it for the student's T distribution. 123 00:10:10,250 --> 00:10:16,220 We just wanted to introduce this idea that the tea distribution is a normal curve symmetric, centered 124 00:10:16,220 --> 00:10:22,010 at the mean of zero, just like the standard normal distribution, but that the exact shape of the tea 125 00:10:22,010 --> 00:10:25,730 distribution will change depending on sample size. 126 00:10:25,730 --> 00:10:31,280 And we will rely heavily on the tea distribution, especially when we're taking small samples or when 127 00:10:31,280 --> 00:10:36,200 population standard deviation is unknown, which it very, very commonly is unknown. 128 00:10:36,200 --> 00:10:40,340 So we'll rely on the tea distribution and therefore the tea table very often. 129 00:10:40,340 --> 00:10:45,680 But now that we understand the difference between the standard normal distribution and the tea distribution 130 00:10:45,680 --> 00:10:51,740 going forward will have the foundation, we need to start investigating hypotheses and making statements 131 00:10:51,740 --> 00:10:58,580 about our level of confidence surrounding the values of the population parameters, specifically the 132 00:10:58,580 --> 00:11:01,340 actual value of the population mean.