1 00:00:00,090 --> 00:00:07,110 We talked briefly before about this idea that a sample mean, let's say we call it X bar might be a 2 00:00:07,110 --> 00:00:11,140 good or a bad estimate of the associated population mean. 3 00:00:11,160 --> 00:00:14,700 MU It really just depends on the sample that we take. 4 00:00:14,700 --> 00:00:21,510 We might get lucky and pull a random sample that does a good job representing the population and therefore 5 00:00:21,540 --> 00:00:24,480 that sample mean is close to the population mean. 6 00:00:24,480 --> 00:00:31,590 Or we might get unlucky or just do a bad job and find a non representative sample where the sample mean 7 00:00:31,590 --> 00:00:36,000 does not do a good job estimating that associated population parameter. 8 00:00:36,000 --> 00:00:40,200 The population mean this is what we call a point estimate. 9 00:00:40,200 --> 00:00:45,780 So the sample mean is a point estimate for the population mean because these are single points. 10 00:00:45,780 --> 00:00:53,040 They're single values in the same way that sample standard deviation is a point estimate for population 11 00:00:53,040 --> 00:00:54,120 standard deviation. 12 00:00:54,120 --> 00:00:59,520 Now, the benefit of using just a point estimate is that it's pretty easy to do, right? 13 00:00:59,520 --> 00:01:05,400 If this is what we're doing here, we can pull a sample, we calculate sample standard deviation, and 14 00:01:05,400 --> 00:01:11,070 we say, okay, based on the value we found for this standard deviation for this single sample, we 15 00:01:11,070 --> 00:01:16,110 estimate that population standard deviation is equivalent to that sample standard deviation that we 16 00:01:16,110 --> 00:01:16,410 found. 17 00:01:16,410 --> 00:01:17,820 It's pretty easy to do. 18 00:01:17,820 --> 00:01:23,610 But the drawback, of course, is that calculating just that one point estimate doesn't really tell 19 00:01:23,610 --> 00:01:26,940 us how good or bad that estimate really is. 20 00:01:26,940 --> 00:01:29,310 Again, the point estimate could be good. 21 00:01:29,340 --> 00:01:30,870 It could be a bad estimate. 22 00:01:30,870 --> 00:01:32,910 We wouldn't really know either way. 23 00:01:32,910 --> 00:01:35,610 So in contrast to a. 24 00:01:36,370 --> 00:01:40,180 Point estimate, which doesn't necessarily do a great job. 25 00:01:40,420 --> 00:01:48,550 What we instead may do, may choose to do is calculate an interval estimate, which gives us a range 26 00:01:48,550 --> 00:01:52,450 of values in which the population parameter may lie. 27 00:01:52,480 --> 00:01:55,860 Of course, finding an interval estimate is a little harder to do. 28 00:01:55,870 --> 00:02:00,880 It takes a little more work than the point estimate, but it gives us a lot more information. 29 00:02:00,880 --> 00:02:06,610 So if we use an interval estimate and we touched on this very briefly in a previous lesson, but if 30 00:02:06,610 --> 00:02:12,250 we use an interval estimate, we're able to make statements like I'm 95% confident that the population 31 00:02:12,250 --> 00:02:14,620 mean lies in the interval A to B. 32 00:02:14,650 --> 00:02:24,100 Now, that confidence we just mentioned, the 95% confidence is what's called the confidence level confidence 33 00:02:24,100 --> 00:02:28,230 level or we'll call it c L for short. 34 00:02:28,240 --> 00:02:35,650 And when it comes to confidence levels, we will very often use either 90% confidence, 95% confidence 35 00:02:35,650 --> 00:02:38,530 or 99% confidence. 36 00:02:38,530 --> 00:02:41,410 Those are the most commonly used confidence levels. 37 00:02:41,410 --> 00:02:47,200 And what the confidence level really tells us is the probability that an interval estimate will actually 38 00:02:47,200 --> 00:02:49,120 include the population parameter. 39 00:02:49,120 --> 00:02:56,830 So for instance, if we choose a 95% confidence level and using that 95% confidence level, the interval 40 00:02:56,830 --> 00:03:00,790 we calculate is let's say 11 to 15. 41 00:03:00,790 --> 00:03:07,480 Then what we're saying is that there's a 95% chance that the population parameter will say the population 42 00:03:07,480 --> 00:03:11,470 mean lies somewhere between 11 and 15. 43 00:03:11,470 --> 00:03:16,660 Or another way to put that is that for 95% of the samples that we could take. 44 00:03:16,660 --> 00:03:24,130 So if we were to take 100 samples, then 95% of them would give us an interval where the population 45 00:03:24,130 --> 00:03:26,210 mean lies inside of that interval. 46 00:03:26,230 --> 00:03:31,870 5% of the samples we take will give us an interval outside of the population mean. 47 00:03:31,870 --> 00:03:37,030 So let's say that the population mean is actually we'll say 13. 48 00:03:37,030 --> 00:03:44,020 What we're saying is that 95% of our samples will give us a confidence interval around 13 like this 49 00:03:44,020 --> 00:03:45,250 111 to 15. 50 00:03:45,250 --> 00:03:53,800 But 5% of our samples might give us an interval like we'll say 14 to 18. 51 00:03:53,800 --> 00:04:01,300 And obviously the interval 14 to 18 does not include the population mean 13, 13 lies outside of the 52 00:04:01,300 --> 00:04:02,960 interval, 14 to 18. 53 00:04:02,980 --> 00:04:07,990 Now, before we go further, you might be wondering why wouldn't we always just choose the largest or 54 00:04:07,990 --> 00:04:09,550 highest confidence level? 55 00:04:09,550 --> 00:04:12,940 Don't we always want to be as confident as we possibly can be? 56 00:04:12,970 --> 00:04:14,560 The answer is yes. 57 00:04:14,560 --> 00:04:14,980 Of course. 58 00:04:14,980 --> 00:04:16,930 We want to be as confident as we can be. 59 00:04:16,930 --> 00:04:21,010 But choosing a higher confidence level comes with drawbacks. 60 00:04:21,010 --> 00:04:28,180 For example, in order to achieve this higher confidence level, we may need to take a larger sample. 61 00:04:28,180 --> 00:04:31,510 And taking a larger sample might not always be possible. 62 00:04:31,510 --> 00:04:36,670 So there are other factors we have to balance against our level of confidence in general. 63 00:04:36,670 --> 00:04:41,770 Yes, we want the highest confidence level we can get, but not at the expense of some other important 64 00:04:41,770 --> 00:04:45,490 aspect of our sampling and hypothesis testing procedure. 65 00:04:45,490 --> 00:04:49,450 And so we kind of have to balance those two things against each other, and we'll talk about that in 66 00:04:49,450 --> 00:04:50,950 more detail later on. 67 00:04:51,100 --> 00:04:57,880 Now, this idea of confidence level is directly related to a really important value in statistics, 68 00:04:57,880 --> 00:04:59,800 which is called the alpha value. 69 00:04:59,830 --> 00:05:06,130 We indicate it with the Greek letter alpha, and the alpha value is simply the percentage that remains 70 00:05:06,130 --> 00:05:08,140 outside of this confidence level. 71 00:05:08,140 --> 00:05:13,600 So the alpha value associated with a 90% confidence level is 10%. 72 00:05:13,630 --> 00:05:20,590 The alpha value associated with a 95% confidence level is 5%, and the alpha value associated with a 73 00:05:20,590 --> 00:05:23,530 99% confidence level is 1%. 74 00:05:23,530 --> 00:05:27,340 We also call this the level of significance. 75 00:05:27,340 --> 00:05:33,040 So the level of significance or and we'll talk about this later on to we can also call this the probability 76 00:05:33,040 --> 00:05:36,430 of making a type one error. 77 00:05:36,430 --> 00:05:44,500 But either way, for now, we'll just recognize that the alpha value is always equal to one minus the 78 00:05:44,500 --> 00:05:45,460 confidence level. 79 00:05:45,460 --> 00:05:52,660 Now we can always visualize the alpha value as the total area under the normal distribution outside 80 00:05:52,660 --> 00:05:54,940 of what's called the confidence interval. 81 00:05:54,940 --> 00:05:57,550 Here, we're going to be talking about the confidence interval for the mean. 82 00:05:57,550 --> 00:05:59,710 We'll get to that confidence interval in a second. 83 00:05:59,710 --> 00:06:05,230 But if we look here at a normal distribution, so this is the standard, normal distribution with the 84 00:06:05,230 --> 00:06:08,110 mean of zero and a standard deviation of one. 85 00:06:08,110 --> 00:06:13,150 We can show here as an example, a confidence level of 90%. 86 00:06:13,150 --> 00:06:20,500 What that means is that 90% of the area under this distribution or the area under the curve is contained 87 00:06:20,500 --> 00:06:26,590 within these boundaries that we've set, which means, of course, that 10% of the area under the curve 88 00:06:26,590 --> 00:06:29,380 lies outside of these two boundaries. 89 00:06:29,380 --> 00:06:35,670 So if 10% is the alpha value, it's the area outside of this 90% that we've set. 90 00:06:35,710 --> 00:06:41,530 Entered here in the middle, then that means that in this lower tail we're going to have this 5% value 91 00:06:41,530 --> 00:06:42,910 alpha divided by two. 92 00:06:42,910 --> 00:06:46,650 And in the upper tail, we're going to have alpha divided by two or 5%. 93 00:06:46,720 --> 00:06:54,130 So we take the total 10% that lies outside of this 90%, and we divide it exactly in half to put half 94 00:06:54,130 --> 00:06:58,300 of the alpha value in the lower tail and half of the alpha value in the upper tail. 95 00:06:58,300 --> 00:07:00,280 So everything's evenly distributed here. 96 00:07:00,280 --> 00:07:05,740 The 90% in the middle, half the alpha value in the lower tail, half the alpha value in the upper tail, 97 00:07:05,740 --> 00:07:11,410 and in this case with a 90% confidence level, that 90% goes in the middle, which means we have 5% 98 00:07:11,410 --> 00:07:13,240 on the left and 5% on the right. 99 00:07:13,240 --> 00:07:17,950 What we can say then, because this is the standard normal distribution. 100 00:07:17,980 --> 00:07:26,950 Z we know that to find this boundary right here between the lower 5% and that middle 90%, we would 101 00:07:26,950 --> 00:07:30,010 need to look up in the body of our Z table. 102 00:07:30,010 --> 00:07:32,230 This 5% value here. 103 00:07:32,230 --> 00:07:41,680 If we locate that 5% value in the Z table, we find a Z score of -1.65 to find this boundary right here, 104 00:07:41,680 --> 00:07:48,100 this upper boundary, we have the 5% in this lower tail and then the middle 90%, which means this boundary 105 00:07:48,100 --> 00:07:54,940 right here separates 95% of the area on the left from this last 5% of the area on the right. 106 00:07:55,030 --> 00:08:01,120 So we would look up 95% in the body of our Z table or 0.9500. 107 00:08:01,120 --> 00:08:07,990 And when we find that value in the body, we see that it's associated with the Z score of positive 1.65. 108 00:08:08,020 --> 00:08:14,440 Now, of course, it makes sense here that we get exactly opposite Z values because this middle 90% 109 00:08:14,440 --> 00:08:16,690 is perfectly centered at the mean. 110 00:08:16,690 --> 00:08:18,400 It's symmetric around the mean. 111 00:08:18,400 --> 00:08:24,940 And so the distance from the mean to this lower boundary here is the same as the distance from the mean 112 00:08:24,940 --> 00:08:26,890 to this upper boundary here. 113 00:08:26,890 --> 00:08:29,080 And so we're going to get opposite Z scores. 114 00:08:29,110 --> 00:08:32,380 That being said, now is a good time to mention. 115 00:08:32,380 --> 00:08:39,190 Remember we said that 90%, 95% and 99% are the most commonly used confidence levels. 116 00:08:39,190 --> 00:08:45,730 Well, because we're always centering that percentage of the area in the middle of this standard normal 117 00:08:45,730 --> 00:08:46,600 distribution. 118 00:08:46,600 --> 00:08:52,060 That means we're going to have these little symmetric lower and upper tails on either side of that middle 119 00:08:52,060 --> 00:08:57,100 90%, 95% or 99%, which means we're always going to get opposite Z scores. 120 00:08:57,100 --> 00:09:05,440 So when we choose a 90% confidence level, we will always find the Z scores -1.65 and positive 1.65 121 00:09:05,440 --> 00:09:08,440 that bound that middle 90% of the area. 122 00:09:08,590 --> 00:09:16,660 If we use a 95% confidence level, our Z scores are positive and -1.96 And if we use a 99% confidence 123 00:09:16,660 --> 00:09:20,890 level, our Z scores are always positive or -2.58. 124 00:09:20,890 --> 00:09:27,340 So because we very commonly use these confidence levels, that means of course that we will very commonly 125 00:09:27,340 --> 00:09:29,200 use these Z scores. 126 00:09:29,200 --> 00:09:37,090 So we'll see these values 1.65, 1.96 and 2.58 come up all the time over and over again as we start 127 00:09:37,090 --> 00:09:39,640 to work through the hypothesis testing process. 128 00:09:39,640 --> 00:09:41,620 So it's worth pointing them out here. 129 00:09:41,620 --> 00:09:47,050 Now, with all that background, let's talk about this main idea here, which is the confidence interval 130 00:09:47,050 --> 00:09:47,710 for the mean. 131 00:09:47,710 --> 00:09:53,620 Going back to this idea of point estimates versus interval estimates, the confidence interval is the 132 00:09:53,620 --> 00:09:55,630 interval estimate that we're talking about. 133 00:09:55,630 --> 00:10:00,610 Now, the formula that we use to find it and we'll break this down is this formula right here. 134 00:10:00,610 --> 00:10:05,650 So this is the interval from A to B, this is the left edge of the interval A or the lower edge. 135 00:10:05,680 --> 00:10:08,200 This is the upper edge or the right edge of the interval. 136 00:10:08,230 --> 00:10:13,900 B So we have this interval from A on the left side to be on the right side, and it's equal to, you 137 00:10:13,900 --> 00:10:20,500 can see here the sample mean plus or minus the T score that we find multiplied by. 138 00:10:20,500 --> 00:10:26,380 And this value here you may recognize as standard error, which is sample standard deviation divided 139 00:10:26,380 --> 00:10:28,240 by the square root of the sample size. 140 00:10:28,240 --> 00:10:34,090 So sometimes you'll see the formula written this way where we have standard error written out as sample 141 00:10:34,090 --> 00:10:36,820 standard deviation divided by square root of sample size. 142 00:10:36,820 --> 00:10:42,100 And sometimes you'll see the formula written this way where we have se sub x bar, which indicates standard 143 00:10:42,100 --> 00:10:42,550 error. 144 00:10:42,550 --> 00:10:45,340 But either way, here's the first thing we want to point out. 145 00:10:45,340 --> 00:10:52,690 We are showing in our confidence interval formula a T score both here and here in both representations 146 00:10:52,690 --> 00:10:53,650 of the formula. 147 00:10:53,650 --> 00:10:56,260 But up to now we've been talking about Z scores. 148 00:10:56,260 --> 00:11:01,390 We talked about the standard normal distribution here with Z, We talked about these Z scores associated 149 00:11:01,390 --> 00:11:02,950 with each confidence level. 150 00:11:02,950 --> 00:11:10,000 Well, we can use a Z score if we happen to know population standard deviation, but whenever population 151 00:11:10,000 --> 00:11:15,010 standard deviation is unknown, which of course, as you can imagine in the real world, it is almost 152 00:11:15,010 --> 00:11:16,300 always unknown. 153 00:11:16,300 --> 00:11:20,350 Then we need to use a T score instead of a Z score. 154 00:11:20,350 --> 00:11:26,530 But if you did happen to know population standard deviation, then instead of using T here in these 155 00:11:26,530 --> 00:11:28,810 formulas we would use Z. 156 00:11:29,630 --> 00:11:30,640 In its place. 157 00:11:30,650 --> 00:11:35,510 But assuming, of course, that population standard deviation is unknown, which is most likely the 158 00:11:35,510 --> 00:11:40,490 case, then we'll need to use a TX score and this will be our formula for the confidence interval. 159 00:11:40,520 --> 00:11:46,820 Basically, the idea for using a PT score when population standard deviation is unknown is that sample 160 00:11:46,820 --> 00:11:53,780 standard deviation is of course a less reliable predictor of population standard deviation than population 161 00:11:53,780 --> 00:11:55,300 standard deviation itself. 162 00:11:55,310 --> 00:12:03,320 So because s here may not be perfectly accurate, we need to use a more conservative t value instead 163 00:12:03,320 --> 00:12:09,260 of a z value to find the confidence interval In the same way, even if we did no population standard 164 00:12:09,260 --> 00:12:15,830 deviation, we would still have to use a PT score if our sample size was small, because in the same 165 00:12:15,830 --> 00:12:22,340 way we would consider that small sample size to be maybe not so reliable and so we would use a more 166 00:12:22,340 --> 00:12:25,060 conservative t score instead of a Z score. 167 00:12:25,070 --> 00:12:30,950 So just to take a simple example here, so we can show how to calculate the confidence interval, let's 168 00:12:30,950 --> 00:12:38,300 say that the mean exam score of a sample of ten randomly selected students is 86.7. 169 00:12:38,300 --> 00:12:47,240 So we'll say that our sample size is n equals ten, that our sample mean is 86.7. 170 00:12:47,240 --> 00:12:51,770 And we'll say that we calculated a sample standard deviation of. 171 00:12:52,650 --> 00:12:54,030 5.72. 172 00:12:54,030 --> 00:13:00,150 And what we want to do now is find a confidence interval for the population mean at a confidence level 173 00:13:00,150 --> 00:13:01,480 of 99%. 174 00:13:01,500 --> 00:13:07,470 So we want to be really confident here in our result and let's see what that looks like. 175 00:13:07,470 --> 00:13:13,680 So because population standard deviation is unknown, and even if we did know it because our sample 176 00:13:13,680 --> 00:13:19,560 size is small and by small we mean GN is less than 30, our sample size is ten. 177 00:13:19,560 --> 00:13:25,620 We will have to use a T score instead of a Z score because we know that our sample size is n equals 178 00:13:25,620 --> 00:13:25,950 ten. 179 00:13:25,950 --> 00:13:31,260 That means degrees of freedom is equal to ten minus one or nine. 180 00:13:31,260 --> 00:13:38,460 And so if we pull up our t table and this is just a section of the T table that we can use, we look 181 00:13:38,460 --> 00:13:41,310 for degrees of freedom of nine. 182 00:13:41,310 --> 00:13:46,440 We see that over here on the left hand side and then we want 99% confidence. 183 00:13:46,440 --> 00:13:53,550 So if we come down along the bottom to 99% confidence, we see 99% confidence here. 184 00:13:53,850 --> 00:13:57,210 We have degrees of freedom of nine. 185 00:13:57,210 --> 00:14:04,410 And so if we find the intersection of that row and column, we get this value here, 3.25. 186 00:14:04,410 --> 00:14:11,280 Now, if we substitute what we know into our confidence interval formula right here, let's say that 187 00:14:11,280 --> 00:14:15,330 we get the sample mean, which we know is 86.7. 188 00:14:15,330 --> 00:14:21,510 So 86.7 plus or minus the T score that we found 3.25. 189 00:14:21,510 --> 00:14:24,180 So 3.25. 190 00:14:24,180 --> 00:14:30,420 And then we multiply that by sample standard deviation, which we calculated from our sample to be 5.72. 191 00:14:30,420 --> 00:14:38,370 So 5.72 divided by the square root of sample size, our sample size is and equals ten. 192 00:14:38,370 --> 00:14:40,500 So we get square root ten. 193 00:14:40,500 --> 00:14:48,510 And if we use a calculator to find this value, what we get here is 86.7 plus or minus. 194 00:14:48,510 --> 00:14:58,440 This 3.25 times 5.72 divided by root ten is approximately 5.88, which means then that our confidence 195 00:14:58,440 --> 00:15:04,200 interval is A to B equals on the lower end of the confidence interval. 196 00:15:04,200 --> 00:15:12,780 We have 86.7 -5.88 and on the upper end of the confidence interval we have 86.7 plus. 197 00:15:13,830 --> 00:15:14,940 5.88. 198 00:15:14,940 --> 00:15:26,370 And the result there is a confidence interval A to B that is equal to on the lower end, 80.82 and on 199 00:15:26,370 --> 00:15:30,810 the upper end 92.58. 200 00:15:30,810 --> 00:15:34,520 And that is how we calculate a confidence interval. 201 00:15:34,530 --> 00:15:41,070 What it says here, the conclusion that we're basically making is that we are investigating some population 202 00:15:41,070 --> 00:15:42,090 of students. 203 00:15:42,090 --> 00:15:49,170 We took a random sample of ten of those students and from those ten students we calculated a mean exam 204 00:15:49,170 --> 00:15:50,700 score of 86.7. 205 00:15:50,700 --> 00:15:53,670 So we added up the exam score for each of those ten students. 206 00:15:53,670 --> 00:15:58,350 And for those ten students, we calculated a sample mean of 86.7. 207 00:15:58,350 --> 00:16:01,260 So the mean test score was 86.7. 208 00:16:01,260 --> 00:16:05,910 And from that sample we calculated a sample standard deviation of 5.72. 209 00:16:05,910 --> 00:16:11,280 Now, if we were just using a point estimate like we talked about at the beginning, we could stop there 210 00:16:11,280 --> 00:16:14,490 and say that we expect that the population mean. 211 00:16:14,490 --> 00:16:19,890 So for all of the students, not just these ten, but for all the students in our population, we expect 212 00:16:19,890 --> 00:16:22,890 that the mean exam score is about 86.7. 213 00:16:22,890 --> 00:16:28,920 But again, going back to the original point, this particular sample of ten students might be a really 214 00:16:28,920 --> 00:16:31,620 bad estimate of the larger population. 215 00:16:31,620 --> 00:16:37,170 It might not be very representative, it might do a bad job estimating population mean it might also 216 00:16:37,170 --> 00:16:40,380 do a really good job, but we don't really know either way. 217 00:16:40,380 --> 00:16:45,960 It's just a quick and dirty way to find an estimate for population mean, but it doesn't say anything 218 00:16:45,960 --> 00:16:49,770 about how confident we are that this is a good estimate or a bad estimate. 219 00:16:49,770 --> 00:16:52,980 So instead we compute this confidence interval. 220 00:16:52,980 --> 00:17:00,000 And because we used a confidence level of 99%, remember we looked up this 99% confidence level in our 221 00:17:00,000 --> 00:17:00,780 t table. 222 00:17:00,780 --> 00:17:05,790 We had to use a T score instead of a Z score because number one, we didn't know the population standard 223 00:17:05,790 --> 00:17:06,480 deviation. 224 00:17:06,480 --> 00:17:09,810 And even if we did know it, we were taking a small sample. 225 00:17:09,810 --> 00:17:11,970 We had a sample size smaller than 30. 226 00:17:11,970 --> 00:17:15,540 So either of those factors would have required us to use a T score. 227 00:17:15,540 --> 00:17:20,520 We had both of them, so we certainly had to use a T score instead of a Z score, but we looked up that 228 00:17:20,520 --> 00:17:24,900 T score and now this confidence interval is an interval estimate. 229 00:17:24,930 --> 00:17:32,040 It tells us that we are 99% certain that the population mean the mean exam score for the entire group 230 00:17:32,040 --> 00:17:32,520 of students. 231 00:17:32,520 --> 00:17:38,820 The entire population falls somewhere between 80.82 and 92.58. 232 00:17:38,820 --> 00:17:45,180 And as you can see in a lot of ways, that gives us a lot more information than just this single point 233 00:17:45,180 --> 00:17:51,270 estimate where we calculated a sample mean of 86.7 off of just one sample of only ten students. 234 00:17:51,270 --> 00:17:56,430 Now, remember, before we talked about trade offs with the confidence level here, we chose a 99% confidence 235 00:17:56,430 --> 00:17:58,380 level and we found an interval. 236 00:17:58,380 --> 00:17:59,280 Let's round here. 237 00:17:59,280 --> 00:18:08,160 We found an interval that said that the population mean was probably between about 81 and 93, just 238 00:18:08,160 --> 00:18:11,310 rounding these values to the nearest whole number to make it a little easier to see. 239 00:18:11,310 --> 00:18:16,680 But going back to those trade offs, what happens is if we're willing to give up some confidence, so 240 00:18:16,680 --> 00:18:21,840 maybe instead of choosing 99%, we're instead willing to settle for a lower level of confidence and 241 00:18:21,840 --> 00:18:23,490 choose 90% confidence. 242 00:18:23,490 --> 00:18:29,430 Decreasing our confidence level will actually narrow this confidence interval. 243 00:18:29,430 --> 00:18:35,190 So maybe the confidence interval associated with 90% and I haven't actually calculated it, but maybe 244 00:18:35,190 --> 00:18:37,950 it would be something like 85. 245 00:18:38,710 --> 00:18:42,670 To 90 instead of 81 to 93. 246 00:18:42,700 --> 00:18:47,920 In other words, the higher the confidence, the wider the confidence interval, the lower the confidence, 247 00:18:47,920 --> 00:18:49,600 the narrower the confidence interval. 248 00:18:49,600 --> 00:18:55,390 And we like a narrower confidence interval because it really hones in on or gets us closer to gives 249 00:18:55,390 --> 00:19:01,090 us a smaller interval in which our population parameter or population mean likely lies. 250 00:19:01,090 --> 00:19:06,070 But we can't be quite as confident about that narrower interval. 251 00:19:06,100 --> 00:19:12,790 To give you another example, if you ask me to find an interval around the mean height of all American 252 00:19:12,790 --> 00:19:18,910 females, so the entire population is every female in America, and we're interested in the mean height 253 00:19:18,910 --> 00:19:27,400 of that population, I might be able to tell you that I am 99% confident that that population mean lies 254 00:19:27,400 --> 00:19:34,390 somewhere between four feet ten inches and let's say six feet three inches tall. 255 00:19:34,420 --> 00:19:39,010 We can be virtually guaranteed that the population mean is somewhere in this interval. 256 00:19:39,010 --> 00:19:40,630 This is a really wide interval. 257 00:19:40,630 --> 00:19:48,490 But if I start to narrow that interval and I maybe find an interval like five feet four inches to five 258 00:19:48,490 --> 00:19:56,110 feet, six inches, well, I really narrowed the interval, but maybe I'm only 90% confident or even 259 00:19:56,110 --> 00:20:01,510 less confident that the mean height of all American females is somewhere in that narrower interval. 260 00:20:01,510 --> 00:20:07,360 So the higher the confidence level, the wider the interval has to be by definition, the narrower the 261 00:20:07,360 --> 00:20:10,780 confidence interval, the lower our confidence level. 262 00:20:10,780 --> 00:20:16,180 So that is how we calculate the confidence interval for the mean of a population. 263 00:20:16,180 --> 00:20:22,120 Next, we'll use all of the information that we've gathered about Z scores and T scores, how to take 264 00:20:22,120 --> 00:20:27,400 samples and calculate confidence intervals, and we will turn toward hypothesis testing and the steps 265 00:20:27,400 --> 00:20:31,510 that we take to investigate a hypothesis from beginning to end.