WEBVTT 00:00.960 --> 00:06.210 In this session, we will discuss about that school, also known as kind of school. 00:07.960 --> 00:15.100 A comparison of the relative standard like what the mean and the standard deviation can we need using 00:15.100 --> 00:17.440 Z score or standard scooter? 00:18.680 --> 00:23.300 So let us see, we have a particular dataset. 00:24.460 --> 00:28.660 So a particular data set will have some distribution as. 00:30.510 --> 00:39.750 Now, let us see in that particular data set or in that particular sample, I want to find out the location 00:39.750 --> 00:48.600 of a particular data point or I want to find out how far that data point should be present in terms 00:48.600 --> 00:56.720 of the standard deviations or in terms of the distance, in terms of mean and standard deviation. 00:57.060 --> 01:01.530 So that is something which is provided by zig zag. 01:01.530 --> 01:12.810 Score is just a value which is calculated and the value of the Z score is four, then calculated in 01:12.810 --> 01:13.490 the Z. 01:14.430 --> 01:17.790 So we have a Z score, which we calculate from the plot. 01:18.390 --> 01:27.330 And based on that Z school, we get value from the Z table, which gives the probability value. 01:28.940 --> 01:35.930 So this probability value will actually give me that what is the probability of this value being present 01:36.110 --> 01:36.870 in the plot? 01:38.450 --> 01:42.070 And in what region of the world should the value be available? 01:43.480 --> 01:50.090 So let us try to understand this more clearly and we will begin a lot about this value. 01:50.860 --> 01:52.050 So what is it? 01:52.240 --> 02:00.430 The number of standard deviations are data value is above or below the mean for a specific distribution 02:00.430 --> 02:01.070 of the value. 02:01.570 --> 02:04.930 So let's say they have a particular distribution. 02:08.390 --> 02:15.650 Let us say we have this distribution and I want to find out the location of the zettl pointed in terms 02:15.650 --> 02:22.940 of the mean of this distribution, then I can find it on the basis of the Z Squaw Valley. 02:25.490 --> 02:34.400 So that is a very useful statistic because it allows us to calculate the probability of a score occurring 02:34.400 --> 02:36.600 within our normal distribution. 02:36.860 --> 02:43.730 So based on the Z score, I can actually find out what is the probability of a particular value occurring 02:43.730 --> 02:46.330 in this distribution, in this normal distribution. 02:46.970 --> 02:48.670 So I can find that opio. 02:50.390 --> 02:52.160 Now, as it's called. 02:53.390 --> 03:00.230 Enables us to compare to schools that are from different normal distribution, so like I say, I have 03:00.230 --> 03:07.280 two different normal distributions and I want to compare these two schools belong to the same population 03:07.520 --> 03:09.070 or the other. 03:09.170 --> 03:11.030 Where do they actually stand? 03:11.030 --> 03:15.180 In front of each other in a particular sample. 03:15.410 --> 03:20.470 So these things, this kind of comparison can actually be made using the zip code. 03:21.410 --> 03:26.010 Now, the value of the Z score will be from zero to one. 03:26.150 --> 03:34.130 So if a Z score is zero, it will indicate that the data point score is identical to the mean score. 03:37.170 --> 03:40.290 If the Z score value, this is the means for. 03:41.250 --> 03:49.020 The mean score will have the value as Zettl now if the Z score value is towards one, it is anything 03:49.230 --> 03:57.150 greater than zero then it is toward the point is actually lying towards the right hand side of the knee. 03:58.020 --> 04:06.410 If a point has a Z score less than zero, it means that the point this line towards the left hand side 04:06.420 --> 04:12.750 of the knee in their distribution and it is always calculated as a part of the normal distribution. 04:13.020 --> 04:18.840 So when we are calculating the score, we are assuming that the distribution is actually a normal distribution. 04:20.310 --> 04:29.220 And because the distribution is normal and it is symmetric in nature, so if we have let us see a value 04:29.220 --> 04:37.800 of minus zero point seven five a Z value of minus zero point seven five, then we will look at the probability 04:38.190 --> 04:39.070 in the Z. 04:39.630 --> 04:47.040 So let's see, we have the value of Z as my minus point seven five and we are looking up in the Z table. 04:47.400 --> 04:54.060 Then in the Z, the value of the probability, which will be their legacy, it is willing to do so. 04:54.060 --> 05:01.580 It means that the area below this point in the normal distribution has 22 percent of the data. 05:01.830 --> 05:06.730 It means that there is a twenty two point six percent probability of the point being here. 05:07.320 --> 05:10.650 So there is a twenty two point six six percent data below this. 05:11.310 --> 05:16.470 And again, if we have the value as point seven five, this is negative. 05:16.500 --> 05:17.320 This is positive. 05:17.340 --> 05:20.380 Then this becomes again twenty two point six percent. 05:20.880 --> 05:23.580 So the percentage of area is also equal here. 05:23.850 --> 05:30.570 So if let's say we want to find out something here, then the probability of the area which is covered 05:30.570 --> 05:33.320 in this region will be equal to the area covered in this region. 05:34.870 --> 05:37.360 So we can simply calculate the value of. 05:38.410 --> 05:38.860 One. 05:39.550 --> 05:43.390 So if we have a point minus zero point eight. 05:44.330 --> 05:51.630 On minus zero point eight one, then we can calculate zero point eight one and it will have seen the 05:51.820 --> 05:52.400 on the board. 05:54.130 --> 05:55.050 To see for the. 05:57.000 --> 06:02.340 As one would indicate the value that is one standard deviation from the mean. 06:03.500 --> 06:11.150 So if your value is minus zero point seven five, it means that it is zero point seventy five in one 06:11.150 --> 06:12.530 standard deviation of the mean. 06:13.190 --> 06:18.320 If it is at one, this means that it is minus one standard deviation from the mean. 06:18.890 --> 06:23.380 If it is minus two, then it means that it is two standard deviation from the mean. 06:23.690 --> 06:28.450 If it is minus three, it means that it is three standard deviations from them. 06:28.910 --> 06:35.300 And you remember from the standard deviations, we have this particular plot here, which means that 06:35.300 --> 06:41.240 if something is one standard deviation away from the mean, then the area is thirty four point one three 06:41.240 --> 06:41.660 percent. 06:42.200 --> 06:47.920 If something is to a standard deviation from the mean, then it is 95 percent of the data. 06:48.530 --> 06:54.470 If something is three standard deviations from the mean, then it contains ninety nine point seven seven 06:54.470 --> 06:55.070 percent of the. 06:56.560 --> 07:00.540 So we can derive this also from the Z value. 07:05.080 --> 07:12.280 The formula for Z being calculated for a particular value is so let's say we have a value zero point 07:12.880 --> 07:23.770 or fifteen, then the Z value will be fifty minus the mean of the plot divided by the standard deviation 07:23.770 --> 07:24.400 of the plot. 07:24.970 --> 07:35.170 So this will be X minus one divided by sampling error in case of sample and X minus population mean 07:35.290 --> 07:39.100 divided by the population standard deviation in the population. 07:39.880 --> 07:43.240 So this is what the formula Z is. 07:46.800 --> 07:48.590 Now using the Z. 07:49.750 --> 07:52.480 We can actually standardize the date also. 07:52.900 --> 08:00.310 So what is that school standardization, Zelikow's scores, standardization is actually used to scale 08:00.310 --> 08:03.610 the data so that all the variables have the same range. 08:04.090 --> 08:07.880 So let's say we have a column each and a column Silalahi. 08:08.200 --> 08:13.390 And the range of age and salary is very different here. 08:13.570 --> 08:19.900 Age is from 20 to 52, and here the salary is from twenty thousand forty five. 08:20.560 --> 08:22.210 So the scale is a lot different. 08:22.930 --> 08:29.890 So in case I want to find out that what is the probability of having this age with respect to the salary, 08:29.920 --> 08:32.950 then we can only find that out because the scheme is a lot different. 08:33.220 --> 08:40.420 So what we can do is we can scale this age in salary and then we scale these using the Z score to the 08:41.680 --> 08:48.400 standard normal distribution will be change in such a way that the mean will be at zero and the standard 08:48.400 --> 08:49.510 deviation will be one. 08:49.510 --> 08:56.530 For this both of these columns and the Z value will be X minus will divided by Sigma, which is actually 08:56.530 --> 08:59.350 the Z value from this particular formula, which we are very. 09:00.340 --> 09:02.650 X minus moderated by Sigma. 09:06.190 --> 09:10.780 So similarly, we can scale the data and we will get the scale. 09:11.470 --> 09:17.410 Now here you can see the values are ranging from minus one point to nine, minus one point one point 09:17.410 --> 09:19.350 six nine two one point four. 09:19.840 --> 09:22.390 So here we can kind of compare these values. 09:24.460 --> 09:34.060 This is about that school and let us look for the now legacy now we have understood this, but we need 09:34.060 --> 09:36.010 to have more clarity on this. 09:36.370 --> 09:38.740 So let us look for the legacy. 09:38.740 --> 09:42.340 We want to find out the probability, which is the area under the curve. 09:42.370 --> 09:51.040 Remember, the probability is idiomatically for a normally distributed variable by transforming it into 09:51.040 --> 09:53.480 a standard normal variable by using the formula. 09:53.500 --> 09:58.160 So we are using this particular formula and we are trying to find out the probability associated. 09:58.390 --> 10:00.880 So let us see a few examples related to this. 10:02.220 --> 10:03.980 So this is the first example. 10:05.280 --> 10:12.730 So what we are doing here is we are seeing that of the average or the mean of a particular value, so 10:12.780 --> 10:15.880 legacy, we are talking about the students study. 10:16.320 --> 10:21.300 So let us say a student is studying three point one hours every day. 10:22.230 --> 10:26.240 So this is the mean average of all the students in the world study. 10:26.760 --> 10:30.010 And we say that the standard deviation is zero point five. 10:30.780 --> 10:36.420 We say there is a standard deviation of zero point five in the average or the mean value of three point 10:36.420 --> 10:38.870 one, as well as where the students study. 10:39.660 --> 10:46.050 And we want to find out the percentage of students studying less than three point five hours. 10:47.470 --> 10:54.730 So we are seeing the average student studies for three point one dollars and the standard deviation 10:54.730 --> 10:56.060 is of zero point five. 10:56.650 --> 11:02.440 And we just want to find out the percentage of students who are studying less than three point five 11:02.440 --> 11:02.690 hours. 11:03.010 --> 11:05.170 So if we have this particular block. 11:06.140 --> 11:09.350 So from this particular plot, let's try to. 11:10.730 --> 11:11.510 Point out. 11:12.550 --> 11:14.530 So this would be the mean value. 11:16.390 --> 11:18.220 This is the mean value would just. 11:19.500 --> 11:21.540 Three point one in this particular case. 11:24.380 --> 11:32.000 This is the mean value and the standard deviation is zero point five, so the zero five zero point five 11:32.000 --> 11:33.080 is a standard deviation. 11:33.500 --> 11:35.630 So zero point five would be somewhere here. 11:40.300 --> 11:44.120 This is zero point five zero point five. 11:44.770 --> 11:49.930 So if we think about it based on the values of let us look at it on the basis of the Belko. 11:50.200 --> 11:51.940 So this is the Belko. 11:53.280 --> 11:56.100 And this is one standard deviation. 11:57.180 --> 12:02.670 We are saying that this is one standard deviation and this point, this zero point five, and we are 12:02.670 --> 12:06.330 asking about zero point eight, which is somewhat higher. 12:07.340 --> 12:12.680 So we are trying to ask the question, we're asking a question like, what is the probability of student 12:12.680 --> 12:18.200 study less than zero point eight hours, less than three point daters. 12:20.390 --> 12:24.390 Less than three point five hours, less than three point five hours. 12:24.620 --> 12:26.900 So three point one is here. 12:26.930 --> 12:28.370 OK, this is three point one. 12:30.260 --> 12:37.760 And the standard deviation is zero point five, so this is if this is three point five, this is the 12:37.760 --> 12:44.780 standard deviation to zero point five, then one standard deviation will be three point six. 12:45.880 --> 12:48.560 One standard deviation will be three point six. 12:49.000 --> 12:49.900 This means. 12:51.230 --> 12:55.910 That this is actually one standard deviation and this value is. 12:56.990 --> 12:57.770 Three point. 13:03.290 --> 13:09.710 We want to find out what is the percentage of people spending less than three point five hours and we 13:09.860 --> 13:11.460 think that this is three point five. 13:11.840 --> 13:15.660 So give me the percentage of people selling for less than three point five hours. 13:15.890 --> 13:21.840 So if we think about it theoretically and just on the basis of Belko, so let us look about it just 13:21.890 --> 13:23.060 on the basis of Valko. 13:24.170 --> 13:30.740 So this is our Belko, and this is one standard deviation, which is three point six. 13:37.460 --> 13:41.630 This is three point six and this is three point five. 13:48.260 --> 13:50.280 So what will be the percentage? 13:51.020 --> 13:53.360 So the percentage will be 2.8. 13:55.280 --> 13:56.750 Plus thirteen point five. 13:58.450 --> 14:00.220 Plus thirty four point five. 14:01.540 --> 14:07.840 Plus, some part of this thirty four point five, not completely this complete thirty four point five, 14:08.080 --> 14:09.440 but some part of it. 14:09.950 --> 14:12.120 OK, let's say to something like that. 14:12.400 --> 14:15.730 So the sum of these values will be. 14:17.170 --> 14:19.060 Do point to wake. 14:20.080 --> 14:30.370 Plus, thirteen point five nine plus thirty four point one three, which is 50, so the Swatch area 14:30.370 --> 14:36.120 is 50, which is already understandable because the idea is 50 percent of the block. 14:36.400 --> 14:39.090 So this area will have 50 percent of the data. 14:39.670 --> 14:44.680 And on top of it, there will be some more percentage, which we have to calculate. 14:45.100 --> 14:46.690 So how do we calculate that? 14:48.530 --> 14:51.360 So we calculate that using this formula. 14:52.250 --> 14:54.200 So what is this formula now? 14:54.200 --> 14:56.210 Our new value is three point five. 14:57.340 --> 15:03.610 New Orleans, three point five, the value of mine is three point one, the standard deviation is zero 15:03.610 --> 15:04.210 point five. 15:04.540 --> 15:06.420 So what is the value of this? 15:06.730 --> 15:08.970 The value of this is zero point eight. 15:09.820 --> 15:11.470 This is the Z School. 15:12.760 --> 15:15.570 This is the Z score, which we have found. 15:16.450 --> 15:23.620 Now we just found out the Z score, this is the Z School and this is one standard deviation. 15:25.000 --> 15:27.100 So one standard deviation will be one. 15:30.000 --> 15:33.540 And this isn't code for zero point eight. 15:34.670 --> 15:36.310 This is a policy to point. 15:36.770 --> 15:43.580 This is a school for one standard deviation is one that's for one standard deviation is one school for 15:43.580 --> 15:44.450 mean a zero. 15:47.550 --> 15:50.400 So let us find out the percentage here. 15:50.430 --> 15:52.260 So how do we find out this percentage? 15:52.480 --> 15:54.750 We go to the vet, they will know. 15:54.820 --> 15:56.610 So now look at the Z. 15:57.420 --> 15:58.910 So this is the Z table. 16:00.070 --> 16:06.160 So what does this table represent, this table value represents that area to the left side of the Z 16:06.160 --> 16:06.520 score. 16:07.000 --> 16:14.490 So if we have a Z score of zero point eight, we have the Z score of zero point B, so four zero point 16:14.530 --> 16:14.740 eight. 16:14.740 --> 16:17.940 That's all we want to find out the area below zero point. 16:18.250 --> 16:20.510 So towards the left hand side of zero point eight. 16:20.530 --> 16:22.390 And that is exactly what we want to find. 16:22.930 --> 16:24.510 So we will go to this table. 16:24.820 --> 16:26.680 We will look at zero point eight. 16:28.230 --> 16:33.510 And what is the next value, zero point eight point zero zero, because there is nothing else in the 16:33.510 --> 16:36.850 decimal point that is nothing else in the decimal point. 16:37.560 --> 16:39.800 So we will look at zero point zero. 16:39.810 --> 16:40.890 So what is the value? 16:40.920 --> 16:44.940 The value is zero point seven eight eight one. 16:45.540 --> 16:49.260 So this is the percentage of. 16:50.540 --> 16:54.260 Students who are study less than three point five hours. 16:56.910 --> 17:00.700 So what is the percentage, the percentage is seventy eight point one percent. 17:01.050 --> 17:03.740 So they stand for less than fifty point five hours. 17:04.230 --> 17:06.040 So that is how we calculate this. 17:06.450 --> 17:07.650 So it is very simple. 17:07.770 --> 17:09.680 All we need is the mean value. 17:10.020 --> 17:15.600 We need the standard deviation value, and we want to find out with respect to what we actually want 17:15.600 --> 17:16.940 to find out the percentage. 17:17.220 --> 17:23.010 So based on this percentage value, we will find out the Z score, the Z score will actually give us 17:23.400 --> 17:26.610 where in the plot we want to find out the probability. 17:26.880 --> 17:33.930 And from the Z score, once we find out the Z score, we go to the Z value and from the Z table, we 17:33.930 --> 17:35.250 look at the value here. 17:36.640 --> 17:42.600 So if the value was zero point eight to one, then what will happen, zero point eight and then we'll 17:42.610 --> 17:49.240 get to zero point eight two and the percentage would have changed to seventy nine point three. 17:49.250 --> 17:49.690 It was. 17:51.010 --> 17:53.870 So this is how we find out the percentage fun. 17:56.350 --> 17:58.450 Now, look at the next problem in hand. 17:59.170 --> 18:00.440 This is the second problem. 18:00.910 --> 18:02.420 Now, what is the problem here? 18:02.710 --> 18:07.000 The problem is that the mean is 2010. 18:07.660 --> 18:12.870 So now the mean value is 2012 and the standard deviation is to increase. 18:14.360 --> 18:20.560 Mean is to when the standard deviation is 12, so what is the mean value? 18:21.890 --> 18:22.950 Mean value is. 18:23.950 --> 18:25.570 Twenty eight at Zettl. 18:29.570 --> 18:31.600 Zero, the mean value is. 18:33.940 --> 18:34.570 Wendy. 18:37.490 --> 18:37.910 It. 18:39.220 --> 18:48.580 And we want to find out what will be the percentage between 27 and 31, what is the value between 27 18:48.580 --> 18:49.200 and 31? 18:49.540 --> 18:51.830 So simply we will apply the formula. 18:52.090 --> 18:55.300 So we want to find out between 27 and 31. 18:55.300 --> 18:57.160 So first we will find out the area. 18:58.080 --> 19:05.640 Above twenty seven, so we will find out what is the area at 27, what is the area at 31, how much 19:05.640 --> 19:07.180 it is there below 31. 19:07.410 --> 19:11.430 And then we will surprise the area below. 19:11.460 --> 19:15.560 Twenty seven from the area below 31. 19:15.780 --> 19:18.030 And this will give us this particular area. 19:18.360 --> 19:21.300 So let us find out the area below twenty seven. 19:21.570 --> 19:28.680 So below twenty seven will be twenty seven, minus 20 divided by the standard deviation we just do so. 19:28.680 --> 19:32.160 Twenty seven minus twenty eight divided by two gives us minus. 19:33.350 --> 19:34.460 Zero point five. 19:36.220 --> 19:44.500 Next, we will find out the area, the Z value at 31 valued 31 will be X minus MU, which is thirty 19:44.500 --> 19:47.410 one minus 20, divided by two, which is one point five. 19:48.390 --> 19:49.770 Now, what do we want to find out? 19:49.800 --> 19:54.470 We want to find out the area between 27 and 31. 19:54.540 --> 19:57.790 We want to find out the area between 27 and 31. 19:57.990 --> 19:59.050 So what will that be? 19:59.220 --> 20:00.000 That will be. 20:01.550 --> 20:03.200 This entire area. 20:04.590 --> 20:06.000 This entire area. 20:07.370 --> 20:09.930 Minus only this Majida. 20:11.310 --> 20:18.120 We want to minus only this small area from this entire area so that we get this shaded portion, which 20:18.120 --> 20:18.870 we actually need. 20:19.230 --> 20:20.140 So what do we do? 20:20.460 --> 20:24.600 We find out the area of the probability of xai. 20:25.570 --> 20:26.650 At one point five. 20:27.800 --> 20:32.020 And we subtract the probability of that minus one point five from it. 20:32.300 --> 20:33.530 So what are these values? 20:35.270 --> 20:39.470 So we want to find out the probability at one point five, so what is eleven point five? 20:39.490 --> 20:43.880 We have one point five and one point five with this ninety three point thirty one. 20:49.220 --> 20:52.400 What is the probability at minus zero, point five? 20:52.790 --> 20:57.980 So we will look at the negative Vados, this is minus zero point five and it is. 20:59.190 --> 21:01.330 Zero point three zero eight. 21:02.190 --> 21:05.940 So we will subtracted from this one, so we get 60. 21:07.420 --> 21:09.230 Zero point six two four seven. 21:09.580 --> 21:16.060 So this shaded area is 62 percent of. 21:17.210 --> 21:19.060 The entire block. 21:20.510 --> 21:25.400 So the area between 27 to 31 is sixty two point four, seven percent. 21:30.860 --> 21:37.820 Now, let us look at the next problem, the next problem is finding specific data values for a given 21:37.820 --> 21:40.790 percentage using the standard normal distribution. 21:41.030 --> 21:44.180 So now the question is entirely different. 21:44.420 --> 21:47.000 So it is the opposite of what we were doing earlier. 21:47.120 --> 21:53.450 Earlier, what we were doing is we had certain values and based on these values, we had to find out 21:53.450 --> 21:54.280 the percentage. 21:54.710 --> 21:56.420 But now what the problem is. 21:56.630 --> 22:02.020 We are given a percentage and we want to find out the standard distribution. 22:02.420 --> 22:03.440 So what we do is. 22:05.210 --> 22:07.940 We are living in the top 10 percent. 22:09.390 --> 22:12.420 So the top 10 percent of data is given. 22:13.350 --> 22:17.250 So in the top 10 person, the meanest two hundred. 22:19.410 --> 22:22.530 And the standard deviation is 20. 22:25.170 --> 22:29.070 Find the lowest possible school to quality. 22:32.440 --> 22:35.280 So we want to find out the lowest possible. 22:39.860 --> 22:41.590 So what is this 10 person? 22:43.090 --> 22:50.530 We want to find out the Z score for this straight, so how do we find out we have this 10 percent value? 22:51.530 --> 22:57.230 This 10 percent value implies that the value is actually zero point one zero. 23:00.530 --> 23:07.640 OK, this is zero point one zero one method, what we could have done is because this is zero point 23:07.640 --> 23:08.240 one zero. 23:13.000 --> 23:14.710 And this plot is symmetry. 23:15.460 --> 23:17.500 So if I plot another line here. 23:19.370 --> 23:21.320 And look at 10 percent here. 23:22.530 --> 23:25.200 Then if I find out this area. 23:33.300 --> 23:37.380 Then the value will be just in the negative of this. 23:41.620 --> 23:48.760 If we have been blessed in this region and we find them in this region, then the Z value will be negative 23:48.850 --> 23:55.450 in here and here, it will be positive that so we can simply find out what is zero point one zero. 23:56.740 --> 23:58.960 We can simply find out this is 10 percent. 23:59.290 --> 24:05.160 So, again, we can find out what this 10 percent in this region and then we can find out the Z value. 24:05.470 --> 24:07.000 So let us look at that post. 24:10.780 --> 24:12.460 So in the negative region. 24:13.500 --> 24:14.670 The Z value is. 24:16.700 --> 24:21.720 Then so this is the zero point one zero zero. 24:21.830 --> 24:23.100 So this is 10 percent. 24:23.630 --> 24:24.760 This is 10 percent. 24:24.770 --> 24:25.610 And what is this? 24:25.610 --> 24:27.800 What this is four minus one point. 24:29.030 --> 24:36.800 So we can simply say that here the Z value is one point to begin to validate the values of one point 24:36.800 --> 24:39.710 to the Z value will come out to be one point. 24:43.990 --> 24:50.170 So how we can calculate this now, similarly from the Z table, you can look at that then and you can 24:50.170 --> 24:53.340 see that this value is zero point zero three. 24:53.350 --> 24:57.430 And on the basis of that, you can find out the minus one point to it. 24:57.440 --> 25:04.210 So you can see it simply see the Z value is one point to it, which is exactly what we just did. 25:04.690 --> 25:10.150 So this is one point two and it is at the eighth column Zettl. 25:15.870 --> 25:16.290 Three. 25:19.100 --> 25:22.550 Five, six, seven. 25:23.430 --> 25:28.110 And so this is one point two eight. 25:29.670 --> 25:32.380 So their value is one point, do it. 25:33.300 --> 25:40.200 We have the mean value, we have the standard deviation so we can find the value here by X is equal 25:40.200 --> 25:41.380 to Z Endou. 25:42.900 --> 25:43.780 God plus. 25:45.090 --> 25:53.160 So it comes out to 26, so we can say the minimum value has to be 226. 25:58.980 --> 26:07.480 We have simply find out that the lower value actually has the same percentage because the plot is symmetrical. 26:07.650 --> 26:13.020 So based on symmetry, we have found out the value of Z to be does the negative of this value. 26:13.260 --> 26:15.290 So we found out it is one point, do it. 26:15.300 --> 26:20.440 And from one point to it, once we put it in the formulative, we get the value to 26. 26:20.700 --> 26:27.870 So the value, which is the actual numeric value here, will be to 26 and the mean will be two hundred 26:27.870 --> 26:29.120 at this particular point. 26:34.300 --> 26:35.880 Now we have another problem. 26:37.290 --> 26:43.860 Now, what this problem is that we want to select the middle 60 percent of the population. 26:46.570 --> 26:49.240 And we say that the mean is 120. 26:50.940 --> 26:53.520 And the standard deviation is eight. 26:55.830 --> 26:58.080 Find the upper and then lower values. 26:59.260 --> 27:02.410 So let us give you the bell curve again. 27:03.400 --> 27:05.220 OK, let us get to the Belfour. 27:09.550 --> 27:10.660 This is the Belko. 27:11.840 --> 27:19.210 And from this vehicle, you can see that one standard deviation is almost 60 percent. 27:20.420 --> 27:24.000 One standard deviation is almost sixty eight percent. 27:24.140 --> 27:25.180 So let us get back. 27:31.900 --> 27:38.830 Now we want to find out in the middle 60 percent or so, because this much area. 27:40.630 --> 27:46.150 Is actually sixty eight percent, one standard deviation is 68 percent. 27:46.480 --> 27:51.460 So 60 percent will anyhow be between these one standard deviations. 27:52.700 --> 27:56.480 So and what is one standard deviation, one standard deviation is eight. 27:58.310 --> 28:00.380 So these the values. 28:01.810 --> 28:04.090 These raised values will actually be. 28:06.210 --> 28:11.430 120 plus it, that is 128, this value will be. 28:12.750 --> 28:13.320 One. 28:16.890 --> 28:17.940 Twenty eight. 28:22.450 --> 28:29.740 Mean my writing here and this value will be 120 minus eight, which is one hundred twelve. 28:31.910 --> 28:35.100 So this much idea is 68 percent. 28:35.360 --> 28:38.580 So 60 percent will be between these two values. 28:39.500 --> 28:41.210 So how do we find that out? 28:41.510 --> 28:47.700 So we can simply kind of see that the values will be between 100 and twelve and 120. 28:47.720 --> 28:49.670 Now we have to find out the actual values. 28:49.860 --> 28:51.440 So how do we find out these? 28:51.860 --> 29:00.110 So we can find that out by simply thinking that this is 60 percent middle 60 percent means that this 29:00.110 --> 29:00.800 area is. 29:02.510 --> 29:03.500 The middle line. 29:04.740 --> 29:07.600 So here we have some 30 percent. 29:10.190 --> 29:17.900 And here also we will have other people of the area which we want to have now, because here we want 29:17.900 --> 29:19.340 to have a 30 person. 29:21.070 --> 29:26.140 This means that this 30 person will again be. 29:28.630 --> 29:30.250 A part of this area. 29:31.840 --> 29:33.660 So we have 30 percent here. 29:34.850 --> 29:36.970 And we have another 20 percent you. 29:38.260 --> 29:45.880 Similarly, we will have a 30 percent year and we have another 20 percent this this that is in totality 29:45.880 --> 29:47.500 will make the one hundred percent right. 29:48.070 --> 29:49.620 This is the end of the goal. 29:49.660 --> 29:52.680 The area under the curve is hundred percent. 29:53.050 --> 29:59.650 So if we have 30 percent, if we have the middle 60 percent, then from the middle line, that is the 29:59.660 --> 30:00.100 mean. 30:00.100 --> 30:02.880 That is the median from this middle line. 30:03.130 --> 30:06.420 This much area will be the 30 percent vaidya. 30:07.730 --> 30:13.380 And this area will again be another 30 percent media, which will totally add to 60 percent. 30:14.160 --> 30:17.090 And further, there will be a 20 percent. 30:17.090 --> 30:19.410 And here also we will have another 20 percent. 30:20.360 --> 30:21.860 So how do we see that? 30:21.860 --> 30:32.070 We can state that in a form that the probability of area between minus area of the probability with 30:32.450 --> 30:36.020 E is minus E is less than say. 30:37.530 --> 30:42.300 Minuses less than, say, and Zeller's, less than a. 30:43.410 --> 30:44.730 Well, that is the mean value. 30:45.450 --> 30:51.670 We have certain value, any value could be there, and that value has to be greater than mine. 30:51.670 --> 30:53.160 MASEY and less than eight. 30:53.370 --> 30:58.320 Then the area between this and will be 60 percent, which is zero point sixty. 30:59.790 --> 31:04.870 Now, because this is symmetric in nature, then we can see because this is symmetric. 31:04.890 --> 31:12.000 So instead of seeing all these infarcts, we can simply see a Z of less than minus. 31:14.420 --> 31:15.140 In total. 31:16.880 --> 31:20.210 Area of Z, less than minus eight. 31:26.600 --> 31:27.290 What is it? 31:27.860 --> 31:29.810 Less than minus eight, this 20 percent. 31:32.390 --> 31:35.420 This 20 percent is nothing but. 31:37.940 --> 31:39.020 This Bizzell. 31:42.200 --> 31:49.220 This feasible minus Z, less than minus eight is simply the probability of area being less than minus 31:49.220 --> 31:50.710 eight, which is this portion. 31:51.080 --> 31:57.500 And when we do the case of this, then it becomes 40 percent and we're approaching it from one. 31:58.500 --> 32:01.470 This means that we are getting the 60 percent of the. 32:02.780 --> 32:11.120 Because this is symmetric, right, so we can equate this, we can equate this entire thing with this 32:11.120 --> 32:16.580 probability of A minus A less than Z, less than a. 32:18.250 --> 32:19.060 Which is this? 32:21.250 --> 32:25.580 And then we are just swapping it and solving the equation. 32:25.930 --> 32:33.070 So what do we get the probability of that less than is equal to one minus probability of this entire 32:33.070 --> 32:35.770 thing, which is 60 percent divided by two. 32:35.980 --> 32:42.310 So when we solve this entire thing, we get one minus zero point zero, which is probability of this 32:42.530 --> 32:46.550 just what we have found out here and divided by two. 32:46.720 --> 32:48.670 So one minus 60 percent this. 32:50.280 --> 32:55.260 Forty percent, but just zero point zero four divided by to give zero point zero to. 32:56.350 --> 33:01.220 And we just want to point out this zero point zero two percent. 33:02.050 --> 33:08.270 So now when we take zero point zero two percent in the table, that is the value. 33:08.620 --> 33:10.720 What is the Z value for the person? 33:10.840 --> 33:11.680 Zero point zero. 33:14.040 --> 33:21.540 So zero point zero, actually, Lezak minus zero point eight. 33:23.170 --> 33:26.210 One zero, one, two, three, four. 33:26.710 --> 33:32.410 So 20 percent lies and zero point eight four minus zero point eight. 33:33.040 --> 33:35.860 This means that this minus eight. 33:38.480 --> 33:42.560 This minus eight is that zero point eight for. 33:44.470 --> 33:51.970 So from the zero point eight four, if we want to find out the value of X1 x2, how do we find then 33:51.970 --> 33:57.490 we simply say Z in the new plus new. 33:57.760 --> 34:02.470 So it will be easy to find each four in do eight, which is a standard deviation. 34:02.830 --> 34:06.500 Plus the mean value is equal to twelve point seven two. 34:06.700 --> 34:10.850 And similarly, we will apply the same formula for finding out X2. 34:11.080 --> 34:14.360 So it comes out to be plus zero point eight four. 34:14.950 --> 34:20.890 So it gives us twelve point seven and one hundred and thirteen point two it. 34:30.310 --> 34:36.160 So these are the values which we get we get one hundred and thirteen point week and one twenty six point 34:36.160 --> 34:40.120 seven, which is the value of minus and plus. 34:44.220 --> 34:46.550 This value is incorrect. 34:48.940 --> 34:53.050 This value has to be one twenty six point seventy. 34:54.150 --> 34:55.440 So please get the. 34:58.660 --> 35:00.600 So this is the solution to this problem. 35:10.340 --> 35:16.310 Now, one last thing which we need to discuss here is that the probabilities of the distribution of 35:16.310 --> 35:17.150 sample mean. 35:18.460 --> 35:21.840 The mean of the sampling will be same as the population. 35:22.660 --> 35:25.240 Secondly, the standard error of the mean will be. 35:26.330 --> 35:29.390 Sigma by a Rueben, which we have derived from these and the. 35:32.640 --> 35:38.640 Also, from the same theorem we can get that they will do X minus MU. 35:39.660 --> 35:40.820 Divided by Sigma. 35:41.430 --> 35:46.860 So, for example, because the sigma, for example, will be. 35:48.320 --> 35:57.260 Sigma upon in so we can replace it, so we can see on the road in X minus mu divided by. 35:58.350 --> 35:58.770 Sigma. 35:59.910 --> 36:02.840 So this will be the Z value, for example. 36:03.930 --> 36:07.560 So this is what we have the I for the sample. 36:08.070 --> 36:11.340 So when we are walking with samples, that will be the. 36:12.860 --> 36:18.020 X minus mu divided by the population standard deviation.