WEBVTT 00:01.570 --> 00:07.930 High in this session, we will discuss about the next wave of Chi Square test, which is Chi Square 00:07.930 --> 00:10.150 test for independence. 00:11.570 --> 00:20.030 Now, in the case of the guys who did this for goodness of fit, we saw that a particular value, a 00:20.030 --> 00:26.750 set of value, which is categorical in nature, they tried to find out the proportion of these values 00:26.960 --> 00:36.660 and try to deduce things about the population distribution using the square based of goodness of the. 00:37.830 --> 00:45.630 Now, what do we do in case of this for independence, so did this for independence can be used and 00:45.630 --> 00:47.630 interpreted in two different ways? 00:48.420 --> 00:55.170 The first way is testing hypotheses about the relationship between two withi within a population. 00:56.480 --> 01:02.700 That is initially we were talking about one single video, but that is one categorical variable and 01:02.730 --> 01:03.650 that category. 01:03.680 --> 01:05.820 The video had multiple values in it. 01:05.990 --> 01:12.390 Let us see the eye color being brown, green, blue and other. 01:12.800 --> 01:14.990 So this was about us in the video. 01:15.620 --> 01:20.740 But here in the square of independence, we are talking about the videos. 01:21.880 --> 01:28.660 So you see the difference between the goodness, if it was about just one particular frequency distribution 01:28.660 --> 01:34.690 people, about one categorical variable, which we were talking about, but here we have two withI. 01:36.160 --> 01:42.400 And the other reading this about differences between proportion four to one world population, so one 01:42.400 --> 01:48.030 is about the relationship between the variables or the differences between the water, though. 01:48.910 --> 01:54.830 Now, if you try to analyze these two statements, they are kind of similar. 01:54.850 --> 02:02.860 Only one is trying to find out if two variables have some relationship or not, and another one is trying 02:02.860 --> 02:07.660 to find out if they do have some independence or differences or not. 02:08.500 --> 02:09.970 So it's one and the same thing. 02:09.970 --> 02:16.420 If they are able to find out about two population variables are having some similarity or differences 02:16.420 --> 02:16.790 or not. 02:16.810 --> 02:21.070 That is that can be considered just one null hypothesis only. 02:22.100 --> 02:25.920 So we don't need to be the sole solution here. 02:26.240 --> 02:28.850 We need just one good problem to solve. 02:30.270 --> 02:37.230 So the only defense would you need to take care of here is that the goodness effect is regarding one 02:37.230 --> 02:38.520 categorical variable. 02:39.210 --> 02:49.560 While the test for independence is regarding two or more populations, two variables present in different 02:49.560 --> 02:50.110 populations. 02:50.130 --> 02:54.930 So either they could be to population and we want to find out if these two populations are different 02:54.930 --> 02:55.310 or not. 02:55.830 --> 02:59.220 Or we could have two variables, insane population. 02:59.370 --> 03:05.140 And we want to find out if there is any relationship between these two of these two peoples. 03:05.690 --> 03:10.200 This could be something like a number of hours someone is studying. 03:11.460 --> 03:18.420 With respect to if they're scoring good or not, so here we have the same population, but we want to 03:18.420 --> 03:25.620 find out if the number of hours a person study has some relationship with the of the marks with the 03:25.620 --> 03:26.970 score they are getting on. 03:28.380 --> 03:29.910 So let us get for the. 03:32.080 --> 03:38.170 Now, although the two versions of the test of independence appear to be different, they are equivalent 03:38.170 --> 03:40.170 and they are interchangeable in nature. 03:41.710 --> 03:50.530 The first version of the test emphasizes the relationship between squit and of coordination because 03:50.740 --> 03:54.090 both procedures examined the relationship between two people. 03:54.130 --> 03:57.810 So here we are having squared all joint coordination models. 03:57.880 --> 04:03.340 It kind of emphasizes between guys when information if we're able to relate it to each other or not, 04:04.180 --> 04:10.860 while it determines whether there is an association between categorical variables, also known as a 04:10.860 --> 04:17.190 squared test of association or a test of independence, is also known as good as Twatt Association. 04:17.440 --> 04:21.220 So we are checking if the city was associated or they are independent. 04:23.300 --> 04:26.780 I'm just trying to find out the relation between two videos. 04:28.160 --> 04:37.010 So a very small square statistic means that the absorbed data puts your expected data extremely. 04:37.280 --> 04:40.430 In other words, there is a relationship between two evils. 04:40.850 --> 04:43.060 If the guy is square, the value is very small. 04:43.070 --> 04:46.160 It will mean that there is a relationship between the variables. 04:46.400 --> 04:51.910 If the square value is very large, then it will mean that the data does not fit very well. 04:51.920 --> 04:54.620 That is, it does it does not have any relationship. 04:54.620 --> 05:00.350 That is those two with equals, which we are trying to compare and we are trying to find out an association 05:00.350 --> 05:01.940 for are not related. 05:02.840 --> 05:04.570 And that independent of each other. 05:06.130 --> 05:14.470 Right, so the subscripts see here is the degree of freedom and the formula remains the same, if you 05:14.470 --> 05:20.700 see the formula is just exactly what we had for the formula, for the goodness of it. 05:21.550 --> 05:22.650 Just have a look. 05:22.650 --> 05:22.840 You. 05:25.010 --> 05:32.370 See, of the value minus expected value will square divided by expected what is the formula here? 05:33.080 --> 05:37.350 The families observe minus expected equals squared divided by expected. 05:37.850 --> 05:40.100 So the formula is exactly the same. 05:40.130 --> 05:41.120 The problem is. 05:42.980 --> 05:44.700 Let us go ahead now. 05:44.900 --> 05:49.660 This guy squared value, which we have this whole value, which we have is value for. 05:50.540 --> 05:56.210 This is an equation political and this is the these are the different goals which are generated from 05:56.210 --> 05:56.900 this equation. 05:57.590 --> 06:06.590 Now, what do these different colors signify, these different goals for different different values 06:06.590 --> 06:07.070 of the. 06:10.700 --> 06:11.760 Degree of freedom. 06:12.230 --> 06:17.600 So here we have different degrees of freedom and you can see four different degrees of freedom. 06:17.630 --> 06:19.490 We have the free golf screed. 06:21.670 --> 06:30.850 And as the degree of freedom increases, it it it it somehow represents a normal distribution, so as 06:30.850 --> 06:37.180 the degree of freedom will increase, the people will move towards being a normal distribution. 06:39.960 --> 06:41.860 So let us have an example here. 06:42.240 --> 06:49.270 So here we have a public opinion poll which is surveyed on a simple random sample of thousand voters 06:49.330 --> 06:52.070 support a number of people, which we have here is a thousand. 06:52.380 --> 06:56.300 And people of people who responded would classify it by gender. 06:56.310 --> 06:58.770 That is, either they were male or female. 06:59.490 --> 07:05.260 Now, this is our first video, which is a categorical variable that a person is a male or a female. 07:05.910 --> 07:07.350 What is the next category? 07:07.650 --> 07:09.030 The next is the. 07:10.850 --> 07:16.640 Voting preference of a person that is a person is Republican, Democrat or independent. 07:17.690 --> 07:23.630 So this is, again, a categorical variable, that is if a person is a public Democrat or independents 07:23.840 --> 07:26.810 and another one is the gender of the bush. 07:27.410 --> 07:31.860 So these are two variables which we have in Square Woodenness office. 07:32.240 --> 07:37.020 We had one right now for this. 07:37.070 --> 07:38.570 This is the baby, which we have. 07:38.580 --> 07:40.340 This is the values that we have. 07:40.350 --> 07:45.530 That is there are four hundred million six hundred females which were taken in the sample. 07:46.540 --> 07:54.050 And out of these four, 450 were Republican, 450 Democrat and hundred were independent. 07:54.460 --> 07:56.260 Now this is the data which we have. 07:57.950 --> 08:02.520 Now we want to find out if there is a particular gender gap. 08:03.140 --> 08:08.130 Do men's voting preferences differ significantly from the women of preferences? 08:08.330 --> 08:15.080 So we are trying to compare if the voting preferences of male and female are different or is there any 08:15.080 --> 08:16.740 relationship between both of these? 08:17.480 --> 08:26.360 Now, we could have seen that if the number of male and female would have been similar, if the number 08:26.360 --> 08:32.060 of male and female would have been five hundred and five hundred, then we could have seen if the values 08:32.060 --> 08:33.400 were significantly different. 08:33.770 --> 08:39.620 But here, the problem is that because it is a random sample, the number of male is willison, the 08:39.620 --> 08:40.490 number of female. 08:41.590 --> 08:47.380 So we cannot really determine anything from this data, which we have, so that is why we will be doing 08:47.380 --> 08:48.810 the squarest. 08:49.400 --> 08:50.610 Now, how do we do that? 08:52.690 --> 08:58.960 So we will be doing that by performing of kriesel steps. 08:59.590 --> 09:02.670 OK, now what is the null hypothesis here? 09:03.430 --> 09:08.970 The null hypothesis is that the gender and voting preferences are independent. 09:09.310 --> 09:15.970 That is, no, there is no relationship between the gender and voting preferences. 09:17.110 --> 09:24.490 That is me do not really vote for Republican or female, don't vote for independence. 09:24.760 --> 09:25.630 So these are. 09:26.880 --> 09:32.210 Different viewpoints, which could have been there, but there is no such dependency, right? 09:32.580 --> 09:34.050 They are completely independent. 09:35.160 --> 09:40.380 While the alternate hypothesis is that gender and voting preferences are not independent, that is, 09:40.410 --> 09:46.230 gender is actually impacted by the voting preferences of voting, preferences are actually impacted 09:46.230 --> 09:46.970 by the gender. 09:47.370 --> 09:54.990 If they have a particular gender voting more, then there is a chance of one particular party winning. 09:56.190 --> 09:59.330 Right, but we want to find out if that is true or not. 10:00.840 --> 10:03.900 So we have created this NALAN alternate hypothesis. 10:05.020 --> 10:12.820 Now, what are degrees of freedom now here, the degree of freedom will be the product of a number of 10:13.000 --> 10:17.830 degrees of freedom, rule of law and degrees of freedom of the column. 10:18.010 --> 10:18.660 What is it? 10:20.200 --> 10:21.700 Here we have two values. 10:22.930 --> 10:25.510 So here, the degree of freedom will be one. 10:26.560 --> 10:33.490 Here we have three values, so here the degree of freedom will be to the total degree of freedom will 10:33.490 --> 10:34.450 be to. 10:36.200 --> 10:39.230 Right, so how do we find out the degree of freedom? 10:39.260 --> 10:42.420 So the first thing which we will be finding out will be degrees of freedom. 10:42.800 --> 10:48.500 Now, here is the number of levels for one categorical variable, and C is the number of level for the 10:48.500 --> 10:50.110 other categorical variable. 10:50.840 --> 10:52.910 This is how we find out the degrees of freedom. 10:53.420 --> 10:56.100 Next value is the expected frequency. 10:56.100 --> 10:58.580 Is that is what is the expected frequency value? 10:59.760 --> 11:00.950 How do we find that out? 11:01.260 --> 11:07.380 The expected frequency count are computed separately for each level of one categorical variable at each 11:07.380 --> 11:08.820 level of the other, categorical. 11:09.510 --> 11:15.110 So we will be finding out the expected value for this, for this, for this. 11:15.120 --> 11:18.870 So for each and every data point, which we have here, we will be finding the. 11:19.840 --> 11:21.220 Expected value for these. 11:22.720 --> 11:24.650 Now, how do we find that out? 11:25.000 --> 11:31.630 It will be the number, the expected number of rule into expect the total number of those, the total 11:31.630 --> 11:34.310 number of values in the column divided by the total number. 11:35.170 --> 11:35.430 Right. 11:35.850 --> 11:42.290 So and that is what is the total number of sample observations I can level out of the variability and 11:42.310 --> 11:50.020 in total number of sample observations at levels of variability and is double the number that the sample 11:50.020 --> 11:50.530 size. 11:52.980 --> 11:53.460 OK. 11:54.690 --> 11:56.370 So this will be nothing but. 11:57.240 --> 12:00.850 For this, what will be the value for me versus representative? 12:00.870 --> 12:05.340 The value will be for 50 in the four hundred divided by thousand. 12:05.700 --> 12:06.970 What will be for this one? 12:07.170 --> 12:12.210 This will again be 450 Endou four hundred divided by a thousand. 12:14.880 --> 12:20.310 Right now, what is or C or C is the opposite of the frequency. 12:20.550 --> 12:21.510 So what will we observe? 12:21.510 --> 12:21.930 Frequency? 12:22.200 --> 12:24.220 Four million devices in the public. 12:24.300 --> 12:26.860 It is two hundred fifty million republic. 12:26.880 --> 12:29.280 It is 250 for female and Democrat. 12:29.280 --> 12:33.810 It is three hundred for male and Democratic is 150 for male and independent. 12:33.810 --> 12:36.740 It is 50 for female and independent of is 50. 12:36.870 --> 12:39.120 So this is how we find ourselves of values. 12:41.950 --> 12:49.270 OK, now we are doing the calculations, so the calculated value will be degree of freedom comes out 12:49.270 --> 12:54.670 to be to how the number of rulers minus one and the number of columns minus one. 12:56.500 --> 13:06.700 That is the minus one one and three, minus one, two, two, plus one is equal to two, then these 13:06.700 --> 13:10.350 are the different values for expected values. 13:10.930 --> 13:17.230 So you can go back to the values here and compare how these values are calculated. 13:17.920 --> 13:25.030 And similarly, we have the observed values already present and based on the expected value and the 13:25.030 --> 13:28.000 observed value, we can find out guys statistics. 13:28.570 --> 13:30.430 So what will be the guys for statistics? 13:30.430 --> 13:38.140 Guys with statistics will be of value minus the expected value squared divided by the expected value. 13:38.470 --> 13:45.430 And then we take a sum of all of these observations, all of these guys good values, what each and 13:45.430 --> 13:46.990 every little column value. 13:48.860 --> 13:54.020 So it comes out to be sixteen point two, you can pause this video. 13:54.930 --> 14:01.220 And verify the values, do the calculations on your own. 14:03.270 --> 14:04.110 And then to. 14:05.540 --> 14:09.600 So these are the values, so you can calculate these values accordingly. 14:09.950 --> 14:15.830 Now here with the degree of freedom is two and square value is sixteen point two. 14:17.280 --> 14:26.640 So for this, if we go to the guys table, then four degrees of freedom do sixteen point two will come 14:26.640 --> 14:27.030 where? 14:30.090 --> 14:35.550 This is the value, this is the degree of freedom do and the value is sixteen point something. 14:36.420 --> 14:41.040 Now you can see that the value sixteen point something is Vitullo. 14:45.040 --> 14:47.530 Here also, you can see four degrees of freedom to. 14:48.670 --> 14:58.150 Sixteen point two comes way above zero point zero one, which means that these we have to reject the 14:58.150 --> 14:59.170 null hypothesis. 14:59.590 --> 15:01.630 So what was the null hypothesis? 15:03.310 --> 15:09.730 The null hypothesis was that the GENDEREN voting preferences are independent, but because we have rejected 15:09.730 --> 15:15.670 the null hypothesis so we can conclude that there is a relationship between gender and voting preferences. 15:16.630 --> 15:23.410 So this means that if a particular agenda votes more than the other agenda, then there is a chance 15:23.410 --> 15:26.090 for a particular party to win, right? 15:26.290 --> 15:30.960 So this is what we have in the square, sort of independent. 15:31.780 --> 15:38.050 So next we will be discussing about a one which is another set of test, which we will be discussing 15:38.050 --> 15:39.340 in the next session. 15:39.820 --> 15:40.230 Thank you.